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is the porosity of the medium, ar, etc are, respectively, the thermal and solutal expansion coefficient, A = (pocp)f/(poc)m is ratio of heat capacity, in which c is the specific heat of the solid and cp is the specific heat of the fluid at constant pressure a , (3 is a positive constant. To (1) we append the boundary conditions T = TL, C = CL (v.) - y>(v') - ¥ > « ) ! < 9tf|v' - v||v* - v'| < |V||n • V | (2.3)
onz = 0
T = Tu,C = Cu
on z = d
(2)
with TL > Tu, CL > Cy(> 0). Equations (l)-(2) admit the motionless state ms = {0,Ts,ps,Cs} given by: Vp s = -Pog[l
T.{z) = -P
- aT(Ts - T0) + ac(Cs - C 0 )]k; Cs{z) = -SC Z- + CL z ^ 4£ +6£ +3£ +2e )Z a
i- 4 €-^ " ^ d
with /?* = ^ g ^
> 0,£1
= g(>
1),
£2
= g
(3)
e [0,1], SC = CL -
Cu(> 0), r = Tu - TL(< 0). We denote by {u = (u,v,w), 9, T, ir} the perturbations to the (seepage) velocity, temperature, concentration and pressure fields, respectively. On introducing the dimensionless quantities
R
_ .
\gaTkd\T\ ukr
V
vk-T
where RT and Re are thermal and solute Rayleigh numbers, respectively. Dropping all asterisks, the dimensionless equations in IR2 x [0,1] are: ' VTT = - u + RT6k - RcTk, V •u = 0 1 + l}w + A9 + 2^VLeN(ex 1 + u • V6> = RT[jF(z)
- z)T+
+ 12 7 ^V
(4)
Re
^ e Le r,t + ie u • v r = i?c™ + Ar. in which 7 = —-, e = — (reduced porosity) and F(z) = 4z 3 - 12eiz 2 + 12£?z - {e\ + 3e? + 2eie 2 ) • a
The subscripts / and m refer, respectively, to the fluid and to the mixture.
(5)
85
To (4) we append the stress free boundary conditions w = 9 = T = 0on z = 0,1. We assume that the perturbation fields are periodic in the x and y directions of periods 2n/ax, 2ir/ay and denote by Q, = [0,2Tr/ax] x [0,2ir/ay} x [0,1] the periodicity cell, a = (a2 + a2)1/2 the wave number and by < • > and || • || the L 2 (fi)-scalar product and the L 2 (fi)-norm, respectively. Finally to ensure that the steady state (3) is unique, we assume that < u > = < v > = 0. 3. Onset of penetrative convection By virtue of p(T, C) = po[l — arT + acC] in view of (3), setting z* = z/d, on ms it turns out that p(Ts,Cs)
= P0aTp*[z*4 - 4elZ*3 + 6e2z*2 - (e§ + 3e? + 2e1e2)z*]+ -po{aTT + ac5C)z* + p0[l - aTTL + acCL].
[
'
Dropping the stars and setting p3(z) = p(Ts,Cs), since f3* > 0, p's{z) is an increasing function and it turns out that if p's(0) < 0 and p's(l) > 0, i.e. 0 < \(3*\(e22 + 3e 2 + 2eie 2 ) + r + ^5C
< 4|/F| (1 - 3ex + 3e2)
(7)
ay
then 3!zi G (0,1) : ps{z\) = mmps(z).
Denoting by IT3 the inner plane
z = z\, S is divided into the two parts (IIi, II3) and (II3, II2), "potentially stable" and "potentially unstable", respectively. The penetrative convection occurs only when ms is unstable 10 ' 11 . Therefore its onset is guaranteed by ms instability conditions. On following the methodology introduced by Rionero 5 - 9 , setting am(z) =
ffj
2
a2n(z) = - ^
^
[1 + jF(z)} - ( n V + a2) [1 + lF(z)) + 2*yy/UN(£l
a2 RT RC eLe(n2ir2 + a2) '
__ "
- z)
a2 R2C eLe(n2TT2 + a2)
(8) n2it2 + a2 sLe
with a procedure completely analogous to the which one of Section 5 of [12], one obtains f dS
™
^(
0 1
A + a * I \ ) - U m - VSW + 1 2
m
t.
at
-'ST(n„.a.±n.,r.\-Ln...
1
7
^ ^
[S^f
(9) r
. \7
86
and the associated system — - = an9i + a 2 i r i , dt
— - = a 3 i0i + a ^ T i at
d9„ ~ — Gl(m-l)0m-l + dt
fl2(m-l)fm-l
(10) dt "
m
0t
— =
+
alm9m+a2mrm-lJm
0-4(m-l)^m-l
• V5W + 1 2 7 ^ ^ [ S £ > ] Ac
3
^r) = E r - um = x;ui
^ E * . i=l
(11)
*=1
Let ( 0 i , . . . , 0 m ; r i , . . . , r m ) be a solution of (10), then (0i + ... + <9m;ri + ..+ r m ) is a solution of (9). Therefore - by virtue of the uniqueness theorem we can evaluate the behaviour of 0$, T, by substituting (10) to (9). In view m
2
m
2
2
of \\s£>\\ = E l N I ' H ^ l l = Ell r «H 2 ' [ t
follows t h a t t h e i n s t a b i l i t
y
is guaranteed by the instability of the zero solution of an
-— = Oii^i + a 2 l F l
dt dt
(12) a3i0i + a4iTi
i.e., if 53zo € [0,1] : either if I\(z0) 011(20)041 - 021(20)031 < 0 where 011(20) = -^TK
= an(zo)
+ 041 > 0 or
AI(ZQ)
=
I1 + 7^(^0)1 - (^2 + o?)
021(20) - - a f f f i 2 C [1 + lF{z0)} + 2 4 7 v ^ i V ( e i - z0) ir* + a* Ti-2 + a 2 a 2 /? 2 , a 2 RT Re 031 0 4 1 eLe(n2 + a 2 ) e Le £Le(7r2 + a 2
(13)
87
Theorem 3.1. Let (7) and Rj< > i?j
1
Rr
1 + 7(6e^ + 4e2 + 1)
£ Le +1 „
-£-
+
-T
.
„
RB } , RB=
eLe eLe hold. Then the penetrative convection occurs.
., . . 4TTJ
14
'
Proof. By straightforward calculations, one easily obtains that by virtue of (14), 3 z b G [ 0 , l ] : 7 1 ( z 0 ) > 0 . r, a2 < ir2 and
Theorem 3.2. Let (7), 7 < ——77 1 J
24iV2(£i - 1) R\ > R\
(15)
with '7r2+a2-247iV2(e1-l)D2 R Rn = c^ 2 1 + 7 (6e2 + 4e 2 + 1) IT2 + a
(TT2 + a 2 ) 2 ' ^
(16)
hold. Then the penetrative convection occurs. Proof. By straightforward calculations, one easily obtains that by virtue of (14), 3 z 0 e [ 0 , l ] : A 1 ( z 0 ) > 0 . Remark 3.1. Setting 2 2
_
2ir\V7LT+T)3 2
2
TT2 + C5 2
2___ 2
37JV (£i-l)£ ie '
3
2
2
7r + a + 2 4 7 i V 2 ( e 1 - l )
K
'
it follows that: i) for {eLe < C%, R2-, < C2}, Theorem 2 furnishes an instability condition more restrictive than Theorem 1; ii) for {eLe > C2, R2-, > C | } , Theorem 1 furnishes an instability condition more restrictive than Theorem 2; hi) in heat source absence (7 = 0), since a2 — n2, C2 = 1, Theorems 1-2 reduce to instability conditions of double diffusive natural convection in porous media 3 . Acknowledgments This work has been performed under the auspices of the G.N.F.M. of I.N.D.A. M. and M.I.U.R. (COFIN2005): "Propagazione non lineare e stabilita nei processi termodinamici del continuo".
88
1. G.I.Barenblatt, V.M.Entov, V.M.Ryzhik: Theory of fluid flows through natural rocks. Kluwer Academic Publishers 1990. 2. J.N. Flavin, S. Rionero: Qualitative estimates for partial differential equations. An introduction. CRC Press, Boca Raton, Florida 1996. 3. S.Lombardo, G.Mulone, B.Straughan: Math. Meth. Appl. Sci. 24, 1229 (2001). 4. D.A.Nield, A.Bejan: Convection in porous media. Springer Verlag, BerlinHeidelberg New York 1992. 5. S.Rionero: A rigorous reduction of the L -stability of the solutions to a nonlinear binary reaction - difusion system of P.D.Es. to the stability of the solutions to a linear binary system of O.D.Es. In press on Journal of Mathematical Analysis and Applications. 6. S.Rionero: A nonlinear L -stability analysis for two species population dynamics with dispersal. Mathematical Biosciences and Engineering 3, n . 1 189 (2006). 7. S.Rionero: Functionals for the coincidence between linear and nonlinear stability with applications to spatial ecology and double diffusive convection. In: Proc. Waves and Stability in Continuous Media, WASCOM 2005, Acireale, June 2005 (to appear). 8. S. Rionero: "L -stability of the solutions to a nonlinear binary reactiondiffusion system of P.D.Es." Rend. Accademia dei Lincei. Fasc.4, 2005 (To appear). 9. S. Rionero: "Global nonlinear L -stability for double diffusive convection in porous media with and without rotation. (To appear). 10. B.Straughan: The energy method, stability and nonlinear convection. 2nd ed. Springer, Appl. Math. Sci. Ser., New York 2004 . 11. G.Veronis: Astrophys. J. 137, 641 (1963).
NON-LINEAR SCHRODINGER EQUATION IN A TWO-FLUID PLASMA*
P. CARBONARO Dipartimento di Matematica e Informatica Cittd Universitaria, Viale Andrea Doria 6, 95125 Catania, ITALY E-mail: [email protected]
It is well known t h a t weakly dispersive and dissipative systems can be reduced by means of asymptotic techniques to model equations like K d V or Burger's 1 . In weakly dispersive systems in general the dispersive terms are represented by third order derivatives with respect to space and time variables. It was remarked some years ago by Taniuti t h a t there are particular systems in which the dispersive terms are represented by second order derivatives. A typical example is given by two-fluid magnetohydrodynamics 2 . To be more precise, let us suppose t h a t a system has the form
where U is a vector, A(U) and B{U) are two nxn matrices. T h e dispersion relation of the linearized system (1) is |— ojJ + kA + ik2B\ = 0. If the matrix A is symmetric and B is antisymmetric t h e n kA + ik2B is Hermitian and Imuj = 0. Taniuti 1 has shown using a suitable extension of the reductive perturbation method t h a t these systems are reducible to a K d V equation. However, the variety of phenomena observed in plasmas seem t o suggest t h a t the plain K d V picture may not give a satisfactory framework t o explain for example the existence of different types of localized structures or instability mechanism 3 . In this work we derive from system (1) a nonlinear Schrodinger equation which describes the propagation of modulated
"This work is supported by the Italian Ministry for University and Scientific Research, PRIN: Problemi matematici non lineari di propagazione e stabilita nei modelli del continue) (Coordinator Prof. T. Ruggeri), by Gruppo Nazionale della Fisica matematica of Istituto Nazionale di Alta Matematica, and by the University of Catania.
89
90
envelopes, we determine the conditions under which different types of envelope solitary waves exist and we study the occurrence of modunational instability. We seek solutions to (1) of the form
i
i
where £ = e(x — Xt), 8 — kx — u>t and r = e2t, with e a small parameter. At the lowest order of approximation we get {-iltul + ilkA0 - l2k2BQ)ul1]
(2)
= °-
If we suppose that det(—iluil + ilkAo — l2k2Bo) = 0, for 1 = 1 and det(-ilujl
+ ilkAo - l2k2B0)
^ 0,
for I ^ 1,
we obtain U[1] =TI>(£,T)R,
[/,(1)=0
for
1^1,
(3)
where ip is an arbitrary function and R is given by {-ilwl + ilkA0 - l2k2B0)R
= 0.
The order e2 of the expansion is (A/- - AA0 0 - - 2ilkB02ilkB0)^{-ilwl + ilkA0 - l221.2 k2B0u^rf) )Utz> - _ {XI
• Ul%) - {l'k)2{VB0 • U&)] = 0,
+ j:i,{ll'k{VA0 and for I = 1
orr(l)
{-iwl + ikA0 - k2Bo)u[2) which yields U{
=
4>{£,T)R
- {XI - A0 - 2ikBo)—^—
— ^(-77-)-^-, where
<^>(£,T)
= 0,
(5)
is an arbitrary
function. For I = 2 eq. (4) becomes 2 (-iuI
+ ikAo-2k2B0)
f/2(2) +ik [{VA0.R)
+ik{VB0.R)}
i ? ^ = 0,
from which we get C/2 in terms of ip. From (4) we see also, setting I — 0, (2) that UQ remains undetermined if {{VA0 • R*) + ik (VBo • IT)] R - [{VA0 • R)-ik
{VB0 • R)} R* = 0.
91
To the next order of approximation (e3) we obtain dU,(2) dU,( i ) 9C + 3T £ { [il'k (VA0 • C/£>,) - (Vkf (VB 0 • U[%)] i f >}
(-ilul + ilkA0 - l2k2B0) U}6' + (-XI + A0 + 2ilkBt a 2 (/ ( 1 )
+ E{[il'k(vA0.uZl)-(l'kf(vB0.U^)} U^} + £ { [(VA 0 • [/«,) - 2il'k (VB 0 • U&)] v i"
l
v
^
'
'
-(/"^(vVBo:^^,^)
E#>=0.
For Z — 0, eq. (6) takes the form {-XI + Ao) dK + {[(VA0- R*)+ 2ik(VB0-
R*)]W$
+
(7)
b.c.}=0,
while for I = 1 eq. (6) yields (2)
2ikB0 dU\
+ ikA0 - k2B0) U\4} - (XI -A0-
(-iwl
_*£>
2
3£
+ S(
9 ^
ae 2
fc (vA0 • t/iV) - 2fc2 (vBo • tf-i) tf
A; (V^o • Kp') + *2 (V5 0 • t/2(2)) v™
+
ifc (VA 0 - C/f>) - k2 (vBo • £^2))
m
ik ( w A o : U™U™) - fc2 (vVBo : tfiV tfi(1)) "z/fc ( w A o : [ f V ^ ) +k2 (vVBo : i f ^ ) ]
t^
= 0.
Left multiplying the foregoing expression by L and taking into account (2), (3) one has .dip d2w d2ip !fr + 2dk^ de (8) + (L • R)'1 L [ik (vAo • f/0(2)) - k2 (VB0 • E^2))] Rip + a \ip\2 ip = 0. l
Combining eqs. (7) - (8) one gets a non-linear Schrodinger equation. The model equations describing a two-fluid isothermal plasma are given by l. — + div{nv) = 0,
(9)
92 dv n ——\- j3 gradn + B x curl B + 7 (curlB.grad) v + n — (curlB/n) . . dt J(10) at +7(1 + 7) [(cwZ£? • grad) (curlB/n)}, — +curl (B xv)+curl
(-^-)
=0,
(11)
where n and u are respectively the density and the plasma velocity, B the magnetic flux vector, j3 is the pressure ratio (plasma pressure/magnetic pressure), 7 is the electron-to-ion mass ratio, d/dt = d/dt + v.grad. The equations are obtained by assuming charge neutrality while the electron flow velocity and the electric field vector are eliminated by means of Amperes's law and the equation of motion for the electron fluid. The effects of the electron inertia appear in eq. (10) in terms containing 7 which is taken small, of order e2 (with e the perturbation parameter). All the quantities are dimensionless and in the following are assumed to depend only on the spatial variable x and on the time t. In order to take into consideration a situation in which the change of the wave envelope is slow in comparison to the carrier wave we define a couple of slow space and time variables 2 £ = e (x — At), r = e 2 i, (e < < 1), and we suppose that theese are independent variables for the envelope motion. We then expand the field variables n, v and B in terms of e as follows £=+00
n = 1 +£ ] T n;(1) (£,r) e«(fcx-u,t)
£=+00 +£
2 j - „(2) ( ^ r ) ji^-ut)
l = — OO
(=—00
?
(=—00
Z=+oo
B=e
_
l = — OO
l=+oo
Y, Bll)(t;,T)eil(kx-^+e2 /=—00
^
B{2) (£,T) e«(**-<")... .
i=—00
Substituting the foregoing expansions in the original system (10) - (11) we obtain first at the lowest order of approximation the dispersion relation for the Alfven wave 2LO — k {k + \/k2 + l) , and the relations
n[1} = vi\] = 0, v$ = iv{£ = -ikW,r),
B$ = iB$> =
2
To the order e , for I = 1 we obtain the compatibility condition du> T k2 + 2 A X — —— — k-\2
dk
Vk +4'
W&T).
93 For I — 1, we also obtain the following expressions for the second order perturbation field
where
M-fl2^.^^), 2
,12,
J2)
«, -/?)^- = 2 » 2 ^ A
^ = - W
(13)
and for Z = 1 to the equation
a^
V
i ^ 9 V _ +
2 d2/fc ^
/
(2)
_ _^k_
^ + kip0
H °
(2)\
_
)y~0-
(14)
Combining eqs. (12) - (14) we obtain the equation for the amplitude
£ + ! ! £ +owl**-<>.
<»>
where fc2
+
6
^ _
P = w"(fc) = l + k ( f c 2 + 4 ) 3r ^ ,Q /2'^
^ /3-V2
fc3(fc2 fc+v/^4'
The sign of the product PQ is the determining factor in deducing the characteristics of the wave propagation described by eq. (15). The non-linear Schrodinger equation (15) admits different type of localized solutions according to whether the product PQ is positive or negative. If we look for solutions of the form V'=V/x(e,r)exp[z(^-5r)]
94 where R and S are real constants, we find what follows. (1) For PQ> x(e,r)
0
= ^sec/
2 1
|v/M
( e
_
V T
)l
R =
l
i S
= L
(^1-PQ^y
which represent a localized pulse travelling with speed v ("bright" envelope solitons ) whose width y/P/ipoVQ depends on the maximum amplitude of the wave ipQ. (2) For PQ< 0 x(e,r) = V o t a n h ^ ^ / - ^ ( C -
U
r)|,i?=p5=i
(~
- PQ^
,
which corresponds to a localized region travelling at a speed v ("dark" envelope solitons). The sign of the product P Q also affects the linear stability analysis. If we seek a solution of the form 4>(£,T) = ip0 exp(zQ|V'o| 2 TJ
[l+e
and substitute in (15), we get , negletting terms of order e 2 , the linearized equation
^ + fg
2 ^ o | ( , + ,*) = 0, +
Taking the perturbation
C2 exp \—i (K£, -
fir)]
we obtain the dispersion relation K2
tf = ^-
(p2K2-APQ
|^0|2).
The wave is stable if the product PQ is negative. When PQ is positive, the condition for the instability is
K
T H E NON-RELATIVISTIC LIMIT OF RELATIVISTIC EXTENDED THERMODYNAMICS WITH MANY M O M E N T S - PART II: H O W IT I N C L U D E S T H E M A S S , M O M E N T U M A N D E N E R G Y CONSERVATION.
M . C . C A R R I S I , F . D E M O N T I S A N D S. P E N N I S I Dipartimento
di Matematica ed Informatica, Universita di Cagliari, Via Ospedale 72,09124 Cagliari, Italy; [email protected]; [email protected]; [email protected]
-mail:
In part I of this article, Borghero, Demontis and Pennisi have obtained the limits for light speed c going to infty, of the balance equations in Relativistic Extended Thermodynamics with many moments. In order to obtain independent equations, they have taken a suitable linear combination of the equations, before taking the limit. What happens with this procedure to the relativistic conservation laws of mass, momentum and energy? Obviously, they transform in their classical counterparts; but proof of this property is not easy and is treated in this part II of the article.
1. Introduction In the paper x , a method has been shown to obtain starting from the balance relativistic equations Q
Aaa2---aN — jcn2---OiN
(1) daBaa*-aM
= I^'aM
with M + N odd and M < N
their non-relativistic counterparts by taking the limit when c —> oo, where c is the speed of light. It is interesting to observe that we have not only transformed eqs.(l) in 3-dimensional notation in order to calculate the limit for c —* oo; because, if we proceed in this way, the equations arising from eq.(l)2 would be a subset of those coming from eq.(l)i. In other words, subtracting from eq.(l)2 some of eq.(l)i we find infinitesimals of higher order with respect to £. Then, we have to make a suitable linear combination with constant coefficients, but depending on c, such that its limit is finite and gives independent equations. There are various linear combinations that satisfy our requirements, but their limits are the same. For example,
95
96
the linear combination that we have chosen in 1 (in the case N = 3, M = 2) is not the same combination used by Dreyer and Weiss in 2 (exposed also in 3 ) , although its limit as c —» oo is the same. We have chosen a different linear combination in order to avoid cumbersome calculations and their difficult justifications arising from the fact that our considerations are valid VW and VM. The problem arises in verifying as the relativistic conservation laws of mass, momentum and energy are transformed in their classical counterparts, through our linear combination and its limit for c —> oo. These results have been obtained here. In order to briefly describe the result, we consider the non-relativistic counterparts of eqs.(l) obtained by the method mentioned above, i.e.,
dtFil"-i'
+ dkFil-i'k
OtF
= P*!-*" 2
il
ir ceiei
+<9fcP '" '
for 0 < s < N - 1
2
[Z)
+ h
'" = Q "
ir
for 0 < r < M - 1
If we start considering only eqs. (l)i with N even, we obtain only the equation (2)i with lim P = 0 (mass conservation) and lim P n = 0 ( m o c—*oo
c—>oo
mentum conservation), but losing energy conservation. Instead, if we consider also eq. (1)2, obviously for M odd, we prove, in this paper, that P " is infinitesimal, obtaining in this way energy conservation. Similarly, if we consider only eq. (l)i with N even, we obtain only eq. (2)i with lim P = 0 (mass conservation), but losing momentum and energy c—>oo
conservation. The presence of eq. (1)2 with M odd affects also the productions in eq. (l)i: we will see that, always as a consequence of eq. (1)2, also Pn and P " are infinitesimal and by this fact we obtain the momentum and energy conservation. Thus, in a relativistic approach, eqs. (l)i and (1)2 cannot be neglected.
2. T h e case with N odd a n d 1V[ even Obviously, in this case is included the 14-moments one. The maximal trace of eq. (l)i gives the mass conservation law; let us express it in terms of the
97 tensor pn"•4s, n _
by using also the notation of paper *:
ra2-••<*Nn
n
N
•*/V
(."'0:20:3
2
^aa'ces )---\rt'aN-iocN
'•aN_1taN)—
/iV-l\
/ yI h=0
) I-''-)
1
-*JV
^2*o:3-"*o:2h^o:2ft4-i"Q:2/ l + 2Q:2h+3---"o!N-lO!iv —
W-l
2,
/ N-l\
0..-0eiei...ejv_i_2heiv_i_2h
JV-1
£2 /JV-1 f— ^
(_1)/lmN+2c2^1p^«1--^=-ei-^t
(3)
h=0
which can be multiplied by c2~N and gives W-3 p==y^
2 / l-2~\ JV-1
^ 0
*
^!±tfilc2h-.N+ipeiei---eN~i-2heN-i-2h
ft
whose non-relativistic limit is
lim P = 0,
(5)
c—>oo
which is the mass conservation law for system (2). Similarly, the maximal trace of eq. (1)2 gives momentum and energy conservation in the relativistic context. It reads: 0 = 1$'"aM ga3ai---9aM~iaM which, with calculations similar to the ones above , becomes M-2 M-2 2 JL^ / M-2\
/M-2\
_o a 2 0...0eiei...eM-2-2h 2
L
eM-2-2h
M
h=0
from which, for a-i = 0 and Q 2 = ii respectively, we obtain M-4
/ OH—£ \
M
M-4
/M=2\
M
,
_ ,
,
eiei...eM-2-2h eM-2-2h
iieiei...eM-2-2heM-2-2> 1
(6)
98
Let us now consider the expression of Q*i•••*'• in ' , with r = 2, and let us compute its trace, thus obtaining: M-4
2
ii±f
ee
(-2c )- ^Q
bq2(-2c2)-qPMei-eqeqeq+ieq+1
= r Y l 62
+
g =o N-3 2
1
+ — V a p2 (_2c 2 )-Pp e i e i- e *> e » e p+i e *>+i 62 P=o
(7)
whose non-relativistic limit is 0 — ( ^ ^ ( ^ - P j w + a02-P * 1 ) , where an overlined term denotes its non-relativistic limit. By using the property 002 = —602 (see ref. 1 ) , we obtain P'iei=Peiei
.
(8)
Note that, in the case M = 2, there isn't the term on the left hand side of eq. (8), so that this eq. is P = 0. In other words, we have energy conservation for eq. (2)i. Let us also consider the expression of Qn-Ar in 1, with r = 0; by writing explicitly the terms with q = 0, p = 0 and using the expressions (6)1 and (4) of PM and P we obtain: M-4
= -£ M - 2 ) ^ E ( T ) <-i>*+^+4-M.
^Q
Af-2
eiei...e M -2-2he M _ 2 _ 2h
2
•PM
2
j
v^
N+M-I-2Q
6
°
0
0
_ _
-
p
^ " ^ c 2 - 2 ^ 6 1 - e*e* +
+rEM-2) 9= 1
JV-l
T
Y
Opo(-2)
5
c 2-2 P pe 1 e 1 ...e,e J) +
Q (
_
2 )
^ ^ _
.
N-3
whose non-relativistic limit is
0= - ^ h
m {
+ ^a10(-2)^P bo
- 2 ) ^ ^
{
- l ) ^ P
'>'* + l a b0
0
hw{-2)N-^P^
™+
o(-2)^
^I(-i)"-ip ™, 2
which, by using eq. (8), becomes 0 = [6 00 (M - 2 ) ( - l ) M - 3 + 610 + 010 + a00(N - ! ) ( - ! ) " ] PM'1
that is
99 0 = [(M - 2 ) ( - l ) M ~ 3 + b10 -(N-M)= P
eiei
b1Q -(N-
1)(-1)"] Peiei =
.
In this way we have obtained energy conservation for the system (2). It remains to prove momentum conservation. To this end, let us consider the expression of QH-lr in 1 with r = 1; by writing explicitly the term with q = 0 and using the expression (6)2 of PJ}, we obtain M-1
2
Q*i(-2c) =3r* = I J2
bgl(-2c2rqPTei-e'eq
9=1 1
JV-3 2
+ - V apl(-2c2)-pPiieiei-^^ + M-4
1
v^
~ ^
/M-2\
he1ei...eM_2-2h eM-2-2h
E ( J J (_1)/1+^C^2-MPM
- ^
r-
whose non-relativistic limit is 0 = (p^oi-P x- but aoi = —601 = —1, so that it remains P
= 0 , i.e. momentum conservation for the system (2).
3. The case with N even a.nd rvl odd Eqs. (4) and (6) still hold, but after exchanging M and N, P and PM, P}} and PZl, i.e., P
A/-3 2_ / M - l \
^ = E /.=o ^ jV-4
p
=
eiei...eM-i-2teM-i-jfc
? ) (-I)"+^C2"-M+IPM
"•
-^—
2
J
_y- f^")
(_i)^+J^c2/'+2-jvpeiei-ew"22-2,'ew"2"2h
h=0
The non-relativistic limit of (10)2,3 can be quickly computed and equals P = 0, P = 0 , i.e., we have mass and momentum conservation for the system (2). It remains to prove energy conservation. Now the passages after eqs. (6) and until eq. (8), of the previous section, can be adapted also to the present case (there is only to substitute the upper values of q and p with ^ r ^ and ^f^ respectively), so that eq. (8) still holds in the present
100 case. Now t h e expression of Q*i•••*'• in 1 with r = 0 , by exploiting the terms with q = 0 , p = 0 and using eqs. (10)1,2, gives M-l
Q ( _ 2 c 2 ) -- ± a
1£ 6
W_^r,+
° 9=1 N-2 2
ap0(-2c2yp+1Peiei-e»e»
1 J2 M-3 . 2 .
1
1
/ M —1\
iV-4 2
/ N - 2 \
1^...«.«. +
eiei...ejvf_i-2h e M ^ ! ^ 2 h
„ „
/i=0
whose non-relativistic limit is
- 5
610 + a10 - 2 6 0 0 ^ - ^ ( - l ) M - 2 + 2 a
0 0
= r - [fcio - (W - M ) - 6 10 + Af - 1 + N - 2] P 00
^(-l)^&1&1
3
•p
eiei
= - ^ [ 2 M - 3]P
eiei
oo
1 from which P = 0, i.e. we have energy conservation for the system (2). In this way all our aims have been accomplished.
References 1. Borghero, F., Demontis, F., Pennisi, S.: "The non-relativistic limit of Relativistic Extended thermodynamics with many moments-The balance equations" . To be published in proceedings of Wascom 2005 2. Dreyer W., Weiss W., "The classical limit of relativistic extended thermodynamics". Annales de I'Institut Henri Poincare 45 (1986). 3. Miiller I., Ruggeri T., Rational Extended Thermodynamics, second Edition, Springer Verlag, New York, Berlin Heidelberg (1998).
E X I S T E N C E A N D E N E R G Y CONSERVATION FOR T H E B O L T Z M A N N EQUATION
CARLO CERCIGNANI Dipartimento
di Matematica del Politecnico di Piazza Leonardo da Vinci 32 20133 Milano - Italy E-mail: carlo. cercignani@polimi. it
Milano
The paper presents a recent result by the author concerning Maxwell molecules, without any cutoff in the collision kernel, in the one-dimensional case. Conservation of energy also holds.
1. Introduction The well-posedness of the initial value problem for the Boltzmann equation means that there is a unique nonnegative solution preserving the energy and satisfying the entropy inequality, from a positive initial datum with finite energy and entropy. However, for general initial data, it is difficult, and until now not known, whether such a well-behaved solution can be constructed globally in time. The difficulty in doing this is obviously related to the nonlinearity of the collision operator and the apparent lack of conservation laws or a priori estimates preventing the solution from becoming singular in finite time. The existence theorem of DiPerna and Lions8 is rightly considered as a basic result of the mathematical theory of the Boltzmann equation. Unfortunately, it is far from providing a complete theory, since there is no proof of uniqueness; in addition, there is no proof that energy is conserved and conservation of momentum can be proved only globally and not locally. It seems rather clear that in order to achieve some progress in the study of the initial value problem for the nonlinear Boltzmann equation and prove that the typical solutions have more properties that those proved in the theorem by DiPerna and Lions, more a priori estimates are needed. We
101
102 review a recent result of the author: when the solution depends on just one space coordinate x (which might range from —oo to +00 or from 0 to 1), it is a weak one without any need for renormalization with the consequence that conservation of energy holds. In order to simplify the presentation x is restricted to the interval [0,1] with periodicity boundary conditions. Easy modifications allow extensions to different boundary conditions 6 . The x-, y- and z- component of velocity v £ R 3 are denoted by £,77, CThen the Boltzmann equation 3,7 reads as follows:
g + fg = «/,/>
(u)
Q(/,/)(a:,v,*) =
/ /
B ( n • (v - v . ) , |v - v.|) ( / ' / : - / / * ) sin 0 d 0 # d v „ .
(1.2)
where v' = v - n[n • (v - v * ) ] v;=v.+n[n-(v-v.)]
(1.3)
n being the unit vector along v — v'. As usual, / ' = f(v', /* =
fl = f{v'*),
/ « V = v - v„
To this end we introduce what we call the weak form of the collision term, Q (/,/)> defined by: Q(f, f){x> v> t)
/ -/[0,T]x[0,l]xR
lb 7[o,r]x[o,i J[0,r]x[0,l]xK3xR3xS2
=
3
B(n-(v-v,),|v-v*|)(
zo
(1.4) for any test function tp(x,v,t) which is twice differentiable as a function of v with second derivatives uniformly bounded with respect to x and t. In Eq.(1.4) we have used the notation d/j, = sm6d9d<j)dv*dvdx We remark that for classical solutions the above definition is known to be equivalent to the usual one. The main reason for introducing it is that it may produce weak solutions (as opposed to renormalized solutions in the
103 sense of DiPerna and Lions) even if the collision term is not necessarily in L1. This also avoids cutting off the small relative speeds, as done in previous papers by the author. Another advantage is that we may consider solutions for inverse power potentials without introducing Grad's angular cutoff. On the other hand we shall not allow a growth for large values of |V| , i.e. we exclude in this paper hard spheres and potentials harder than the inverse fifth power. This is an important technical simplification, which perhaps might be be removed by much harder work. For a function / to be a weak solution of the Boltzmann equation the derivatives in the left hand side are distributional derivatives and the right hand side has been defined above. The objective of this paper is to show the initial value problem for the Boltzmann equation has a global weak solution in the sense defined above. The main step in proving this is a proof that the weak collision term Q(f, / ) exists and is finite. 2. A useful identity In this section we want to prove that the definition of the weak form of the collision term makes sense for inverse power potentials without introducing an angular cutoff, as first shown in a paper of the author 4 . To this end we consider the following identity:
f
Jo
ds Jo
d2 dt d^di ^ ( V
f
+ S(V
' ~~ V )
+ i(V
* ~ V'))]
y » ( v , ) - V ( v ' ) - v ( v : ) + v(v)
(2.1)
Hence V?(v) + ¥>(v») -
where the argument of
104 Hence if the kernel B diverges for 8 = 7r/2, but Bcos9 is integrable, then the integral with respect to 9 does not diverge. We recall that, if the intermolecular force varies as the n-th inverse power of the distance, then B(n • (v — v*), |v — v*|) varies as shown in We conclude that for power-law potentials, B cos 6 behaves as the power —2/(n — 1) of \TT/2 — 6\ and the definition of a weak solution given above makes sense for n > 3. Henceforth we shall consider just Maxwell molecules, for which we state the main result of this section as Lemma 2.1. The following estimate holds [ Q(f,f)(x,v,t)
<0oK
[
|V|2//»dvdv,.
(2.4)
JR3XR3
where K is un upper bound for the second derivatives ofjf> and /?o a constant that only depends on molecular parameters.
3. Basic estimates We now set out to prove the crucial estimates for the solution of the initial value problem and for the collision term. It is safe to assume that we deal with a sufficiently regular solution of the problem, because this can always be enforced by truncating the collision kernel and modifying the collision terms in the way described in earlier work, in particular in Ref. 8. If we obtain strong enough bounds on the solutions of such truncated problems, we can then extract a subsequence converging to a renormalized solution in the sense of DiPerna and Lions; and the bounds which we do get actually guarantee that this solution is then a solution in the weak sense defined above. Consider now the functional I[W)=
I
I
I (Z-Z.)f(x,v,t)f{y,v.,t)dv.dvdxdy
J x
(3.1)
where the integral with respect to x and y is over the triangle 0 < x < y < 1. This functional was in the one-dimensional discrete velocity context first introduced by Bony 1 . The use of this functional is the main reason why we have to restrict our work to one dimension; no functional with similar pleasant properties is known, at this time, in more than one dimension ( for a discussion of this point see a recent paper of the author 5 ). Notice that if we have bounds for the integral with respect to x of p — JR f(x, v, t) dv
105 and for
E{t) = [ [ |v|2/ dvdx,
(3.2)
then we have control over the functional I\f](t). A short calculation with proper use of the collision invariants of the Boltzmann collision operator shows that -/[/] = - / " a t
/
[ (£-^)2f(x,v*,t)f(x,v,t)dvdV*dx
(3.3)
J[0,1] JR3 J R 3
Notice that the first term on the right, apart from the factor (£ — £*)2, has structural similarity to the collision term of the Boltzmann equation, and the integrand is nonnegative. This is the reason why the functional /[/] is a powerful tool. After integration from 0 to T > 0 and reorganizing, i-T
r
r
r
II II' JO
J[0,l]Jv
(£ — £*) 2 /(x,v*,i)/(x, v,
t)dvdv*dxdt
Jv.
I[f](0)-I[f\(T).
(3.4)
According to a previous remark, the right-hand side of (3.4) is bounded. Since the total energy is conserved, we have proved Lemma 3.1. If f is a sufficiently smooth solution of the initial value problem given by (3.1) and (3.2) with initial value /o, then /o io1 Jv /v. (£ " £*) 2 /( x > v*> T)f(x>v'r)
dvdv*dxdr
(3.5)
is bounded. We have now the following Lemma 3.2. Under the above assumptions, we have, for the weak solutions of the Boltzmann equation for noncutoff Maxwell molecules: \v-v*\2f(x,v,t)f(x,v*,t)B(9)dtdLi
f 7K3XK3XS2X[0,T]X[0,1]
< K0
(3.6)
where KQ is a constant, which only depends on the initial data (and molecular constants). In fact, we can take if = £2 as a test function and remark that the contribution of the left hand side is bounded in terms of the initial data
106
because £2 < |v| 2 . Hence the right hand side is also bounded. We can now replace v by c = v — u in the right hand side (since the extra terms vanish thanks to mass and momentum conservation). Then:
(3.7)
When we integrate the first term with respect to n the contributions containing c 2 and C3 vanish by symmetry and the weak collision term that remains to evaluate is Q{f,f)(x,x,t)(i2dvdxdt
/
=
J[0,T]x|0,l]xK 3
- / B(6>){[m(e-e*)]2-|n-V|2})//.d/xdt. 3 3 2 J[o,r]x|o,i]xi xi xs
(3.8)
We can separate the contributions from the two terms, since they separately converge and obtain Q(fJ){x,v,t)!i2dvdxdt
/
=
>/[0,T]x[0,l]xK 3
-3£0 /
(£, 3
J[0,T]x[0,l]xK xK
+B0 f
£,*)2ff*dvdv*dxdt
3
\V\2ff*dvdv*dxdt.
(3.9)
J[0,Tjx[0,l]xR 3 xIR 3
where if the force between two molecules at distance r is nr~5, then B0 = a J ~
(a = 1.3703...).
(3.10)
The constant a was first computed by Maxwell10; the value given here was computed by Ikenberry and Truesdell9. Since we know that the left hand side of Eq. (3.9) is bounded and the first term in the right hand side is bounded, it follows that the last term is also bounded by a constant depending on initial data (and molecular constants, such as m and K).
107 4. Existence of weak solutions for noncutoff potentials In order to prove the existence of a weak solution, we shall assume that this has been proved for Maxwell molecules with an angular cutoff2; actually to make the paper self-contained an the proof more explicit, we shall assume that the proof is available when a cutoff for small relative speed is introduced. In this case, in fact the proof immediately follows from the DiPerna-Lions existence theorem with the estimate of Lemma 3.2; it is enough to remark that a solution exists when we renormalize by division by 1 + ef (/ independent of e > 0 and we pass to the limit e —> 0 thanks to (3.6). In the noncutoff case we approximate the solution by cutting off the angles close to TT/2 and the small relative speeds. In this way we can obtain a sequence / „ formally approximating the solution / whose existence we want to prove. • Lemma 4.1. Let {/"} be a sequence of solutions to an approximating problem. There is a subsequence such that for each T > 0 i) J 7 n dv -* / / dv a.e. and in tf^T) x
/
JR3
|V| 2 /„»dv* -> /
|V| 2 /*dv,
JM3
in L^CO.T) x R 3 x BR) for all R > 0, and a.e., iii) ,
.>
IR3XR3 | V | 2 / „ / „ * d v d v ,
gn(x, t) =
/
• 1 + J /„ dv
s
3
lV| 2 //»dvdv«
1 + J f dv
= g{x, t) (4.1)
weakly in ^ ( ( O , T) x (0,1)). Proof, i) is immediate, ii) uses an argument well-known in DiPerna-Lions proof with the estimate sup„ / / n ( l + |v| 2 ) dv < oo to reduce the problem to bounded domains with respect to v*. For iii) we use i) and the fact that / „ converges weakly, but the factor multiplying it in the integral converges a.e. because of ii). Now we remark that gn(x,t) converges weakly to g(x,t) and pn(x,t) converges a.e. to p(x, t) and the integral / pngndxdt is uniformly bounded to conclude with the following Lemma: Lemma 4.2. Let {/„} be a sequence of solutions to an approximating problem. There is a subsequence such that for each T > 0 f \V\2fnfn*dpdt - / \V\2ff*dpdt (4.2) i(0,T)x(0,l)xK 3 xR 3
J(0,T)x(0,l)xE 3 xK 3
108 We can now prove the basic result: Lemma 4.3. Let {fn} be a sequence of solutions to an approximating problem, weakly converging to / . There is a subsequence such that for each T>0 / i(0,T)x(0,l)xR 3
4>Qn(fn,fn)dtdxdv^
/
<{>Q(f,f)dtdxdv
i(0,T)x(0,l)xR 3
(4.3) where Qn and Q are given by the weak form of the collision operator, as defined in Eq. (2.1). Proof In fact the integrand in the left hand side of Eq. (4.3) is, thanks to (2.4), uniformly bounded by the integrand of Eq. (4.2) which weakly converges. Thanks to this result, we can now pass to the limit in the approximating problem to obtain Theorem 4.4. Let f0 e L ' ( R x R 3 ) be such that [ f0(-)(l
+ \v\2)dvdx
f f0\lnf0(.)\dvdx
(4.4)
Then there is a weak solution f(x,v,t) of the initial value problem (1.1), (1.4), such that f <E C ( R + , L 1 ( R x R 3 ) ) , / ( . , 0 ) = / 0 . This solution conserves energy globally. 5. Concluding remarks We have surveyed the existence theory of the nonlinear Boltzmann equation with particular attention for a recent result of the author concerning Maxwell molecules, without any truncation on the collision kernel, in the one-dimensional case. To the best of our knowledge, this is the first result for the noncutoff Boltzmann equation. The solution conserves energy globally. Acknowledgments The research described in the paper was supported by MURST of Italy. References 1. M. Bony, " Existence globale et diffusion en theorie cinetique discrete" In Advances in Kinetic Theory and Continuum Mechanics, R. Gatignol and Soubbarameyer, Eds., 81-90, Springer-Verlag, Berlin (1991).
109 2. C. Cercignani, "Global Weak Solutions of the Boltzmann Equation", Jour. Stat. Phys. 118, 333-342 (2005). 3. C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag, New York (1988). 4. C. Cercignani, "Weak Solutions of the Boltzmann equation without angle cutoff', submitted to Jour. Stat. Phys. 2005. 5. C. Cercignani, "Estimating the solutions of the Boltzmann equation", submitted to Jour. Stat. Phys. 2005. 6. C. Cercignani, and R. Illner, "Global weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions", Arch. Rational Mech. Anal. 134, 1-16 (1996). 7. C. Cercignani, R. Illner, and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer-Verlag, New York (1994). 8. R. DiPerna, and P.L. Lions, "On the Cauchy problem for Boltzmann equations: Global existence and weak stability", Ann. of Math. 130, 321-366 (1989) 9. E. Ikenberry and C. Truesdell, "On the pressures and the flux of energy in a gas according to Maxwell's kinetic theory, I". Jour. Rat. Mech. Anal. 5, 1-54 (1956). 10. J. C. Maxwell, "On the dynamical theory of gases", Phil. Trans. Roy Soc. (London) 157, 49-88 (1866).
F R O M THE BIG B A N G TO F U T U R E COMPLETE COSMOLOGIES
YVONNE CHOQUET-BRUHAT Academie des Sciences, Paris.
[email protected]
1. Introduction A spacetime of General Relativity is a differentiable manifold V endowed with a pseudo Riemannian metric g of Lorentzian signature. The manifold is usually of dimension four, but higher dimensions are being considered in the aim of unification of gravitation with the other fundamental forces of nature, electromagnetism, weak and strong interactions. The spacetime is called Einsteinian if its metric satisfies the Einstein equations Einst(g) +Ag = T,
(1)
where Einst(g) is the Einstein tensor, given in terms of the Ricci tensor Ricc(g) and the scalar curvature R{g) of the metric g by Einst(g) := Ricc(g) - -gR{g).
(2)
A is a number called the cosmological constant. Its presence in the Einstein equations, and its possible values are under discussion. T is a symmetric 2-tensor, called the stress energy tensor, which models all the non gravitational energies, momenta and stresses. It is zero in vacuum. The covariant divergence of the Einstein tensor is identically zero, as a consequence of the Bianchi identities satisfied by the Riemann curvature tensor. The tensor T has covariant divergence zero when the sources satisfy the relevant equations (Maxwell and Yang - Mills equations for electromagnetism and other fundamental interactions, or appropriate equations in the case of macroscopic matter sources). These equations, called conservation equations, V.T = 0, make the system of equations 1.1, 1.3 coherent. 110
(3)
Ill When the source T is known, the Einstein equations are a system of quasilinear partial differential equations of the second order for the metric g. These equations are geometric equations, invariant by diffeomorphisms of V. Their resolution involves many problems linked with the theory of both hyperbolic and elliptic partial differential equations on manifolds. It has become usual to call "cosmological" those Einsteinian spacetimes which have compact spacelike sections, in contradistinction from those spacetimes which have asymptotically Euclidean space sections and are used to model the motions of isolated bodies. However the cosmos we live in may well have non compact spacelike sections. In fact very little is known about our universe as a whole. It is legitimate for the mathematician to study all possible models, with arbitrary topology as well as arbitrary Lorentzian metric. It is still believed by the majority of physicists and astronomers that such a manifold does model the cosmos, and moreover its metric satisfies the Einstein equations, eventually in dimensions greater than four. This hypothesis opens a vast field of investigations for the mathematician, where remarkable conjectures have been proposed. Many results have been obtained, sometimes surprising, but many fundamental questions remain open. Among the most interesting questions debated today, are the questions of the beginning and the end of our universe. It is generally believed that it had a beginning. Most modern astronomers tell us the beginning was a big bang, that is, for the mathematician, a singularity of the spacetime. The ultimate future fate of the cosmos we live in cannot now really be predicted, in spite of the most recent observation data. For the mathematician the question is of the future global existence of a solution of the Einstein equations. Since the cosmos is a physical as well as a geometrical object, where the elapsed time depends on the trajectory of the observer, the definition of the word "global" in the context of Einstein equations requires some thought. Since observers in free fall follow geodesies, the requirement for the future global existence of an Einsteinian spacetime is usually taken as equivalent to the completeness of all future directed timelike geodesies.
2. Obstructions to future completeness It has been known since the seventies, specially due to the work of Penrose and Hawking, that completeness of future complete geodesies in an Einsteinian spacetime encounters many obstructions when the sources satisfy some "positive energy condition", essentially because gravitation is
112
then attractive and therefore has a tendency to focus the timelike, or null, geodesies. I will give below a sample of the Penrose-Hawking singularity theorems, in the context of the solution of the Cauchy problem. The Einstein equations are geometric equations, a proper formulation of the Cauchy problem for these equations is also geometric. An initial data set for the Einstein equations is a triplet (M, g, K) where M is an n-dimensional manifold, g a properly Riemannian metric and K a symmetric 2-tensor on M. A development of an initial data set is an n + 1 dimensional manifold V with a Lorentzian metric g such that M can be identified with a submanifold of V, g is the metric induced by g on M and K is identified with the extrinsic curvature of M as submanifold of (V,g). A development is called Einsteinian if the metric g satisfies the Einstein equations. It is called globally hyperbolic (Leray) if the set of timelike paths joining two points is compact (in the Frechet topology of the set of paths). Global hyperbolicity is equivalent (Geroch) to the existence of a Cauchy surface, n-submanifold such that all timelike paths cut it once and only once. The spacetime manifold V is then a product M x R. It has been known for a long time that the Cauchy data cannot be arbitrary, they must satisfy, on M, a scalar and a vector equation, called constraints. Modulo some arbitrary data, representing in a vague sense initial radiation data, the constraints can be put under the form of a semilinear elliptic system. On the other hand it has been proved long ago, in the vacuum case as well as for usual non dissipative sources, that an initial data set satisfying the constraints admits a globally hyperbolic Einsteinian development, unique in the class of maximal globally hyperbolic Einsteinian developments (CB and Geroch). The Hawking - Penrose singularity theorems give conditions under which the maximal future Einsteinian development of Cauchy data is incomplete for timelike or null geodesies, when the sources satisfy some positive energy hypothesis. A sample of these conditions is the following: 1. M has a constant positive mean extrinsic curvature. 2. The initial 3 - manifold M contains a closed trapped surface S, that is there exists an open relatively compact set in M whose boundary S is such that outgoing null geodesies at points of S are converging. The foundation of the proofs of these theorems is the so called Raychauduri inequality, derived from the. Jacobi identity satisfied by the second variation of the arc length in a Lorentzian manifold with Ricci tensor zero or satisfying some positivity condition. The problem with the application of the second singularity theorem to
113 the global existence of a solution of the Cauchy problem is that it is not kown how to control the formation of a trapped surface for generic initial data. 3. Cosmic censorship conjectures Inspired by the case of the Schwarzschild solution whose singularity at the center cannot be seen by a far away observer, Penrose has formulated the conjecture (weak cosmic censorship) that the non existence of what he called "naked singularities" is a generic properties of physically realistic Einsteinian spacetimes. The strong cosmic censorship conjecture (which in fact does not imply the weak one) also originates from Penrose. It says that the globally hyperbolic Einsteinian development of complete, generic, physically realistic initial data is inextendible, even as a Lorntzian manifold. The Taub spacetime is a counter example to the strong cosmic censorship, but it is non generic, due to its symmetry group. The validity of the cosmic censorship conjectures has been studied in detail for Einstein equations with source a scalar field, by Christodoulou in the spherically symmetric case, with M = R3. He has shown that for small initial data, i.e. g near to the Euclidean metric, K near to zero as well as the scalar field initial data, the globally hyperbolic Einsteinian development is smooth and complete. For large initial data Christodoulou shows the existence of a global generalized solution, and proves that it may develop naked central singularities, but these singularities are unstable under small variations of the data. 4. Results for global existence 4 . 1 . Vacuum
case
The famous Christodoulou - Klainerman 1992 theorem proves the existence of futue and past complete vacuum Einsteinian developments with initial data on R3 near to the Minkowskian initial data. Previous results by H. Priedrich 1986 proved the existence of future complete developments of hyperboloidal initial data, using his conformal formulation of the Einstein equations. In the case of a compact initial manifold, no complete future and past development is known, except the flat solution T3 x R. a
In fact Christodoulou takes initial data on a light cone and proves global existence in retarded time.
114
It is expected that, in the case of a compact space M, there will, generically, exist a singularity in one time direction, physically in a finite past. Various possibilities will exist for the future. The future complete existence has been proved in the following vacuum cases: -Space T 3 with T2 symmetry, without smallness assumption on the initial data (Berger, Chrusciel, Isenberg, Moncrief 1997) -Space M a principal fiber bundle M —> E with group and typical fiber U(l) and basis E, a 2-manifold with genus grater than 1, with 1 parameter spacelike isometry group defined by the action of U(l). The initial data are supposed to be near initial data of a flat spacetime (polarized case, i.e. orbits orthogonal trajectories to surfaces E x {£}, C.B.-Moncrief 2001, general case, C.B.2004). - Initial data near initial data obtained by quotients with discrete isometries of a flat space time (Andersson - Moncrief 2004). 4.2. Equations
with
sources
In the presence of field sources, the deep study of the spherically symmetric Einstein scalar equations by Christodoulou has been deepened by Dafermos. The CB - Moncrief theorems (U(l) symmetry) have been extended to the Maxwell - scalar sources CB 2005. In the presence of kinetic matter sources various results have been obtained by Rendall and collaborators for spacetimes with 2 parameters isometry group (see references in Rendall 2005). 5. S1 symmetric vacuum Einsteinian spacetimes In the following sections we will review briefly the proof of the future competeness in the case of spacetimes with an S 1 spacelike isometry group. Though our proof is valid only for small data, the future completeness for large data is an open question which may have a positive answer. 5.1.
Equations
A spacetime (V^^g) is said to possess a spacelike S1, i.e. U(l), isometry group if V\ = M x R, and M is an S 1 fiber bundle over a surface E (then Vi is an S 1 fiber bundle over Vjj = E x R), and ^g is invariant under the action of the group S1. Then, in a local trivialization of V4 the metric reads: Wg = e-2^g
+ e2i(d6 + A)2,
(4)
115 where 7, Wg, A are respectively a scalar, a Lorentzian metric, and the representant of a S1 connection 1 form, all on V3. Straightforward calculation shows that the vacuum Einstein equations on V4, Ricd(^g) = 0, imply the following: 1. The curvature F := dA of the S1 connection is such that: d(e^*F)
= 0.
(5)
This equation is solved by setting F = e'^doj
(6)
w is a scalar function on V3 called the twist potential. 2. The pair (j,ui) satisfies a wave map equation from {V3,^g) into the Poincare plane (R2,P), where: P = 2(d 7 ) 2 + (l/2)e- 4 T(dw) 2 .
(7)
3. The Lorentzian metric ^g satisfies the Einstein equations on V3 with source the wave map: Ricci(sg) = p = P(du ® du).
(8)
In 3 space dimensions the Ricci tensor is linearly equivalent to the Riemann tensor, one says sometimes that the Einstein equations in 2+1 dimensions are not dynamical. However the equivalence of the resolution of these 2+1 equations with the resolution of the constraints on each spacelike surface St depends on the topology of S. Indeed it can be shown that the resolution of the constraints on each surface S ( for the induced 2 - metric g and extrinsic curvature k gives that the projections on St, and the mixed projection on St and the normal to St, of the Einstein equations 5.5 are zero. The Bianchi identities show then that the projection on Sf of Ricci^g) — p is traceless, and also transverse, that is has a vanishing divergence in the metric gt of S t . In the case of a compact surface S, the space of such 2-tensor fields, called TT tensors, is finite dimensional, isomorphic to the space of classes of conformally equivalent metrics on S, called Teichmuller space, a manifold diffeomorphic to R6G~6, if G > 1, G the genus of S. On the sphere (G = 0) there are no TT tensors, on the torus (G = 1) there are 2 linearly independent TT tensors. In the case G > 0 some equation must be added to the constraints to insure the verification of 5.5. We consider the case where M is compact and G > 1. We find then that the 3 dimensional Einstein equations 5.5 are equivalent to the following:
116 a. Constraints on each S4, which will be solved by the conformal method: one sets gt = e2Xat, with at some t dependent Riemannian metric on S. One finds that the constraints are equivalent to a semilinear elliptic system on each S t , which determines g, and k, modulo gauge conditions equivalent to elliptic linear equations, we choose in particular constant mean curvature for k, i.e. trgk = r(i). b. O.D. equations for the time evolution of the conformal geometry of (S 4 , at) in space insure the solution of 5.5. 5.2. Local existence
theorem
The formulation given above, and known theorems for elliptic and hyperbolic equations lead to the following theorem. Theorem 5.1. 1. Suppose the Cauchy data on E t o are a C°° metric &o and TT tensor q0, together with scalar functions, (70,^0) = «o = u(to,.) € H2, UQ =(N~1e2Xdou)(to,.) € Hi. Then there exists a solution of the 2+1 Einstein -wave map equations onH x (to — ti,to + £2) taking these cauchy data, ;if ti and £2 are small enough. 2.There exists a corresponding, vacuum, S1 symmetric Einsteinian spacetime if the initial data satisfy an integrability condition for the construction of A. 5.3. Global
existence
We have denoted by r the mean extrinsic curvature of Sj. Hence St expands when r increases from To < 0 to zero. Set t = — T _ 1 , t increases from to > 0 to infinity, E t collapses when t tends to zero. We choose the future to be the expanding direction. The spacetime will manifest a singularity in the contracting direction, that is when t = —r - 1 tends to zero. The future global existence is proved for small data by using the classical methods of energy estimates and bootstrap. A difficulty arises due to the fact that the metric at, used in elliptic estimates, is itself an unknown which must be shown to be uniformly equivalent to its initial value. The proof of this equivalence requires the introduction of corrected energies. Energies estimates. One defines the first energy by using the Hamiltonian constraint of the 2+1 Einstein equations. One denotes by h the traceless part k — \gr of k, and one sets: e2 = E(t)=
I {\Du\2 + \u'\2 +
\\h\2)ng
117 For the second energy, which has no obvious physical meaning, we take the usual second energy of a wave map, a hat denoting a gauge covariant dervative: e2 = r - 2 EW(t)=
f
(\Agu\2
+
\Du'\2)fig
Elliptic estimates deduced from the constraints and gauge equations give: 1 < -=\T\ex V2
< l + C,B,CT(e + ei),
0<2-N
~E(t)=r f N(\uf + hh\2)»g<0 dE( i) - - 2TE^ dt
r
= T /
N\Du'\2
fig + Z
where Z satisfy the inequality: |Z|
(u-
it).u'fig
where u is the mean value of u, namely: u=-—— Ufia, Volat J-zt
u = (7,w)
and EW(t) = E^(t)
+ar
f JT.,
kgu.u'ng
118 It can be shown that, for appropriately chosen a, the quantity Ea + T~ Ea bounds the total energy e2 +e\, and that elliptic estimates lead to inequalities for the corrected energies, which imply the following decay of the total energy: 2
(e2+e21)(t)
+ £21)(t0).
This decay, in its turn, implies a bound of at. The coefficient M in this inequality depends on the a priori bounds CE,
BKL
singularities
The BKL definitions were inspired by the Kasner spacetime metrics. A Kasner metric is defined on R3 x {t > 0} by the formula -dt2 + t2pi(dx1)2
+t2p2(dx2)2
+ t2p*(dx3)2
The space sections t = constant are isometric to the Euclidean space, but the spacetime metric is not flat if the p\s are not all zero. This spacetime metric is a solution of the vacuum Einstein equations if the pi belong to the Kasner circle
E K= E rf = 1i=l,2,3
i=l,2,3
(9)
119 These equations imply that necessarily one of the pi is negative, hence t = 0 is a singularity of the Kasner metric. In all cases the volume of Mt expands from zero to infinity when t increases from zero to infinity. When two of the p's are zero the spacetime isometric to the wedge t > |x| of the Minkowski spacetime. When two p's are not zero, then one at least is negative. If Pi < 0 , p2 > 0, P3 > 0, when t tends to zero the spacetime shrinks in the direction of x2 and x3, expands indefinitely in the direction of xx. The behaviour of a general spacetime near its singularity is interpreted by BKL as a succession of "Kasner epochs". A model is given by the Bianchi type IX spacetimes (V^ g) such that V = S3 x R and <4>ff := -dt2 +W
g
(10)
with (3)
The Einstein equations written for a Bianchi IX metric are analogous to the Einstein equations written for a Kasner metric, but with the addition of the Ricci tensor of the space metric. The asymptotic properties as t tends to zero of a Bianchi IX spacetime are, heuristically, studied as follows (see a deeper study in Ringstrom 2000). Suppose that, in some range of t, the following approximation holds: 2vi
ai=t
.
The Einstein equations give then i
i
Suppose pi < 0, 0 < p2 < P3- Then, for small t, 02 and 03 can be neglected 1
in comparison with aj. Set a ? = eai, the functions a; satisfy approximately the differential equations < + |e4ai=0.
(11)
a'2' = « 3 ' = ^ 4 One deduces from 6.3 that a\ is an oscillating function. The functions a 2 and az start to increase after some time, and the approximation ceases to
120
be valid. Belinski, Lifshitz and Kalatnikov [BKL] introduce the so called Kasner eras, succession of approximations by Kasner solutions with the 3 axes are alternatively shrinking in a random way. These heuristic considerations initiated by BKL have given birth to many interesting studies on the oscillatory - or not - behaviour near the initial singularity. It is believed that the generic big bang singularities of the vacuum 3 + 1 Einstein equations are oscillatory. The most recent and complete study, for general spacetimes in arbitrary dimension, done with the consideration of the Hamiltonian and the walls it creates, is the work of Damour and collaborators.
6.2. AVTD
behaviour
Another situation, which is amenable to more rigorous mathematical study, is the case where to a VTD solution it can be associated a spacetime whose metric converges, in some sense, to this VTD solution when t tends to 0, the singularity. The singularities of such AVTD solutions have then be called quiescent by Andersson and Rendall 2000 when the VTD solution itself admits an attractor at the singularity. They have shown that the singularities of Einstein - scalar 3 + 1 equations are in general quiescent. The same result has been proved to hold for the vacuum Einstein equations in dimension greater than 9 + 1 by Damour, Henneaux Rendall and Weaver 2002. The idea of the proof is to show that the difference U between a VTD solution and a solution of the full equations satisfies a system of partial differential equations of the Fuchsian type, that is: tdtU + LU = t>MF(t,x,U,dxU)
(12)
with L a linear operator independent of t with non negative eigenvalues, (j, a positive number and F a set of tensor fields linear in dxU, continuous in t, analytic in x and U and uniformly Lipshitzian in all its arguments in a neighbourhood of U = 0, for t small enough. Such a system has one analytic solution tending to zero when t tends to zero. Isenberg and Moncrief have used this method to prove the AVTD behaviour of S1 symmetric spacetimes when S is a torus, in the polarized and half polarized cases, that is when the geodesies in the Poincare plane representing the VTD solution converge to the same point as t tends to zero. CB, Isenberg and Moncrief have extended the result to an arbitrary manifold S.
121
7. Open problems Among the open problems is the rigorous mathematical study of the possible oscillating behaviour of the spacetimes near the big bang. A problem for Sx symmetric spacetimes with AVTD behaviour, is to match the solution deduced from a VTD given solution with the future complete solution. The future evolution problem for arbitrary initial data and formation of singularities, together with the verification of the cosmic censorship conjectures, are still largely open. Bibliography Berger B.K., Chrusciel P.T., Isenberg J., Moncrief V. 1997 Global foliations of vacuum spacetimes with T2 isometry. Ann. Phys. 260, 117-142. Choquet - Bruhat Y. and Geroch G. 1969 Global aspects of the Cauchy problem in general relativity. Comm. Math. Phys. 14 329-335. Christodoulou D., Klainerman S. 1992 The non linear stability of Minkowski space. Princeton University Press. Friedrich H. 1986 Comm. Math. Phys 107 587-609. Leray J. 1953 Hyperbolic differential equations Lecture Notes, Princeton. Choquet-Bruhat Y and Moncrief,V. 2001 , Ann. Henri Poincare, 2, 10071064 (2001) See also "Non linear stability of einsteinian spacetimes with U{1) isometry group", gr-qc/0302021. Choquet -Bruhat Y 2004, "Future complete U(l) symmetric einsteinian spacetimes, the unpolarized case", in "50 Years of the Cauchy Problem", eds. P. Chrusciel and H. Friedrich. Rendall A. 2005 gr-qc 0505133 VI Ringstrom H. 2000, Ann Inst. Poincare 2, 405-500. Andersson L., Rendall A. 2001 Quiescent cosmological singularities Comm. Math Phys. 218 479-511, gr-qc 0001047 Isenberg J. Moncrief V., "Asymptotic behavior in polarized and halfpolarized U(l) symmetric spacetimes", Class. Qtm. Grav. 19, 5361-5386 (2002). Choquet-Bruhat Y. J. Isenberg and V.Moncrief 2005 II Nuovo Cimento B, Vol. 119, issue no. 7-9, 2005. Damour T. , Henneaux M, Rendall A, and Weaver M. , 2002, Ann. H. Poin. 3, 1049-1111 . Damour T, Henneaux M., Nicolai H. 2003 Class Quantum Grav 20 145 20 Choquet-Bruhat Y. , Isenberg J. 2005 to appear J. Geom. and Phys. 2005.
FIRST O R D E R W E A K NONLOCALITY IN E X T E N D E D T H E R M O D Y N A M I C S OF RIGID HEAT C O N D U C T O R S *
V. A. CIMMELLI Department of Mathematics, University of Basilicata, Campus Macchia Romana, 85100 Potenza, Italy E-mail: [email protected] P. VAN Department of Chemical Physics, BME, Budafoki ut 8, 1521 Budapest, Hungary and MTA, KFKI, RMKI, H-1525 Budapest 114, P- 0. Box 49, Hungary E-mail:[email protected]
The role of gradient dependent constitutive spaces is investigated on the example of Extended Thermodynamics of rigid heat conductors. First order nonlocality is developed and the consequences of some additional constitutive assumptions are analyzed.
1. Introduction Weakly nonlocal thermodynamic theories introduce the space derivatives of t h e basic variables into the constitutive functions 1 ' 2 ' 3 , 4 . Second Law restricts considerably the form of the constitutive quantities and gives a genuine insight into the structure of the theories. In this paper we investigate extended thermodynamic theories of rigid heat conductors from the point of view of nonlocality assumptions. T h e present work is based on the results of our previous researches 5 ' 6 . In Section 2 we consider first order nonlocal constitutive equations and apply t h e Liu procedure 7 ' 8 for the exploitation of Second Law. In Section
"This work was supported by Progetto COFIN 2002: Modelli matematici per la scienza dei materiali, and by the grant OTKA T048489. The University of Basilicata is acknowledged as well.
122
123 3 we investigate the different solutions of the Liu equations under different additional constitutive assumptions. The results are discussed in Section 4. 2. Second Law and weakly nonlocal state spaces In a rigid heat conductor at rest we start from the following local balance equation of the internal energy e + qi>i=0,
(1)
where e is the density of internal energy, q>j i = 1,2,3 are the components of the heat flux, / = -g£, /<, = g^-, Xi i = 1,2,3 are the Cartesian coordinates and the Einstein convention has been applied. According to the postulates of Extended Thermodynamics 9 ' 10 the basic state space11 in our investigations will be spanned by the variables (e, g,). A first order weakly nonlocal constitutive space11 is spanned by the basic state and its spacial derivatives, that is the fields (e,qi,e>i,qi>j). We assume the evolution equation for the heat current
(2)
where Qi is a constitutive function. With the assumption of nonlocality the spacial derivatives of the above equations give the further restrictions 12-13. e>i + qj'ji = 0,
(3)
Qi'j - 9i,j = 0,
(4)
These equations are sometimes referred to as prolonged forms of the evolution equations (1) and (2). The necessity of forming the derivatives of the constraints is the most important speciality of weakly nonlocal theories. The local balance of entropy is given by s+ji'i=os,
(5)
with s standing for the entropy density, ji for the entropy current and as for the density of entropy production. Second Law of thermodynamics forces as to be nonnegative. Both the entropy and the entropy flux are constitutive quantities. Let us introduce the Lagrange-Farkas multipliers8 T 1 , rf, Tf and T^ for the evolution equations (1), (2), (3) and (4) respectively. Now, Liu procedure gives dise + (d2s)iqi + (d3s)ie,i + {d4s)ijqVj + {d\ji)e>i + (d2ji)jqj'i + (93Ji)j e 'v + {d4Ji)jkqj'ki - T 1 (e + qvi) - r f (qi - gi) - T? {eH + qrji)
-
~ ?ij (Qi'j ~ (di9i)&j - (d29i)kqk'j - (dz9i)ke>kj - (d4gi)kiqk'ij) > 0.
124 Here dn, n = 1,2,3,4 denotes the partial derivative of the constitutive functions according t o the variables {e,qi,e.n-,
+ (d4s)u(d3gi)j
+ (d4s)mi(d4gm)jk
= 0,
(6)
= 0.
(7)
Finally the residual dissipation inequality can be written in the following form [diji + ( f t t s ^ i d i c ^ k i + [(d2Jj)i - disdij + (d4s)kj(d2gk)i\qiij + (d2s)igi
+
> 0.
(8)
In the following sections we are looking for special simplifying assumptions to solve the Liu equations (6)-(7) and t h e dissipation inequality (8).
3 . S o l u t i o n s o f t h e e n t r o p y i n e q u a l i t y i n c a s e of l o c a l s t a t e T h e local state is defined by an entropy which is independent of the gradients: s:=s(e,qi)
(9)
However, solution of the dissipation inequality can be achieved only with further assumptions. In Extended Thermodynamics 9 ' 1 0 the constitutive space is local b u t the evolution equations are balances, i.e. they have a special nonlocal form.
3 . 1 . Local state nonlocality
and nonlocal
evolution
with
linear
In this case the evolution equations depend linearly on the gradients: gt := AijOj
+ Bijkqk,j:
(10)
where Aij,Bijk are local constitutive functions. Now the dissipation inequality (8) reduces t o a solvable form as {diji + {d2s)jAjk)e>i
+ ({d2jj)i
- dis5ij + (9 2 s)fc J B f c j i)^j > 0.
(11)
125 Since the quantities in the parentheses are independent of the gradients they should be zero respectively. Therefore we get diji = -(d2s)jAjk,
(12)
{(hjj)i = disStj - {d2s)kBkji.
(13)
These equations cannot be solved without any further ado. However, they result in strong correlations on the entropy derivatives and the evolution equation, as the mixed partial derivatives of ji should be equal. Let us observe that in this case the entropy production is zero, there is no dissipation. 3.2. Local state and local evolution
in balance
form
If we assume that the evolution equations have a balance form, then we find that the fields (Aij,Biji<:)(e,qi) are not independent. The evolution equation can be written as 9i ••= diQij&j
+ (d2Qij)kqk'j-
(14)
Here Qij is the current 14 of qi. The conditions (12)-(13) are diJi = -{d2s)jdiQij,
(d2jj)i = dis8ij ~(d2s)k(d2Qzj)k-
(15)
The above system of equations can be solved, as the entropy current is a potential of the field (qi,Qij), therefore it can be conveniently written as ii( e i9i) = 3i(
(16)
bodies
Eq.s (15) does not give an explicit form for Qij. Therefore we have lost one of the basic flavors of irreversible thermodynamics, that the requirements of the Second Law can be exploited constructively to get the appropriate evolution equations. Now the dissipation inequality was solved, but the evolution equations cannot be determined. Jou, Lebon, Mongiovi and Peruzza gave some simplifying conditions to have a solution of (15) on the phenomenological level15. They have assumed a local state with balance form evolution equations. From (15) one can deduce {d22s)kjdiQki = dnsSij +
(d12s)k(d2Qik)j-
126 If the material is isotropic t h e flux of t h e heat current and t h e entropy current can be written as Qij = /?(e,g 2 )<% + iP(e,q2)qiqj,
j t = ^(e,q2)qi.
(17)
T h e potential structure imposes several restrictions on the constitutive functions f3(e,q2), ip(e,q2) and ^ ( e , q2) introduced above. For example the entropy current will be ji = ( 9 e s + 2dq2 stpq2) qi. After some calculations one can get explicit solutions for the functions (3 and tp together with some additional restrictions on the form of the entropy function. 3.4. Local state
and given
entropy
current
A different solution of the dissipation inequality can be given with the help of specific assumptions for t h e entropy and t h e entropy current without requiring the balance form. One can consider a local entropy in the following form 16 s(e, q^ = s0 - -rriijqiqj.
(18)
This general expression is motivated by t h e requirement of the thermodynamic stability. Therefore, m ^ is a positive definite constitutive function 1 4 . Moreover, let us write the entropy current in t h e form 1 7 Ji = NijQj,
(19)
and let us introduce t h e convenient notation m as follows (d2s)i = -rriijqj
- - ( ^ m y ) ^ - = -m^qj
(20)
If rriij is constant, then rhij = m ^ . Let us emphasize again t h a t the entropy (18) and the entropy current (19) are rather general assumptions written in a convenient form, as long as the corresponding inductivities m ^ and current multiplier Nij are general constitutive functions. T h e dissipation inequality in our case can be written as [Nji'j - rhijg^qi
+ [Ni:j - dis5ij\qri
> 0.
(21)
Let us observe t h a t in the second t e r m the coefficient of qyi is local, therefore it should be zero. Hence the inequality reduces t o {{dis)'i - rhijgj) ^ > 0.
(22)
This is a force-current system, with t h e following Onsagerian 1 4 solution -rhijqj
+ (dis)'i
= Lijqj
(23)
127 This form of t h e evolution of the heat current results in a C a t t a n e o ' s type heat conduction equation 1 8 .
4.
Conclusions
In this paper we have investigated the consequences of nonlocality assumptions in Extended Thermodynamics of rigid heat conductors. We have seen t h a t t h e balance structure of the evolution equations requires t h e potential structure of t h e currents. On the other hand, suitable assumptions on the form of the entropy current make possible t o build all requirements of the Second Law into the evolution equations in general. In this case Second Law is a material property. Second order nonlocality is investigated in 6 .
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
P. Van. Annalen der Physik (Leipzig), 12(3), 142 (2003). K. C. Valanis. Acta Mechanica, 116, 1 (1996). K. C. Valanis. Acta Mechanica, 127, 1 (1998). P. M. Mariano. Adv. Appl. Mech., 38, 1 (2002). V. Ciancio, V. A. Cimmelli, and P. Van. cond-mat/0407530, 2004. V. A. Cimmelli and P. Van. J. Math. Phys., accepted, (2005). I-Shih Liu. Arch. Rat. Mech. Anal., 46, 131 (1972). R. A. Hauser and N. P. Kirchner. Continuum Mech. Thermodyn., 14, 223 (2002). D. Jou, J. Casas-Vazquez, and G. Lebon. Extended Irreversible Thermodynamics. Springer Verlag, Berlin-etc, 2001. I. Miiller and T. Ruggeri. Rational Extended Thermodynamics, Springer Verlag, New York-eta, 1998. W. Muschik, C. Papenfuss, and H. Ehrentraut. J. of Non-Newtonian Fluid Mech., 96, 255 (2001). V. A. Cimmelli. J. Non-Equilib. Thermodyn., 29, 125 (2004). P. Van. Continuum Mech. Thermodyn., 17, 165 (2005). J. Verhas. Thermodynamics and Rheology. Akademiai Kiado and Kluwer Academic Publisher, Budapest, 1997. D. Jou, G. Lebon, M. S. Mongiovi, and R. A. Peruzza. Physica A, 338, 445 (2004). I. Gyarmati. J. Non-Equilib. Thermodyn., 2, 233 (1977). B. Nyiri. J. Non-Equilib. Thermodyn., 16, 179 (1991). C. Cattaneo. Atti Sem. Mat. Fis. Univ. Modena, 3, 83 (1948).
A SEMIGROUP OF SOLUTIONS FOR T H E DEGASPERIS-PROCESI EQUATION *
G. M. C O C L I T E t A N D K. H. K A R L S E N * Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO-0316 Oslo, Norway E-mail: [email protected], [email protected]
We prove the existence of a strongly continuous semigroup of solutions associated with the Cauchy problem for the Degasperis-Procesi equation with initial conditions in L 2 n L4.
1. I n t r o d u c t i o n Our aim is to investigate well-posedness in the class L2 D L4 for the Degasperis-Procesi equation 5 dtu — dfxxu + \udxu = 3dxudlxu
+ udx'xxu,
(t, x) e (0, oo) x R,
(1)
endowed with the initial condition uo: u(0,x)=uo(x),
x&R,
(2)
where we assume that u0EL2(R)nL4(R).
(3)
Degasperis, Holm, and Hone 7 proved the exact integrability of (1) by constructing a Lax pair. In addition, they displayed a relation to a negative flow in the Kaup-Kupershmidt hierarchy by a reciprocal transformation and derived two infinite sequences of conserved quantities along with a bi-Hamiltonian structure. They also showed that the Degasperis-Procesi * Partially supported by the BeMat A program of the Research Council of Norway. tCurrent address: Department of Mathematics, University of Bari, Via E. Orabona 4, 1-70125 Bari, Italy. * Supported by an Outstanding Young Investigators Award from the Research Council of Norway.
128
129 equation possesses "non-smooth" solutions that are superpositions of multipeakons and described the integrable finite-dimensional peakon dynamics, which were compared with the multipeakon dynamics of the CamassaHolm equation. An explicit solution was also found in the perfectly antisymmetric peakon-antipeakon collision case. Lundmark and Szmigielski 8 presented an inverse scattering approach for computing n-peakon solutions to (1). Mustafa 9 proved that smooth solutions to (1) have infinite speed of propagation, namely they loose instantly the property of having compact support. Regarding well-posedness (in terms of existence, uniqueness, and stability of solutions) of the Cauchy problem for (1), Yin has studied this within certain functional classes in a series of recent papers n > 12 > 13 . 14 . In 4 the authors, proved the well-posedness of the Cauchy problem (1)(2) in the functional setting L1 n BV. Moreover, in Section 4 of the same paper they showed the existence of solutions under the assumption (3). Here we want to complete the proof of the well-posedness in the functional framework L2 D LA, in the sense that we show the uniqueness and stability of the solutions of the Cauchy problem (l)-(2) under the assumption (3). Motivated by the fact that, at least formally, (1) is equivalent to the elliptic-hyperbolic system dtu + udxu + dxP = 0,
-dlxP
+ P=^u2,
(4)
following 4 , we have the definitions Definition 1.1. We say that u : [0, oo) x R —> R is a weak solution of the Cauchy problem (l)-(2) provided u £ L°° ((0, oo); L 2 (R)) and u satisfies (4)(2) in the sense of distributions. Moreover, we call u entropy weak solution of (4)-(2) if: u is a weak solution of (l)-(2), u G L°°((0,T);i 4 (M)), T > 0, and for any convex C2 entropy n : R —> R with n" bounded there holds dtr](u) +dxq(u) +r)' (u)dxPw < 0, in the sense of distributions on (0,oo) xR, where q : R -> R is defined by q'{u) = n'{u) u and Pu solves -dlxPu+Pu = 2 \u . Of course the definition of entropy weak solution cames form the strong analogy between (1) and the Burgers equation, indeed (4) can be rewritten as a Burgers equation with a nonlocal source. Our main results are collected in the following theorem: Theorem 1.1. There exists a strongly continuous semigroup of entropy weak solutions associated to the Cauchy problem (l)-(2). More precisely, there exists a map S : [0,oo) x (L2(R) n L4(M)) -> i°°((0,oo);i 2 (R)),
130 with the following properties: for each u0 € L 2 (E) D L4(M) the map u(t,x) = St(uo)(x) is an entropy weak solution of (l)-(2) and it is strongly continuous with respect to the initial condition in the following sense: if u0,„ -> u0 in L2(R) then 5(w0,n) -> S(u0) in L°°((0,T);L 2 (E)), for every {u0in}nen C L 2 (E) n L 4 (E), u 0 € L2(R) n £ 4 (R), T > 0. 2. Viscous Approximations and Compactness Our approach is based on proving the compactness of a family of smooth functions {us}e>0 solving the following viscous problems (see 2 ) : dtue + uedxue + dxPs — edlxue, 2
2
< -d xxPe + Pe = | « , u £ (0,z) = uEi0(a;),
t > 0, i £ l , t > 0, x G K,
(5)
i £ l
We shall assume that u 0 , £ € Hl(R) n BV(R),
u 0 , £ -* n 0 inL 2 (R) n L4(M).
(6)
We know from Theorem 2.3 2 that for each e > 0 there exists a unique smooth solution ue £ C([0,oo);i? 1 (M)) f)C([0,oo);BV(R)) to (5). The starting point of our argument is the following compactness result for (5) (see Theorem 4 4 ) . Lemma 2.1. Assume (3) and (6). Then there exist an infinitesimal sequence {e^ke'N C (0, oo) and an entropy weak solution u £ L°°((0,oo);L 2 (R)) nL o o ((0,T); J L 4 (E)), T > 0, to (l)-(2) S«C/J i/ia< u E , -> u, in LP((0, T) x E), 1 < p < 4, T > 0. The proof is based on e uniform bounds of ue in L°°((0, oo);L 2 (E)) and L°°((0,T);L 4 (E)), T > 0, (see Lemma 2.2 4 and Lemma 2.10 4 ) and on a compensated compactness argument 10 . In order to construct the semigroup of solutions we need to prove that the vanishing viscosity limit (see Lemma 2.1) is unique and depends continuously on the initial condition. The key point of our argument is the L2 stability estimate stated in the next section. 3. Vanishing Viscosity: L2 Stability Estimate Let?;, io G C([0,co);iJ 1 (E))nC([0,oo);5y(E)) be the smooth solutions
131 to (see Theorem 2.3 2 ) L 2 <*» <9tw +> -» wdxw- + dx™ W -=- and xxw,
'dtv + vdxv + dxV = Xdlxv,
3
- ^ , V + V = |i;a, k u(0,x)
u / j - i/i/ —
= v0(x),
KW(0,X)
2
=w0(x),
respectively, where we shall assume A, fi>0,
v0,w0eH1(R)nBV(R).
(7)
T h e o r e m 3 . 1 . Assume (7). Then for any t > 0, IK*, •) - w(t, •)|| i 2 ( R ) < e a ^ / 2 | | V o - WO||L2 (R )
(8)
+ e a ( t ) / a ^}(ll«»ll^) + IKII^)). a(t) ^ ( M ^ , 2 24(||9 3! i;o|| L i( R )
2 + |K||L2(R)) +
01^11^+11^1^))"
+ HdsWol|£i(R)) (|l^ol|L2(R) + | K | U 2 ( R ) ) min{/x, A}
32
6
3(||t;o|| L 2 ( R ) + ||wo|| z ,2 (R) )
i
min{/i, A} Our approach, as in 1,2 5 is based on the following homotopy argument. Let 0 < 8 < 1. The function u>g interpolates between the functions v and w. More precisely, denote by tug the smeoth solution of the initial value problem 'dtue + uJedxcjg + dxVt9 - (9n + (1 -
6)X)dlxoje,
2
-d xxne + ne = ^ l ^ue(0,x) =6w0{x) +
(9) (l-6)vo(x).
Clearly UQ — v, w\ = w. Indeed 6 i->- uig(t,x) is a curve joining v(t,x) and w(t, x), and, for each t > 0 ||u(i, •) - io(t, •)||L2 ( R ) = dist L 2 (H) (t;(t, •), w(t, •)) < length L 2 (R) (we(£, •))• Employing the Implicit Function Theorem, as in Lemma 3.2 * and Lemma 3.2 2 , we have the following: L e m m a 3 . 1 . Assume (7). The curve 6 £ [0,1] H> wfl(t,-) G ^ W BV(R) is of class C1. In particular, we infer length L 2 (R) (we(V)) = / \\deue(t,-)\\L2(u)dd, Jo
t > 0.
n
132
Using the notation zg := dgcog, Zg :— dgQg and differentiating the equations in (9) with respect to 6, we have dtZg + ZgdxUJg + UJgdxZg + dxZg = (9fi + (1 - 6)\)dlxZg
+ (fj, -
\)dlxU>g,
—dxxZg + Zg = 3u)gZg, Zg(0,x)
= W0(x)
-VQ(X).
Finally, L2 — energy estimates on zg(t, •) similar to the ones in 1'2 give (8). 4. Proof of Theorem 1.1 Finally we are ready for proving Theorem 1.1. The first step consists in the existence of the semigroup. Lemma 4.1. There exists a strongly continuous semigroup of solutions associated with the Cauchy problem (l)-(2) S : [0, oo) x (L 2 (E) nL 4 (E)) -> £°°((0,oo);L 2 (E)), namely, for each u0 G L 2 ( R ) n L 4 ( l ) the map u(t,x) = St(uo)(x) is a weak solution of (l)-(2). Clearly, this lemma is a direct consequence of the following one and of the ones in the previous section. Lemma 4.2. Let {en}n€N, {/"n}neN C (0, oo), en, fin -> 0, and u, v £ L°°((0,oo);L2(R)) n L o o ((0,T);L 4 (E)), T > 0, be such that uEn -> u, «Mri -4 v, strongly in Lp((0, T) x E), T > 0, 1 < p < 4, then u = v. Proof. Let t > 0. From Theorem 3.1, we have that I K f t , •)-«**ft> -)ll^(R) < eW'W)
(\\u0,eh -«o,„ J | L » ( R ) +
l£h
•' ^
) ,
with 0 < A(t, eh, nh) <6t+
— — : — ^ - + ——:—?-— + mm{eh,(ih} min{eft,/ift}
mm{£h,fih}'
where S depends only on supfc(||uo,e-fc l|i=(R) + I K M J I L 2 ( R ) ) a n d mh := H^eUo.eJIi^flR) + ||3xuo,MhlliHR)" Choosing suitable subsequences as in Lemma 7.2 3 we get u = v. • The third and last step is the stability of the semigroup. Lemma 4.3. The semigroup S defined on [0, oo) x L 2 (E) fl L4 (E) satisfies the stability property stated in Theorem 1.1.
133 P r o o f . Fix e > 0 a n d denote by S£ t h e semigroup associated with t h e viscous problem (5). Choose {u0t„}neN C L 2 (E) n £ 4 ( E ) , u0 € L 2 (M) n i 4 ( E ) such t h a t u 0 ,n -> «o in £ 2 ( E ) . T h e initial d a t a u0,e,n, uo,e G -H" 1 (E)nfiy (E) satisfy condition (6). Finally, write ue
Employing L e m m a 2 . 1 , we get
\\un(t, •) - u{t, -)lli 2 (R) = h m \\ue
e—>-0
t > 0.
Using Theorem 3.1, we get t h e inequality \\u6,n(t,
0 - Ue(t, -))lli 2 (R) < eA^'e'n^\\u0,e,n
-
UQ,S\\L^(R),
with ln '•—I , Sit2m£,n •—I , Sit3 , e e e where o\ depends only on s u p £ > 0 ||uo,e||z,2(R) a n d m e ,„ : = ||9a;Uo,e,n||n(R) + ||^a; u o,e,n||L 1 (R)- Now, choosing a suitable sequence {e n }n6N as in Lemma 8.1 3 t h e claim follows. a n0 ^< A(t, ,./* e, n)\ ^< JC bit* H.
Sltm
References 1. G. M. Coclite and H. Holden, J. Math. Anal. Appl. 308, 221 (2005). 2. G. M. Coclite, H. Holden, and K. H. Karlsen, Discrete Contin. Dynam. Systems 13, 659 (2005). 3. G. M. Coclite, H. Holden, and K. H. Karlsen, to appear on SIAM J. Math. Anal. 4. G. M. Coclite and K. H. Karlsen, to appear on J. Fund. Anal. 5. A. Degasperis and M. Procesi, Symmetry and perturbation theory (Rome, 1998), World Sci. Publishing, River Edge, NJ, 23 (1999). 6. A. Degasperis, D. D. Holm, and A. N. W. Hone, Nonlinear physics: theory and experiment, II (Gallipoli, 2002), World Sci. Publishing, River Edge, NJ, 37 (2003). 7. A. Degasperis, D. D. Holm, and A. N. I. Khon, Teoret. Mat. Fiz. 133, 170 (2002). 8. H. Lundmark and J. Szmigielski, IMRP Int. Math. Res. Pap., 53 (2005). 9. O. G. Mustafa, J. Nonlinear Math. Phys. 12, 10 (2005). 10. M. E. Schonbek, Comm. Partial Differential Equations 7, 95 (1982). 11. Z. Yin, J. Math. Anal. Appl. 283, 129 (2003). 12. Z. Yin, Illinois J. Math. 47, 649 (2003). 13. Z. Yin, Indiana Univ. Math. J. 53, 1189 (2004). 14. Z. Yin, J. Fund. Anal. 212, 182 (2004).
NUMERICAL SIMULATION OF MAGNETICALLY DEFLECTED TRANSFERRED ARC *
V . C O L O M B O , E . G H E D I N I , A. M E N T R E L L I Department of Mechanical Engineering (DIEM) and Research Centre of Applied Mathematics (CIRAM), University of Bologna Via Saragozza 8, 40123 Bologna, Italy; E-mail: [email protected]
A 3-D time-dependent numerical model for the analysis of a dc transferred argon arc is presented. The model allows to investigate the influence on the behavior of the arc and on the evolution of the temperature inside the anode of an external magnetic field induced by a current flowing in an external conductor. If the currents of the arc and of the external conductor have opposite directions, the magnetic fields mutually repel and the arc is deflected. Such a practice to obtain a deflected arc finds applications in surface treatment of metallic substrates.
1. I n t r o d u c t i o n The aim of this work is to investigate by means of a 3-D time-dependent numerical model the behavior of a transferred arc plasma operating at atmospheric pressure, for the treatment of a metallic substrate material 1 , u s i n g a c u s t o m i z e d v e r s i o n of t h e C F D c o m m e r c i a l c o d e F L U E N T © 2 - 3 ' 4 .
Pertinent equations are solved for an argon plasma in local thermodynamic equilibrium, over a computational domain that includes the region between the two electrodes, where the discharge takes place, and the anodic region. The developed model allows also to evaluate the effects of the presence of an external magnetic field generated by a electric current flowing in a leading wire whose axis is parallel to the axis of symmetry of the cathode on the shape and behavior of the arc and on the evolution of the temperature inside the metallic substrate on which the arc is attached. It is well known that, if the current of the arc and the current flowing in the wire have opposite ' T h i s work was performed with partial financial support from the Italian Ministry of Education, University and Scientific Research (MIUR) national project cofin-2002 and from the National Group for Mathematical Physics (GNFM) oi the Italian Institute of High Mathematics.
134
135 directions, the magnetic fields repel reciprocally. As a consequence, the electric arc is deflected. Such a practice to obtain a deflection of the electric arc has found in recent years application in the industrial processing of materials, like treatment of metallic substrates. With this regard, even though many papers have been published in recent past on the modelling of transferred arc in both undeflected and deflected arc configurations 5 ' 6,7,8,9 , a time-dependent 3-D numerical study of the transferred arc including the anodic region in the computational domain has not been developed so far, to the authors' knowledge. The aim of this paper is the evaluation of the transient effects of an imposed external magnetic field on the shape of the arc and on the evolution of the temperature inside the anode, with the final aim of optimizing the treatment process of the metallic substrate. The effects on the substrate temperature of different external current intensities and positions of the leading wire have been investigated. 2. Modelling a p p r o a c h The 3-D time-dependent model developed at the University of Bologna 2 ' 3,4 for the simulation of atmospheric pressure transferred arc plasma torches both in undeflected and magnetically deflected configurations relies on the following main assumptions: the plasma is supposed to be optically thin; the plasma is supposed to be in local thermodynamic equilibrium (LTE); the viscous dissipation is neglected; the plasma flow is assumed to be turbulent and it is modelled by the FLUENT©-implemented RNG k-e model; electrode sheaths are not taken into account. Under these assumptions, the governing equations of mass, momentum and energy may be written as follows: ^ d(pu)
at
+ V-(pu) = 0
+ V • Ouu) = -Vp + V • [fj, (Vu + V u T ) ] + J x B + F d< ph)
" + V • (pu/i) = V • f - V / i ) + J x E - 3 * dt \ cp / and the scalar and vector potential equations are the following: V • (
V 2 A = -MoJ
(1) (2) (3)
(4)
In the previous equations u, h, T, p, p, k, a, are, respectively, the velocity, enthalpy, temperature, pressure, density, thermal and electrical conductivities, specific heat and viscosity of the plasma; E, B are the electric field
136 and magnetic induction, respectively; J is the arc current density; QR is the plasma volumetric radiative losses; V, A are the scalar and vector potentials, respectively. The magnetic induction B is obtained from the vector potential A by means of the relation B = V x A. Finally, F is the deflecting force acting on the arc due to the current in the conductor, that may be obtained by means of the relation F = J x B e once the external magnetic induction B e has been calculated with the following relations: Bc,x — Vole Bc,y = Vole Bc,z
cos (arctan (y/x)) 27iVz 2 + y2 sin (arctan (y/x)) 2TTy/x2 + y2
= 0
where Bc,x, Bc,y and Bc,z are the Cartesian components of the external magnetic induction generated by the current (whose intensity is Ic) flowing in the leading wire, parallel to the z-axis (the axis of symmetry of the cathode) 7 . The selected turbulence model is the FLUENT®-implemented RNG-based k—e one 4 ' 10 .The governing equations described so far are solved by the FLUENT© solver on a structured grid made up of hexahedrons and prisms. The User-Defined Scalar (UDS) approach 11 has been adopted for the treatment of the electromagnetic field, while extra source terms in transport equations have been implemented by means of User-Defined Functions10. The number of cells making up the grid is 1.8xl0 6 and the fixed time-step is At = x l O - 5 s. The thermodynamic and transport coefficients of the argon plasma used for the simulations are those calculated by Murphy 12 . All the calculations have been performed by PlasMac, a cluster of workstations available at CIRAM & DIEM of the University of Bologna. Such a resource allows for a large reduction in computational time as well as for the treatment of complex computational domain otherwise not manageable with traditional personal computers. The arc configuration and the implemented boundary conditions have been presented elsewhere7 and will not be repeated here. 3. Selected Numerical Results In Fig.1(a) and in Fig.1(b), the plasma temperature fields on two planes perpendicular one to each other and passing through the axis of the cathode are shown for the case of undeflected arc configuration. In Fig. 1(c) and in Fig. 1(d) the temperature fields inside the metallic substrate are shown, for
137 the same two planes of Fig.l(a) and Fig.1(b), respectively, and under the same operating conditions, while in Fig. 1(e) the temperature field at the plasma-anode interface is shown. It may be noticed that the temperature field is both the discharge region and the anodic region is axisymmetric because of full axisymmetric configuration of the torch. If Fig.2(a) and
in Fig.2(b) the plasma temperature fields on the same planes as before are given for the case in which the electric arc is deflected by the presence of a external current with intensity Ic = 50 A. In Fig.2(c) and Fig.2(d) the corresponding temperature fields inside the substrate are shown. Finally, in Fig.2(e), the temperature field on the upper surface of the anode is given. In this case, the symmetry of the discharge and of the temperature distribution in the substrate is lost because of the presence of the deflecting external magnetic field. In Fig.3 and in Fig.4 the results of a time-dependent simulation are presented. In this case, the external conductor is fixed for t < 0 and it starts moving around the axis of the cathode with an angular velocity u> = 2w at t = 0, keeping its axis parallel to the axis of the cathode. In Fig.3(a-d) and Fig.4(a-d) the temperature fields in the discharge and in the anodic regions are shown for the times t = 0 and t = 0.186 s, respectively. The temperature fields are given in Fig.3(a-d) and Fig.4(a-d) for two planes orthogonal one to the other and both passing through the axis
138
Figure 2. Temperature distribution [Kj in the discharge region (a.b) and inside the anode (c,d) on two planes perpendicular to each ether and passing through the axis of the cathode, and or. the upper surface othe anode (e). The wire is represented by the white line in Fig 2(a) and by the white dot in Fig.2(e).
of the cathode, one of these two planes containing the wire and, in Fig.3(e) and Fig.4(e), for the upper anodic surface.
Figure 3. Temperature distribution [K in the discharge region (a.b) and hieicie the anode (c,d) on two planes perpendicular to each other anci passing through the axis of the cathode, and on the upper surface of the anode (e) at i — 0. The wire is represented by the white line in Fig. 3(a) and by the white dot in Fig.3(e).
139
References 1. V. Colombo, A. Mentrelli and T. Trombetti, Eur. Phys. J. D 27, 239 (2003). 2. S. Melini, Analisi fisica e modellazione tridimensional per la caratterizzazione stazionaria e dipendente dal tempo finalizzata al progetto di torce al plasma a corrente continua ad arco trasferito, thesis, Univ. of Bologna, Italy (2004). 3. D. Bernardi, V. Colombo, E. Ghedini, S. Melini and A. Mentrelli, IEEE Trans. Plasma Sci. 33, 2, Part I, 428 (2005). •1. V. Colombo, E. Ghedini, A. Mentrelli, Time Dependent 3-D Numerical Simulation of Transferred Arc Plasma Treatment, 17th International Symposium on Plasma Chemistry (ISPC-17), Toronto, Canada, 7-12 August 2005 (2005). 5. G. Speckhofer and H. -P. Schmidt, IEEE Trans. Plasma Sci. 24, 1239 (1996). I). P. Freton, J. J. Gonzales, A. Gleizes, J. Phys. D.: Appl. Phys. 33, 2442 (2000). 7. A. Blais, P. Proulx and M. I. Boulos, J. Phys. D: Appl. Phys. 36, 488 (2003). 8. F. Lago, J. J. Gonzales, P. Freton, A. Gleizes, J. Phys. D: Appl. Phys.37, 883 (2004). *.»- ,1. J. Gonzales, F. Lago, P. Freton, M. Masqure and X. Franceries, J. Phys. D: Appl. Phys. 38, 306 (2005). HI. FLUENT 6.1 User's Guide, FLUENT Inc., Lebanon, NH (2003). II. D. Bernardi, V. Colombo, E. Ghedini, A. Mentrelli. Eur. Phys. J. D 27 55 (2003). VI. A. B. Murphy, Plasma Chem. Plasma Process. 20, 279 (2000).
A N O T E O N B A L A N C E LAWS FOR SLOW A N D FAST CHEMICAL R E A C T I O N S *
FIAMMETTA CONFORTO Dipartimento
di Matematica, Universita di Messina, E-mail: [email protected]
Italy
MARIA GROPPI Dipartimento
di Matematica, Universita di Parma, E-mail: [email protected]
Italy
Two sets of hydrodynamic equations describing fast and slow chemical reactions in a mixture of four gases are discussed in the framework of dissipative quasi-linear hyperbolic systems compatible with an entropy principle.
1. Introduction The kinetic modelling 1 of chemically reacting gas mixtures represents a fundamental step towards the physical understanding of the process, and also a starting point for a consistent derivation of adequate hydrodynamic equations at a macroscopic level 2 . In the present work, we consider the reactive Euler equations following as zero order asymptotics from the kinetic scheme proposed in 3 for a simple bimolecular chemical reaction A\ +A2 ^ A3 + A\, in a mixture of four gases As, s = 1 , . . . ,4. We assume that the typical relaxation time for mechanical encounters (elastic scattering) and/or the characteristic time relevant to chemical interactions are much shorter than the other typical macroscopic times. Hydrodynamic variables are hence determined by the dominant operator driving the process, and quite different scenarios occur in different physical situations. In particular, we examine here those which appear as most significant, namely the case "This work is supported by MIUR (Projects "Mathematical problems of kinetic theories" and "Nonlinear mathematical problems of wave propagation and stability in models of continuous media"), by INdAM-GNFM, by t h e Universities of Parma and Messina, and by Politecnico of Torino (Italy)
140
141 of dominant elastic scattering only (slow chemical reaction), and the one in which the dominant role is played by the whole, mechanical plus chemical, collision term (fast chemical reaction) 4 ' 5 . The two physical situations imply different hydrodynamics, and the relevant equations can be reconsidered within the context of the theory of hierarchies of hyperbolic systems 6 . In this paper, the macroscopic equations modelling the case of fast chemical reactions, which is a set of conservation laws preserving entropy, is reconsidered as the equilibrium subsystem of the one for slow reactions. Moreover, the latter is a genuinely dissipative quasi-linear system of balance equations, compatible with an entropy principle and of hyperbolicparabolic type. Such kind of systems have been widely studied and, recently, important results have been obtained on existence and uniqueness of smooth solutions for systems of hyperbolic-parabolic type 7 ' 8 and their long-time behavior. In particular, in 9 a conjecture states that the solution of the Riemann problem for such kind of systems converges to the one of its equilibrium subsystem. This suggests future investigation of either the steady shocks 10 or the Riemann problem n for both slow and fast reactive systems, in order to show the relations between their solutions.
2. Governing equations and thermodynamics features The hydrodynamic equations for a slow reacting gas mixture, derived in 12 from the kinetic model 3 by performing a closure at Euler level in the elastic collision dominated regime, in 1-D read as dtfii + dx (riiu) = Qi, 2
dt(pu) + dx(pu tot
i = 1,2,3,4
(1)
+ nkT) = 0 ,
(2)
tot
dt (pe ) + dx (pe u + nkTu) = 0,
(3)
where the total energy is given by the sum of kinetic, thermal and chemical components: tot
pe
1 3 1 3 = ~pu2 + -nkT + £ch = -pu2 + -p + ^
4
Eini.
Here 7ij, i = 1,2,3,4, are the number densities of the four species, u is the mean velocity, T the temperature. Moreover, n = J2i=i ni a n ^ P = J2i=i mini a r e the total number and mass densities, respectively, being TO; the molecular masses (mass conservation yields rn\ + m,2 —TO3+ VTM), Ei are the constant energies of chemical bond (and we may always assume
142 AE = £3 + E4 — Ei — E2 > 0). The state equation is the one of perfect gases p = nkT, and k is the Boltzmann constant. In Eqs. (1), Qi = OiQ, with &\ = a2 = —03 = —
(mlm2V/2
(AE
exp —— - m n 2 \ kT
\m3m4J
7rfC0, (4)
where 7 j | (T) is a suitable Gaussian-weighted collision frequency 13 . Eqs. (1) can be re-arranged in such a way to involve the conservation equation of total mass and, thus, system (l)-(3) is re-written in the following equivalent form, containing the greatest number of conservation equations: dtni + dx(niu) = Q, dtVj + dx(yjU)
= 0,
2
dt(pu)+dx(pu +p)
(5) j = 1,2,3, = 0,
(6) (7)
dt {petot) + dx {petotu + pu)=0
(8)
where y\ = ni - n 2 , y2 = ni + n 3 , y 3 = «i + n 4 . When both elastic collisions and chemical reactions are dominant processes (fast chemical reaction), the hydrodynamic equations obtained at Euler level are given by the system of conservation laws 10 dtVj + dxiyju) = 0, 2
dt(pu)+dx(pu +p) tot
j = 1,2,3, = 0,
(9) (10)
tot
dt {pe ) + dx {pe u +pu)=0,
(11)
coupled with the transcendental equation
™ = («V/2exp(^) n3n 4
\m3m4J
(12)
\ kl J
representing the well known mass action law of chemistry which characterizes the chemical equilibrium. It's worth noticing that both slow and fast reactive systems are hyperbolic and their eigenvalues may be found in 10 . As regards the thermodynamic properties of such reactive gas mixtures, following 14 let us observe that, from the state equation p = nkT, the total pressure p can be written as p — Yli=iPi> where pi — UikT is the pressure of the i—th species. Moreover, the internal energy density e takes into account both thermal and chemical contributions, pe = J2i=i Piei> internal energy density e,, enthalpy per unit mass hi and chemical potential pi of
143 the i— th species are respectively given by: 3 ~ ei = -kiT + Ei,
v hi = ei + ^ ,
^
ni = hi-siT,
(13)
Pi
where fc; = fc/m, and Ei = Ei/m,i. Prom the kinetic theory 13 , the expression of the entropy density s, of each species can be obtained as (up to an additive constant): Si = \h log (kT) - ki log ( - ^
J ,
(14)
and the entropy density of the reacting mixture, S, is given by: pS = Y2PtSi=
-kn log (kT) - k ^
i=l
m log
i=l
- ~
.
(15)
.TO.
In what follows, we choose as field vector u = (ni,yi,y2,y3,u,T)T terms of the field variables we have:
and in
n-2 = ni - y i , n 3 = ?/2 - " l , ™4 = 2/3 - " i , « = -2/i + 2/2 + 2/3 , P = -TO22/1 + "132/2 + m42/3 P = (-2/1 + 2/2 + 2/3) fcT, £ch = - A £ m - £22/1 + -B32/2 + -B42/3 •
(16)
3. Symmetrization and equilibrium subsystem As well known 6 , both systems (5)-(8) and (9)-(ll) can be put into symmetric form d2h'°
d2hn
where h'° = nk = p/T and /z'1 = nku = pu/T, in terms of the main field u', since they are both compatible with an entropy principle 13 dth° + dxhl = £ < 0
(18)
with h° = —pS, h1 = —pSu and with entropy production £ given by 3
E, = - | u r i o g
123124 1 m\rri2
n^n^
+ AE},
E/
=
0,
(19)
\m3m4
where indices s and / denote slow and fast reaction cases, respectively. Observe that at equilibrium, as Q = 0, also S s = 0.
144
In particular, system (5)-(8) for slow reactions can be symmetrized in terms of the main field u^ = (A1, A2, A3, A4, A", A T ), being
nin2 _ m2 / u2\ A - -Y- ^ 2 - Y )
Vm 3 m4/
2
A" = -
34
,
A
+ AE
m3,4 / u2 = — ^3,4 - y
AT = - -
whereas system (9)-(ll) for fast reactions is symmetrized in terms of u^ = (A2, A 3 , A 4 , A", A T ). Equilibrium states for slow reactions are characterized by the condition A1 = 0. Within the context of the theory of hyperbolic principal subsystems 6 , main field u^ is split into two parts u's = ( A \ w i ) ,
wi = (A 2 ,A 3 ,A 4 ,A",A T ) =u'f
the source term F = (Q, Ogs) splits too and finally we get the following: Proposition. The equilibrium subsystem of the system for slow reactions is
and it coincides with the symmetrized system for fast reactions. Remarks. It is worth noticing that, as proved in 6 , system (20) is compatible with a sub-entropy law (18) with zero sub-entropy production, and sub-characteristic condition holds, since the eigenvalues of the equilibrium subsystem (20) are strictly included between the minimum and the maximum eigenvalues of the full system 10 . It can be easily shown that the system for slow reactions is genuinely dissipative ' ' since Q(0, w^) = 0 and (dQ/dA1) {0,w's) is negative. Moreover, Kawashima-Shizuta condition 7,S ' 9 (K-condition) holds, i.e. in the equilibrium manifold any characteristic eigenvector does not belong to the null space of V U F. According to the general theory 7 ' 8 , the system for slow reactions admits global smooth solutions and, for large t, the solution converges to equilibrium. Concerning the Riemann problem, there is a conjecture 9 that, for large t, solutions to the equilibrium subsystem result to be the asymptotic solutions to the full system.
145
In 10 steady detonation waves are investigated and numerical simulations show the convergence of the solution for slow reactions to the one for the fast reaction case.
References 1. Cercignani, C.: Rarefied Gas Dynamics. From Basic Concepts to Actual Calculations. Cambridge University Press, Cambridge (2000). 2. Giovangigli, V.: Multicomponent Flow Modeling. Birkhauser, Boston (1999). 3. Rossani, A., Spiga, G.: A note on the kinetic theory of chemically reacting gases. Physica A 272, 563-573 (1999). 4. Bisi, M., Spiga, G.: Hydrodynamic limit for a gas with chemical reactions. In: Monaco, R., Pennisi, S., Rionero, S., Ruggeri, T. (eds) Proceedings XII International Conference on Waves and Stability in Continuous Media (WASCOM 03), 85-93. World Scientific, Singapore (2004). 5. Bisi, M., Groppi, M., Spiga, G.: Fluid-dynamic equations for reacting gas mixtures. Applications of Mathematics, 50 (2005) 1, 43-62. 6. Boillat G., Ruggeri T.: Hyperbolic principal subsystems: entropy convexity and subcharacteristic conditions. Arch. Rat. Mech. Anal. 137, 305-320 (1997). 7. Hanouzet B., Natalini R.: Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Arch. Rat. Mech. Anal. 169 (2), 89-117 (2003). 8. Ruggeri T., Serre D.: Stability of constant equilibrium state for dissipative balance laws system with a convex entropy. Q. Appl. Math. 62 (1), 163-179 (2004). 9. Brini F., Ruggeri T.: On the Riemann problem in extended thermodynamics. To appear in: HYP2004 Conference Proceedings. Yokohama Publishers, Inc. (2005). 10. Conforto F., Groppi M., Monaco R., Spiga G.: Steady detonation problem for slow and fast chemical reactions. In: Modelling and Numerics of Kinetic Dissipative Systems, 127-140. Nova Science, New York (2004). 11. Conforto F., Jannelli A., Monaco R., Ruggeri T.: On the Riemann problem for a gas mixture undergoing bimolecular reactions. Internal report, Department of Mathematics, Politecnico of Torino, n. 23 (2005), submitted. 12. Conforto F., Monaco R., Schiirrer F., Ziegler I.: Steady detonation waves via the Boltzmann equation for a reacting mixture. J. Phys. A: Math. Gen. 36, 5381-5398 (2003). 13. Groppi M., Spiga G.: Kinetic theory of a chemically reacting gas with inelastic transitions. Transport Theory Statist. Phys. 30 (4-6), 305-324 (2001). 14. Giovangigli V., Massot M.: Entropic structure of multicomponent reactive flows with partial equilibrium reduced chemistry. Math. Meth. Appl. Sci. 27, 739-768 (2004).
INTEGRATION OF PARTIALLY I N T E G R A B L E EQUATIONS
R. C O N T E Service
de physique de I'etat condense (CNRS URA no. 2464) CEA-Saclay, F-91191 Gif-sur-Yvette Cedex, France E-mail: Robert. [email protected]
Most evolution equations are partially integrable and, in order to explicitly integrate all possible cases, there exist several methods of complex analysis, but none is optimal. The theory of Nevanlinna and Wiman-Valiron on the growth of the meromorphic solutions gives predictions and bounds, but it is not constructive and restricted to meromorphic solutions. The Painleve approach via the a priori singularities of the solutions gives no bounds but it is often (not always) constructive. It seems that an adequate combination of the two methods could yield much more output in terms of explicit (i.e. closed form) analytic solutions. We review this question, mainly taking as an example the chaotic equation of Kuramoto and Sivashinsky vv!" + bu" + fiu' + u 2 / 2 + A = 0, v ^ 0, with (u, b, fi, A) constants.
1. Introduction Phenomena in continuous media are often governed by a partial differential equation (PDE), e.g. in one space variable x and one time variable t
E
({|S§H)=O-
p)
in which u and E are multidimensional, the integers m,n take a finite set of values. Our interest is the nonintegrable or even chaotic case, for which the powerful tools of Lax pairs, inverse spectral transform, etc * are inapplicable. The derivation of analytic results must then use other methods. Let us quote a few examples. (1) The one-dimensional cubic complex Ginzburg-Landau equation (CGL3) iAt + pAxx + q\A\2A ~ i-yA = 0, pq Im(p/g) + 0,
(2)
(and its complex conjugate, i.e. a total differential order four), in which p, q are complex constants and 7 a real constant, a generic
146
147 equation which describes many physical phenomena, such as the propagation of a signal in an optical fiber 2 , spatiotemporal intermittency in spatially extended dissipative systems 19'10>23. For two coupled CGL3 equations, see analytic results in Ref. 6 . (2) The Kuramoto and Sivashinsky (KS) equation, ft + WPxxxx + bifxxx + Wxx
+
(3)
in which u, b,/j, are real constants. This PDE is obeyed by the variable ip — arg A of the above field A of CGL3 under some limit 22 17 ' , hence its name of phase turbulence equation. (3) The quintic complex Ginzburg-Landau equation (CGL5), iAt + pAxx + q\A\2A + r\A\4A - ijA = 0, pr Im(p/r) ^ 0,
(4)
in which p, q, r are complex constants and 7 a real constant. (4) The Swift-Hohenberg equation iAt + bAxxxx
24 18
'
2
+ PAXX + q\A\ A + r\A\4A - i^A = 0, br ^ 0,
(5)
in which b,p,q,r are complex constants and 7 a real constant. The autonomous nature of (1) (absence of any explicit dependence in x and t) allows the existence of travelling waves u = £/(£), solutions of the ordinary differential equation (ODE) u(x, t) = U(£), i = x~ct,
E(U{N\
C/(JV-1}, ...,U',U)
= 0.
(6)
For the CGL3, KS, CGL5 and Swift-Hohenberg equations (with one exception, KS with b2 = 16/xy), all the solitary wave solutions \A\2 = KOif = ^(f)i£ = x — ct, which are known hitherto are polynomials in tanhfc£ (or cotanh,tan,cotan, which are the same in the complex plane), and such solutions are easy to find by taking advantage of the singularity structure of the PDE (see, e.g., the summer school lecture notes 5 ) . Hence the natural questions: (i) Can other solitary waves u = f(x — ct) exist (in closed form)? (ii) If yes, please find them all, not just a few ones. The present paper introduces to the methods in principle able to answer both questions. They will mainly be exemplified with the KS equation (3). The paper is organized as follows. In section 2, we give a mathematical formulation of the problem. In section 3, we prove the inexistence of an analytic expression representing the general solution, and we compute the gap between the differential order N of the ODE (6) and the maximal number of integration constants in a singlevalued solution. In section 4, we give hints (not proofs) that some analytic result still has to be found. In
148 section 5, we review the consequences of the assumption of singlevaluedness for a solution of the ODE (6), and present an algorithm to implement them. In section 6, we present the consequences of the assumption of meromorphy for a solution of (6). The last section 7 states the open problems. 2. Mathematical formulation of the problem The successive steps of the announced program are (1) To perform the traveling wave reduction from the PDE to an ODE. The KS PDE (3) depends on three fixed constants (v, b, /z) (fixed means: which occur in the definition of the equation), the reduction tp(x,t) = c + u(£), £ = x - ct,
(7)
introduces in the ODE one more fixed constant A (the second constant c cancels out because of the Galilean invariance) u2 vv!" + bu" + nu' + — + A = 0, i / ^ O ,
(8)
and the general solution of (8), if it exists, depends on the four fixed constants (v, b, /j,, A) and three movable constants (movable means: which depends on the initial data), which are the origin £o of £ and two other constants C\,ci(2) To count the number of constants which survive in the general solution of (8) when one requires singlevaluedness. (3) To find this largest singlevalued particular solution in closed form. Indeed, its representation as a series can be misleading, as shown by classical authors like Poincare and Painleve. 3. Local separation of singlevaluedness and multivaluedness Because the ODE (6) is nonintegrable, the number of integration constants present in any closed form solution is strictly smaller than the differential order of the ODE. This difference, an indicator of the amount of integrability of the ODE, can be precisely computed from a local analysis. Two local representations of the general solution of (6) exist. The first one, also the most well known, is useless for our purpose. This is the famous Taylor series near a regular point, whose existence, unicity, convergence, etc is stated by the existence theorem of Cauchy. The reason why it is useless is its inability to make a distinction between chaotic ODEs such as (8) and integrable ODEs such as v!" - Yluvl - 1 = 0.
149 The second one, less known than the Taylor series of Cauchy, is a Laurent series (or more generally psi-series and/or Puiseux series) near a movable singularity XQ. This one does provide the expected information. The technique to compute it is just the Painleve test (see Ref. 4 for the basic vocabulary of this technique). Let us present it on the KS example (8). Looking for a local algebraic behaviour near a movable singularity XQ u~x->x0 uoXp, u0 7^0, x = x~xo,
(9)
one first balances the highest derivative and the nonlinearity, p - 3 = 2p, p ( p - l ) ( p - 2 ) i / u o + ^ = 0 ,
(10)
a system easily solved as p = - 3 , u0 = 120i/.
(11)
The resulting convergent Laurent series, (0)_
120^ ~ x3
15b X2
15(16/^-62) Axl9uX
1 3 ( 4 ^ -b2)b 32xl9^2 +
U
, >' [ '
lacks two of the three arbitrary constants. They appear in perturbation 7 , u
=
u(0)+£u(D+£2u(2)
+
_^
( 1 3 )
in which e is not in the ODE (8). The linearized equation around u^
has then the Fuchsian type near XQ, with an indicial equation (q = —6 denotes the singularity degree of the lhs of (8)) \imX-J-q(vd3x+u0xp)xJ+P x->o
(15)
= v(3 ~ 3)0' - 4)(j - 5) + 120i. - v(j + l)(j2 - 13j + 60). (16) The resulting local representation of the general solution, u (xo, £c__ i, £C+, ec_ ) = 120i/x~ 3 {Regular(x) +£[c_ 1 x" 1 Regular(x) +
c + x ( 1 3 + i v ^ T ) / 2 Regular(x)
+ c _ x ( 1 3 - l V 7 I ) / 2 R e g u l a r ( x ) ] + 0 ( £ 2 ) } , (17) in which "Regular" denotes converging series, depends on 4 arbitrary constants (xo,£c_i,£c + ,£c_) but, as shown by Poincare, the contribution of
150 ec_i is the derivative of (12) with respect to xo, so c_i can be set to zero. The dense movable branching due to the irrational indices reflects 25 the chaos, and to remove it one has to require ec+ = £c_ = 0, i.e. e = 0, making the analytic part of (17) to depend on the single arbitrary constant xo. The ODE (8) admits other Laurent series in the variable (u — V ~ 2 A ) _ 1 , but they provide no additional information. The question is then to turn this local information into a global one, i.e. to find the closed form singlevalued expression depending on the maximal number (here one) of movable constants. We will call unreachable any constant of integration which cannot participate to any closed form solution. The KS ODE (8) has two unreachable integration constants, the third one x§ being irrelevant since it reflects the invariance of (8) under a translation of x. We will also call general analytic solution the closed form solution which depends on the maximal possible number of reachable integration constants, and our goal is precisely to exhibit a closed form expression for this general analytic solution, whose local representation is a Laurent series like (12). The above notions (irrelevant, unreachable) belong to an equation, not to a solution. Let us introduce another integer number, attached to a solution, allowing one to measure its distance to the general analytic solution. The distance of a closed form solution to the general analytic solution is defined as the number of constraints between the fixed constants and the reachable relevant constants. For the ODE (8), the fixed constants are v, b, /i, A, the movable constant XQ is irrelevant, the movable constants c\ = ec+,C2 = ec_ are unreachable, so the distance d is the number of constraints among the fixed constants. The closed form singlevalued solutions known to date are (1) one elliptic solution (distance d—1)
9 15
'
in which p is the elliptic function of Weierstrass, p> = Ap3-g2p-g3, (2) six trigonometric solutions (d = 2) u = 1 2 0 ^ 3 - 15fer2 + f-/, , 5L,2 13&3 bk 2 ~ 32^19^
+
+
(19)
16 13
' , rational in ek^,
- mvk2 - - ^ - \ 7fib 4-xlQV
T
T
k , Jfc ^ ,_N = 2 t a n h 2 « " *>U*»
151 the allowed values being listed in Table 1, (3) one rational solution (d = 3), 6 = 0, (i = 0, A = 0 : u = 120^(C-Co)" 3 ,
(21)
which is a limit of all the previous solutions.
b2/((iv)
0 144/47 256/73 16
vk2 /(i
vA/fj,3
-4950/19 3 , 450/19 3 -1800/47 3 -4050/73 3 -18, - 8
11/19, - 1 / 1 9 1/47 1/73 1, " I
All those solutions admit the representation 15(16/^ -b2) d (22) 76^ d£' in which ip is an entire function. This linear operator 2>, which captures the singularity structure, is called the singular part operator. The Laurent series (12) yields another information 12 . If its sum is elliptic, the sum of the residues of the poles inside a period parallelogram must vanish. Since the only poles of (8) are one triple pole, a necessary condition 12 for the sum to be elliptic is to cancel the residue of (12), i.e. b2 = \&(iv. For this equation, the condition is also sufficient, see (18). T> Log I/J + constant, V ••6 0
+1
^ V
+
4. Experimental and numerical evidence of missing solutions Experiments or computer simulations display regular patterns in the (x, t) plane (see 2 3 ) , some patterns being described by an analytic expression. For the other patterns, the guess is that there should exist matching analytic expressions. For the equation (3), one has observed a homoclinic wave 26 tp = f (£),£ = x — ct, while all known solutions are heteroclinic. The Laurent series (12) only provides a local knowledge of the general analytic solution. Rather than obtaining a global knowledge of the solution, which is the ultimate goal, it is easier to look at its singularities, by computing the Pade approximants 3 of the Laurent series (12). Pade approximants are a powerful tool to study the singularities of the sum of a given Taylor series, and more generally to perform the summation of divergent series.
152 Given the first N terms of a Taylor series near x = 0, N C XJ
( 23 )
SN = J2 J > 3=0
the Pade approximant [L, M] of the series is the unique rational function
[^^=v^°rl'6o=1'
(24)
obeying the condition SN-[L,M}
= 0{xN+1),
L + M = N.
(25)
The extension to Laurent series presents no difficulty. In particular, for L and M large enough, Pade approximants are exact on rational functions . The advantage of [L, M] over SN (which has no poles) is to display the global structure of singularities of the series. Prom a thorough investigation 27 of the singularities of the sum of the Laurent series (12) one concludes (this is not a proof): for generic values of (v,b,n,A), no multivaluedness is detected, no cuts are detected, and the singularities look arranged in a nearly doubly periodic pattern, the elementary cell being made of one triple pole and three simple zeroes. 5. Consequences of singlevaluedness (Painleve) 5.1. Classical
results on first order autonomous
equations
The failure to detect any multivaluedness in the unknown general analytic solution by no means implies the singlevaluedness of this general analytic solution, because the Painleve test only generates necessary conditions, and the Pade approximants are a numerical investigation. It is however worthwhile to examine in detail the consequences of an assumed singlevaluedness. Given the iV-th order autonomous algebraic ODE (6), any solution is « = /(€-&),
(26)
in which £o is movable. Provided the elimination of £o between the equation (26) and its derivative is possible, one obtains the first order nonlinear ODE F(u,u') = 0,
(27)
in which F is as unknown as / . However, /(£ — £0) is now the general solution of (27), and there exist classical results on first order autonomous ODEs which are in addition
153 algebraic. Let us therefore assume from now on that the dependence of / on £o is algebraic (this is a sufficient condition for F to be algebraic). Let us summarize. Given the iV-th order ODE (6) and its particular solution / Eq. (26), and assuming the dependence of / on £o to be algebraic, one is able to derive a first order ODE (27) which is algebraic. Conversely, given an algebraic first order ODE F = 0 Eq. (27), is it possible to go back to / ? This question has been answered positively by Briot and Bouquet, Fuchs, Poincare and put in final form by Painleve 21 . Theorem 5.1. Given the algebraic first order ODE F = 0 Eq. (27), if its general solution is singlevalued, then (1) Its general solution is an elliptic function, possibly degenerate, and its expression is known in closed form. (2) The genus of the algebraic curve (27) is one or zero. (3) There exist a positive integer m and (m + l ) 2 complex constants ajtk, with aQm ^ 0, such that the polynomial F has the form m 2m—2k
F(u,u') = Y, fc=0
Yl
a
o^uju'k = 0, a0,m ^ 0.
(28)
j=0
Then, assuming / singlevalued with an algebraic dependence on £o> (1) It is equivalent to search for the solution / or for F. (2) The solution / can only be elliptic (i.e. rational in p and p'), or a rational function of eax with a constant, or a rational function of x. The explicit form (28) of F makes it much easier to look for F than / . 5.2. Method to obtain the first order subequation
autonomous
The input data and assumptions are: (1) a N-th order algebraic ODE (6), N > 2, (2) a Laurent series representing its general analytic solution, (3) a first order algebraic ODE sharing its general solution with (6). Then, by the classical results of section 5.1, F has the form (28), and there exists an algorithm 20 yielding the solution / in the canonical form u = R(p', p) = .Ri(p) + p'Ri{&),
(29)
154 in which Ri,R? are two rational functions, with the possible degeneracies R{p',p)—>
R{ek^)-^
R(Z),
(30)
in which R denotes rational functions. This algorithm is 20 : (1) Compute finitely many terms of the Laurent series,
u = XP {i2^jXj + 0(XJ+l) ) , X = £ - So-
(31)
(2) Choose a positive integer m and define the first order ODE m
F(u, u') = J2 fc=0
l(m-k)(p-l)/p]
Yl
aj,kuju'k
= 0, a 0 , m ^ 0,
(32)
j=0
in which [z] denotes the integer part function. The upper bound on j implements the condition m(p — 1) < jp + k{p — 1), identically satisfied if p = — 1, that no term can be more singular than u'm. (3) Require the Laurent series to satisfy the Briot and Bouquet ODE, i.e. require the identical vanishing of the Laurent series for the Ihs F(u.u') up to the order J F = x"^"1) [ £
FjXj + 0(XJ+1)
) , Vj : Fj = 0.
(33)
If it has no solution for dj^, increase m and return to first step. (4) For every solution, integrate the first order autonomous ODE (32). The main step is to solve the set of equations (33), i.e. a linear, overdetermined system in the unknowns a^fc. This is quite an easy task. An upper bound on m will be established in section 6.
5.3. Results
of the method on the KS
equation
The Laurent series of (8) is (12). In the second step, the smallest integer m allowing a triple pole (p = —3) in (32) is m = 3. With the normalization a o,3 = 1J the subequation contains ten coefficients, which are first determined by the Cramer system of ten equations Fj = 0,j — 0 : 6,8,9,12. The remaining overdetermined nonlinear system for (y, b, /j,, A) contains as
155 greatest common divisor (gcd) b2 - 16/iz/, which defines a first solution 62
i«
3&3
a.
ftv = 16,' UsS = u + 32*/ 2 ' —
After division by this gcd, the remaining system for (u, b, n, A) admits four solutions (stopping the series at J = 16 is enough), namely the first three lines of Table 1, each solution defining a subequation, 6 = 0, , , 180fi2y
360^ 2 \
f ,
9 /
2
30^ ,
30V i
L n /3 9 / 2 30/x , 30 V V „ 6 = 0, « ' H u 2 H -u' H ^— = 0, 40i/ V 19 19 3 ^ / 2 3 6 _ 144 _ 56 2 / , b \3 9 <4 = 0, /xi/ 47 ' 144i/ ' V 4i/ V 40i/ b2 256 4563 s liv 73 ' u, 2048i/2'
u/ +
b
V f ,
u+
b
\
+
9 /
&;"•) [ to"') wA
Us 2+
563
T^
Ws +
562
u
^)
(36) (37)
A2
=
°' (38)
To integrate the subequations (34), (35)-(38), one first computes their genus a , which is one for (34), and zero for (35)-(38). Therefore (34) has an elliptic general solution, listed above as (18). The general solution of the four others (35)-(38) is the third degree polynomial (20) in tanh A;(£ — £o)/2These four solutions, obtained for the minimal choice of the subequation degree m, constitute all the analytic results currently known on (8). 6. Consequences of meromorphy (Nevanlinna) If the solution / is meromorphic, much can be said from the study of its growth at infinity (Nevanlinna theory). For the KS ODE, the meromorphy requires c+ = c_ = 0 in (17), restricting the solution to the series (12). By direct application of the Nevanlinna theory, one can prove the a
For instance with the Maple command genus of the package algcurves ments an algorithm of Poincare.
n
, which imple-
156 Theorem 6.1. 8 If a solution of (8) is meromorphic, then it is elliptic or degenerate of elliptic. Furthermore, (1) Elliptic solutions only exist ifb2 = 16fiv, and their order is three. (2) Exponential solutions have the necessary form P(tanhfc(£ — £o))> with k constant and P a polynomial of degree three. (3) The only rational solutions are u = 120z/(£ — £o)~ 3 , they exist for b = /j, = A = 0. Consequently, the value m = 3 is an upper bound to the algorithm of section 5.2, which has therefore found all the meromorphic solutions of (8). 7. Summary and open problems Let us represent the solutions of (8) by the following inclusions, elliptic C meromorphic C singlevalued C multivalued.
(39)
One has seen the various implications (1) (2) (3) (4)
(Singlevalued, algebraic dependence on x0) = > elliptic (thm 5.1), Meromorphic = > elliptic (8 using Nevanlinna theory), Elliptic = > (b2 = IQfiu) (residue theorem 1 2 ), (Elliptic or degenerate) =>• (order three) (8 using Nevanlinna theory) ==>• (all such solutions in closed form 2 0 ) .
The problem is open to find the general analytic solution in closed form for arbitrary (i/, b, fj,, A), which would be the sum of the Laurent series (12). Pade approximants and Painleve analysis find no multivaluedness nowhere. Two and only two possibilities remain about this general analytic solution for generic values of (v, b, fi, A), (1) either it is multivalued, and strong efforts have then to be made to uncover this multivaluedness with both the Painleve test and the Pade approximants. This event is unlikely; (2) or it is singlevalued. In this case it cannot be elliptic, and the dependence on XQ is necessarily transcendental. Solving this open problem would solve ipso facto many similar problems for nonintegrable equations such as CGL3, CGL5 or Swift-Hohenberg. Acknowledgments The author warmly thanks the organizers for invitation.
157
References 1. M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Stud. Appl. Math. 53, 249 (1974). 2. G. P. Agrawal, Nonlinear fiber optics, (Academic press, Boston, 2001). 3. C. Brezinski and J. van Isegehm, 47, Handbook of numerical analysis, Vol. Ill, eds. P.G. Ciarlet and J.-L. Lions (North-Holland, Amsterdam, 1994). 4. R. Conte, The Painleve property, one century later, 77, ed. R. Conte, CRM series in mathematical physics (Springer, New York, 1999). http://arXiv.org/abs/solv-int/9710020 5. R. Conte, Direct and inverse methods in nonlinear evolution equations, 1, ed. A. Greco, Lecture notes in physics 632 (Springer Verlag, Berlin, 2003). http://arXiv.org/abs/nlin.SI/0009024 6. R. Conte and M. Musette, Reports on mathematical physics 46, 77 (2000). 7. R. Conte, A. P. Fordy and A. Pickering, Physica D 6 9 , 33 (1993). 8. A. E. Eremenko, h t t p : / / a r x i v . o r g / a b s / n l i n . S I / 0 5 0 4 0 5 3 9. J.-D. Fournier, E. A. Spiegel and O. Thual, Nonlinear dynamics, 366, ed. G. Turchetti (World Scientific, Singapore, 1989). 10. M. van Hecke, C. Storm and W. van Saarlos, Physica D 1 3 3 , 1 (1999). 11. Mark van Hoeij, package "algcurves", Maple V (1997). http://www.math.fsu.edu/~hoeij/algcurves.html 12. A.N.W. Hone, Physica D205, 292 (2005). 13. N. A. Kudryashov, Prikladnaia Matematika i Mekhanika 52, 465 (1988). 14. N. A. Kudryashov, Matematicheskoye modelirovanie 1, 151 (1989). 15. N. A. Kudryashov, Phys. Lett. A147, 287 (1990). 16. Y. Kuramoto and T. Tsuzuki, Prog. Theor. Phys. 55, 356 (1976). 17. J. Lega, Physica D 1 5 2 - 1 5 3 , 269 (2001) . 18. J. Lega, J. V. Moloney and A. C. Newell, Phys. Rev. Lett. 73, 2978 (1994). 19. P. Manneville, Dissipative structures and weak turbulence (Academic Press, Boston, 1990). French adaptation: Structures dissipatives, chaos et turbulence (Alea-Saclay, Gif-sur-Yvette, 1991). 20. M. Musette and R. Conte, Physica D 1 8 1 , 70 (2003). 21. P. Painleve, Lecons sur la theorie analytique des equations differentielles (Legons de Stockholm, 1895) (Hermann, Paris, 1897). 22. Y. Pomeau and P. Manneville, J. Physique Lett. 40, L609 (1979). 23. W. van Saarloos, Physics reports 386, 29 (2003). 24. J. Swift and P. C. Hohenberg, Phys. Rev. A15, 319 (1977). 25. O. Thual and U. Frisch, Combustion and nonlinear phenomena, 327, eds. P. Clavin, B. Larrouturou and P. Pelce (Ed. de physique, Les Ulis, 1986). 26. S. Toh, J. Phys. Soc. Japan 56, 949 (1987). 27. Yee T.-l., R. Conte and M. Musette, From combinatorics to dynamical systems, 195, eds. F. Fauvet and C. Mitschi, IRMA Lectures in math, and theor. physics 3 (de Gruyter, Berlin, 2003). http://arXiv.org/abs/nlin.PS/0302056
O N T H E STABILITY A N D A S Y M P T O T I C STABILITY OF S T E A D Y SOLUTIONS OF T H E NAVIER-STOKES EQUATIONS I N U N B O U N D E D D O M A I N S
F. CRISPO Seconda
Universitd degli Studi di Napoli, Via Vivaldi 43, 81100 Caserta, Italy E-mail: francesca. [email protected]
We are concerned with the problem of the energy stability of steady motions of an incompressible viscous fluid. It is assumed that the motion takes place in an unbounded smooth region of the three-dimensional Euclidean space with a priori non-compact boundary and it is governed by the Navier-Stokes equations. We study the asymptotic behavior in time of the kinetic energy of perturbations; the class of unperturbed motions is, in some sense, physically reasonable and certainly non empty. Precisely, we prove that the kinetic energy of the perturbation goes to zero and, moreover, we give a decay rate.
1. Introduction The aim of the present note is to show some results on the asymptotic behaviour of the perturbation to a Navier-Stokes stationary motion in unbounded regions with not necessarily compact boundaries. This choice of the region of motion is very interesting in the context of dynamic of fluid flow, since from the physical point of view many important problems are related to such a geometry. For instance, it models the region of motion of a fluid flow in channels or pipes which, with respect to suitable coordinates system, has variable sections (see Refs. 14, 15, 17 in the references below). Another example is given by the so-called aperture domains, that, roughly speaking, is an unbounded domain consisting of two half-spaces separated by a wall and connected by a hole in this wall (see, for instance, Ref. 5). The L 2 -decay problem for Navier-Stokes flows, starting from the famous work by Leray 7 , has provoked a remarkable interest in fluid-dynamics even if one has to wait until 1983 for contributions in such sense (see Refs. 8, 9, 12, 13, 16). Asymptotic stability results in unbounded domains are
158
159 given in Ref. 10, for small value of the initial perturbation and, for weak solutions, in Ref. 1. As far as the L 2 -decay problem for perturbations to stationary motions is concerned, results are already known for exterior domains (cf. Refs. 1, 11), but, as we are aware, no contributions have been given for domains with a priori non-compact boundaries. In this connection, this is the subject of paper Ref. 4. We are interested in studying the asymptotic behaviour of the perturbation to a stationary motion v by showing its decay rate. The main difficulty is connected to the circumstances that the geometry chosen does not enable us to apply results of unbounded domains such as the whole space, the half-space and the exterior domain. To overcome these problems we use some inputs given, for exterior domains, in Ref. 11, developed in an original way. What is remarkable is that the decay rate we found is exactly the same found for perturbations to steady motion in exterior domains (cf. Ref. 11).
2. N o t a t i o n s Throughout this paper ( I c R 3 will denote an unbounded domain, whose boundary d£l is assumed of class C2'5, 5 G (0,1). For T e (0, oo], let ftT = ft x [0,T). By LP(Q) and Wm'p(Q), m nonnegative integer and p £ [l,oo], we denote the usual Lebesgue and Sobolev spaces, with norms | • \p and | • | m , p , respectively. By CQ°(Q.) we mean the space of infinitely many times differentiable functions defined in ft and with compact support. We set Co(ft) = {V'GC 0 00 (ft): V - ^ = 0}, J(ft) = completion of e 0 (ft) in I/ 2 (ft), J 1>2 (ft) = completion of C0(ft) in W 1,2 (ft). By the symbol (f,g) we mean, as usual, (/, g) = Jnf(x)g(x)dx, for any / , g such that the integral is finite. By L p ((0,s);X) we denote the set of functions f(r) from (0, s) C R into a Banach space X, such that Jo l / ( r ) l x ^ T < °°> similarly, by C([0,s];X) we denote the Banach space of all bounded and continuous functions / ( T ) from (0, s) into X, with the norm |/|c([o, s ] ; x) = m a x | / ( r ) | x .
160
3. Results Let us consider a steady motion governed by the nonlinear stationary Navier-Stokes system. If v (x) is the kinetic field and p{x) is the pressure field of the motion, then the pair (v,p) satisfies the following boundary value problem (v • V)i> + Vp = vAv,
in Q,
V - v = 0, in Q,, v(x) = 0 on dfi,
n\ lim v(x) = 0,
where (v • V)f = f j t ^ S i = 1,... ,n, v > 0 is the kinematic viscosity. We assume that the kinetic field v(x) satisfies the following conditions: ' U ^ ' ~ (l + \x\)1+c" |Vu| 3 = M2.
a G
'°' ^
(2)
As shown in Refs. 2, 15 the set of solutions of system (1) satisfying conditions (2) is a nonempty set. We explicitely observe that the regularity requested on d£l seems necessary in order to ensure the validity of the assumptions on v. However, if we consider perturbation to the rest state it is enough a weaker assumption, such as the cone property. Let u0 be the perturbation to the kinetic field v(x) at t — 0. Then the perturbation flow (u(x,t),ir(x,t)) is solution of the following initialboundary value problem ut + (u- V)u + (v • V)u + (u • V)u +
VTT
= uAu,
in Q x (0, T),
V - u = 0, i n f i x (0,T), M(i,t)=0onffix(0,T),
lim u(x,t)=0
^
|a:|—*oo
u(x, 0) = u0(x)
in Q,.
where ut = J^u. In this note we state some results studied in paper Ref. 4, related to the energy stability. However, as it is well known, another approach to the stability problem can be the pointwise stability, that studies the pointwise decay rate, in space and time, of the perturbations; in this regard we refer, for instance, to Refs. 3, 6. This kind of approach, for this particular geometry, will be the topic of a future research.
161 Definition 3.1. Let u0 £ J(fi). A field u: fi x (0,T) -> M3 is said a we«& solution of system (3) if (i) u{x,t) G L°°(0,T; J(fi)) n L 2 (0,T; J 1 ' 2 ^ ) ) ; (ii) / Jo
[-(u,^ T )+»'(V«,VV')+((«-V)u,^)+((u-V)w,V)+((wV)u,^)]dr
= (uo,^(o)),
w{x,t) e e0(nT),vt e (o,r];
(iii) there exists L > 0 such that \u(t)\22 + L f |Vu(r)||
=
0(t-i(r-^);
then lim |u(i)| 2 = 0.
(4)
£—>oo
We observe that in the case where a G (0,1] in the assumption (2)i on v, we can prove the existence of weak solutions using a variational formulation also. In this case, if we adimensionalize the Navier-Stokes equations (1), then L = 2 ( ^ — ^ - ) , in the definition of weak solutions, where R is the Reynolds number and Rc > R is the critical Reynolds number. If, instead, a = 0, we have to give up a variational formulation and prove the existence of solutions in a different way, requiring 2M\ < v. In this case L — 2{v — 2Mi). To achieve our main result first we deal with the linearized perturbation system. In this connection, by the pair (u 0 ,^ 0 ) we denote the kinetic field and the pressure field of a perturbation, solution of the following system u°t + Vo° = vAu° - (u° • V)t> - (v • V)u°, in fi x (0, T), V - u ° = 0, i n f i x (0,T), u° = 0 on m x (0, T),
lim u°(x, t) = 0,
w
\x\—>oo
u°(x, 0) = u0(x) in Q. For the Galerkin approximations {u°"(x, i)} related to system (5) we show the following result.
162 L e m m a 3 . 1 . Ifu0 G J(Q) D L p ( ^ ) , for some p € [|, 2), i/ien the {u0n} satisfies the following inequality
sequence
| u 0 " | 2 < c | u o | p H ^ \ Vt>-1, uniformly
in n G N, wit/i M = M(|v|oo, |Vu|3).
T h e following step is to consider the Galerkin approximations {un(x, t)} related t o system (3) corresponding to the same d a t a u0. Then, setting uln{x,t)
= un(x,t)
-
u0n(x,t),
the field w l n satisfies the following integral relation
(uin(t),m) = A ( u l n ( T ) , V v ( T ) ) - K V u l n ( T ) , VV^(r)) ./o
((«n(r).V)«n(r),V(r))
-((«1''(r).V)D(T)^(T))-((t;(r).V)u1''(r)^(rj](lr, W> G C ( [ 0 , T ) ; J 1 > 2 (fi)), w i t h ^ t G L2(0,T,
J ( f i ) ) , V T > 0.
For the sequence {u1™} it is possible t o prove t h a t |M 1 T I (£)| 2 (£ + 1 ) ~ 5 G L r ( 0 , oo), for any r > 4. Then, it is easy to prove t h a t if p G ( | , 2 ) ,
then |un(t)|2 = 0 ( t ~ * ( i ~ * ) ) ;
i f p = 2, t h e n lim | u n ( * ) | 2 = 0.
(6)
£—•00
Finally, estimates (4) are consequence of (6) once one observes t h a t Ki)|2
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
W. Borchers and T. Miyakawa, Arch. Rational Mech. Anal. 118, 273 (1992). W. Borchers and K. Pileckas, Arch. Rational Mech. Anal. 120, 1 (1992). F. Crispo and P. Maremonti, Zap. Nauchn. Semin. POMI318, 147 (2004). F. Crispo and A. Tartaglione, work in progress. J. Heywood, Acta Math. 136, 61 (1976). G. H. Knightly, Approximations methods for Navier-Stokes problems, Lecture Notes in Math., Springer, Berlin 771, 287 (1980). J. Leray, Acta Math. 63, 193 (1934). P. Maremonti, Proceeding of the workshop on mathematical aspects of fluid and plasma dynamics, Trieste, 369 (1984). P. Maremonti, Annali di Matematica Pura ed Applicata 142, 369 (1985). P. Maremonti, Math. Z. 210, 1 (1992).
163 11. P. Maremonti, XXVIII Scuola Estiva di Fisica Matematica, Ravello, (2004). 12. K. Masuda, Nonlinear functional analysis and its applications, Proc. Summer Res. Inst, Berkeley/Calif. 1983, Proc. Symp. Pure Math. 45/2, 179 (1986). 13. K. Masuda, Tohoku Math. Journ. 36, 623 (1984). 14. S. A. Nazarov and K. Pileckas, Rend. Sem. Mat. Univ. Padova 99, 1 (1998). 15. K.Pileckas, Rend. Sem. Mat. Univ. Padova97, 235 (1997). 16. M. E. Schonbek, Arch. Rational Mech. Anal. 88, 209 (1985). 17. V. A. Solonnikov, Colle de France Seminar, Vol.4; Pitman Research Notes in Mathematics 84, 240 (1983).
REFLECTION A N D T R A N S M I S S I O N OF ACCELERATION WAVES IN ISOTROPIC SOLIDS W H E N A STRONG DISCONTINUITY OCCURS *
C. C U R R O A N D G . V A L E N T I Department of Mathematics, University of Messina Contrada Papardo, Salita Sperone 31, 98166 Messina, Italy M. S U G I Y A M A Graduate
School of Engineering, Nagoya Institute of Showa-ku, Nagoya 466-8555, Japan
Technology
The propagation of acceleration waves in two isotropic solids in contact with one another is studied in the case of plane symmetry. The transmitted and reflected waves across an interface are evidentiated and their amplitudes are determined.
1. Introduction In order to analyze 3D isotropic solids, a new continuum model, derived statistical-mechanically from a three-dimensional anharmonic crystal lattice 1, has been proposed 2 . This model is a finite-deformation continuum one incorporating explicitly the microscopic thermal vibration of constituent atoms and it is valid in a wide temperature range up to the melting temperature TMBy using this model, both linear and nonlinear wave propagation have been extensively investigated 3 _ 6 and both thermal and mechanical properties have been analyzed. Furthermore, in the linear case, the study of reflection and refraction of elastic waves across an interface has been carried on 7 . *This work is supported by Cofin 2003 Progetto di interesse nazionale INDAM-GNFM, Problemi Matematici Non Lineari di Propagazione e Stabilita nei Modelli del Continuo. (coordinator T. Ruggeri).
164
165
The aim of the present paper is to analyze the propagation of an acceleration wave in isotropic solids when there are two different solids in contact with one another. In this case the propagation of an acceleration wave moving along the wave front trace is influenced by the strong discontinuity across the interface line so that transmitted and reflected discontinuities are generated 8 _ n . The analysis carried on herein is restricted to the case of plane symmetry. The amplitude of the transmitted and reflected waves are esplicitly determined.
2. Field equations The one dimensional equations describing the 3D isotropic solids at finite temperature incorporating microscopic thermal vibration of costituent atoms in plane symmetry can be obtained from the general system 2 , 4 by assuming that all the field variables depend only on X\ — X and on t so that they can be written in a matrix form as follows n_1Ut-a_1A(U)Ux =0
(1)
where Q, is the characteristic microscopic frequency, a l the microscopic characteristic length, the field vector U and the matrix A are given by:
U
0 V P l (V F l o-) V r (V F l cr) 1 0 0 0 0 A
q Fi r
(2)
where I is the identity matrix and the following relations hold: (3)
Pk =
da dgpq dgpq dFki'
A=
3
2
do- dgpq dgpq dr
The field variables are the dimensionless velocity q, the deviation of the temperature r induced by a wave from the reference equilibrium state and the column vector F i of the the deformation gradient tensor F . In fact, owing to particular plane symmetry assumption , F reduces to 'Fn 0 0 F21 10 ^31 0 1
166 The simmetric tensor g, namely the deviation of the atomic thermal vibration, is related to F and r by the following state equation: (AI+g)Vg<7=
kBT D 1+ k T 2D V B
(4)
where D is the depth of the atomic pair potential, kB the Boltzmann constant, T the absolute temperature and a = a (F, g) the dimensionless potential energy. In what follows we adopt for a an expansion form such that a (F, g) = fo + fall + foil + foh + fohh + foh + foil +foh + foil + foh + fooh + foih + £12/9 +£13/1/2 + £14/1/5 + £15/1/7 + £16/2/4 + £17/! + £ 1 8 / 4 / 5 + £19/4/7 + £ 20 /i 2 /4 + £21/1/1
(5)
with respect to the basic invariants defined by
h
h = 9ik9ki ,
h = gik9kj9ji,
h = Bss — 3,
/s = (Bst —fist)(Bst — 5„t), h = (Bst —fist)(Btp — 5tp) (Bps — 5ps), (6) I7 = 9st (Bts — $ts), h = 9ps (Bst —fist)(Btp — fitp), h = 9tpgps (Bst — fist) being B — F F the left Cauchy-Green tensor and % the Kronecker symbol. The expansion coefficients £'s occurring in eqn (5) are function of T and they can be evaluated in terms of the pair potential. The system (1) admits a conservative form, that is
fl^G"
a
-1
Gv=0
with q Pi e
G°
G
(VFla) q _q-(V Fl o-),
(7)
where e is the dimensionless total energy density defined by e
1 2 3 = 2 9 + 2
(kBT D
+ r)+a.
(8)
Now we consider two different isotropic solids in contact with one another and separated by a stright line D of (X, t) —plane. We suppose D at rest and orthogonal to the X-axis so that it has the equation X = X^. Such a situation can be described by assuming that the coefficients of the field equations are piecewise continuous functions with their discontinuities occurring across D.
167 On the right of Xd we suppose that the governing system is n-1U(*-Q-1A*(U*)Ui=0,
fi-1G°*-a-1G^=0
(9)
with U*, A*, G°* and G* similar to the corrisponding quantities defined in (2), (7). The strong discontinuity across D modifies the propagation of acceleration waves moving along the wave front trace and generates transmitted and reflected discontinuities 8 , 10 , n . We assume that the two solids are in thermal equilibrium Uo = UQ = [0, Sn,0] with the same temperature T. In this case the dimensionless characteristic velocities associated to (1) 5.
are
V = 0, VT
V? = 4
= ( 4/37
2
2X01
+
2 A ).
xma+b)
(a - b) (a + 2b)
with multiplicity 2
a—b
(10)
with a = f/?i + 2A/32 + 2A/33 , b = |/?i + 2A&. The (10) have been evaluated for different materials at different values of the reference equilibrium state temperature and it turns out that: H+) < ^ Tv^ ?(+) v£-} < vT"] < o < vf
(11)
The corresponding right d and left 1 eigenvectors are given by: do
n n n
(3/3 4 +fe) (a-b+6A/3 2 )
n n
_T/2
,
4A/3,(3^ 4 +fe) 3(a-b+6A/3 2 )(a+2b)
io=[ooo-^f|fiooi;
(13) lT
VL
3Vi(o+26) 2
_ 4A/3i(3g4+/35) (3/34+fe) 00 V_ (a-b+6\0 3V L (a-b+6A/3 2 )(a+2b) L 2) T 0 0 , l l T = 0 1 0 0 -VT 0 0
li
1 0 0 -VL +
diT
0 100-^-
d2r
00 1 0 0 - J - 0 VT
, iIT
(12)
(14) (15)
ooioo -yTo
3. Transmitted and reflected waves We consider the fastest acceleration wave propagating with velocity V^ . At a certain time t this characteristic meets the interface D which represents a shock line. As result of this interaction we shall have reflected
168
and transmitted waves that, as the governing system is hyperbolic, must propagate along the chracteristics. Since the line D is at rest, the Rankine -Hugoniot relations connecting the field vectors across the shock line, reduce to: (G)D.
= (G*)D+ -
(16)
It is easy to ascertain from (11) that the evolutionary Lax conditions are automatically satisfied being the shock velocity equal to the characteristic velocity V = V* = 0. Therefore we have two reflected waves along the chracteristic propagating with the velocities V^ ,Vj-~ and two transmitted ones propagating with the velocities VL , VT . The amplitudes of the reflected and the transmitted waves in terms of the incident one can be determined by following the procedure given in n . We denote by \j(R^ and U*(T) the reflected and transmitted field vectors, respectively so that we have
u<«> =Uo+7r0+7ri*VL + 4 f l W ,
u*
(R\
(R\
where TT, irL , iTj, are the incident and reflected discontinuity vectors transported along the characteristics
Furthermore, since Uo and UQ are constant states and the system (1) is homogeneus and autonomous, the discontinuity vectors are constant too, so that, after differentiating (17)i,2with respect to X, we obtain:
U =n + *f) + 4 R ) ,
U^(T) = ^ T ) + 4T)
(18)
provided we choose, without loss of generality, J ^ = ^^- — ^ £ = %£L — ax ~ *•• In order to connect Ux' and Ux ' across D we differentiate the Rankine-Hugoniot relations (16) along the shock and, taking into account (1) and (9), we obtain ( V u G ) ( A U £ > ) = (Vu-G*) (A*U^ ( T ) ) .
(19)
As it is well known, the discontinuity vectors are proportional to the corresponding right eigenvectors, so that (18) become:
U ^ = a 4 + ) + aidP
+ a2d[p + a 3 d £ \
169 which inserted in (19) give a set of conditions for the unknown coefficients a>i and /?,, whose solution is:
y*(+) _ y{+) V*L{+) + Vt]
yl VlW
(y«V
+
V™)
T h e coefficient a represents the amplitude of t h e known incident discontinuity and satisfies t h e Bernoulli differential equation. In passing we remark t h a t , in agreement with B r u n ' s condition, the interaction between an acceleration wave and a strong shock is uniquely determined 1 2 . References 1. Sugiyama M., Goto K., Statistical-thermodynamic study of nonequilibrium phenomena in three-dimensional anharmonic crystal lattices. I. Microscopic basic equations, J. Phys. Soc. Jpn. 72, 545-550, (2003). 2. Sugiyama M., Statistical-thermodynamic study of nonequilibrium phenomena in three-dimensional anharmonic crystal lattices. I . Continuum approximation of the basic equations, J. Phys. Soc. Jpn., 72, 1989-1994 (2003). 3. Sugiyama M., Goto K., Takada K., Valenti G., Curro C., Statisticalthermodynamic study of nonequilibrium phenomena in three-dimensional anharmonic crystal lattices. IH . Linear waves, J. Phys. Soc. Jpn., 72, 3132-3141 (2003). 4. Valenti G., Curro C , Sugiyama M., Wave features for a new continuum model of isotropic solids, in Proceedings WASCOM 2003, Villasimius 1-7 June 2003, Word Scientific, 547-554, (2003). 5. Valenti G., Curro C , Sugiyama M., Acceleration waves analyzed by a new continuum model of solids incorporating microscopic thermal vibrations, Continuum Mech. Thermodyn., 16, 185-198 (2004). 6. Curro C , Sugiyama M., Suzumura H., Valenti G., Weak shock waves in isotropic solids at finite temperatures up to the melting point, submitted 7. Sugiyama M., Chaki M., Reflection and refraction of elastic waves in solids at finite temperatures up to melting point, Jpn. J. Appl. Phys., 44, 6B, (2005). 8. Jeffrey A., Quasilinear hyperbolic systems and waves, Pitman Research Note, 5, London, 1976. 9. Donato A., On a magneto-elastic system with discontinuous coefficients and the propagation of a weak discontinuities, Meccanica, 12, 127-133, (1977). 10. Donato A., The propagation of weak discontinuities in quasilinear hyperbolic systems when a characteristic shock occurs, Proc. Roy. Soc. Edinburg, 78A, 285-290, (1978). 11. Boillat G., Ruggeri T., Reflection and transmission of discontinuity waves through a shock wave. General theory including also the case of characteristic shock, Proc. Roy. Soc. Edinburg, 83A, 17-24, (1979). 12. Brun L., Ondes de choc finies dans les solides elastiques, Mechanical waves in solids, Eds. J. Mandel and L. Brun, Springer, Vienna, 1975.
INSTABILITY OF A P R E - S T R E S S E D SOLID-FLUID MIXTURE
FRANCESCO DELL'ISOLA Laboratory of Smart Structures and Materials of Cisterna di Latina, Palazzo Caetani (Ala Nord). Cisterna di Latina (LT), 04012, Italy E-mail: [email protected] NICOLETTA IANIRO Department of Mathematical Modelling for Applied Science, Via Scarpa, 16. Roma, 00185 Italy E-mail: nicoletta. ianiro@uniromal. it LUCA PLACIDI Laboratory of Smart Structures and Materials of Cisterna di Latina, Palazzo Caetani (Ala Nord). Cisterna di Latina (LT), 04012, Italy E-mail: [email protected] A solid-fluid mixture is generally modelled assuming that the state of stress in the reference configuration is identically equal to zero. However, such an assumption is not always appropriate to take into account some instability phenomena occurring in Nature. In this contribution, the continuum mechanics point of view is used and the reference configuration of the solid-fluid mixture has a state of stress, i.e. the pre-stress is different from zero. The instability of the mixture with respect to the perturbation fields given by a general plane wave is then studied.
1. Introduction In the present macroscopic continuum theory, the balance equations of mass and linear momentum are derived in the form given by Truesdell 7 ' 8 . Moreover, a set of constitutive equations is given in order to characterize the material and to close the problem, see also 1 ' 2 . To do this, we assign the energy per unit mass of the mixture and then we derive the constitutive quantities. The solid part is considered as an elastic material and the fluid part is non viscous. The interactions between the two species is also considered through a non dissipative and a dissipative (Darcy) terms. The Cauchy stress tensor of a given species depends on the constitutive variables of the
170
171 other species. We note that this interaction is not in contraddiction with the second law of thermodynamics: the interaction force, in this formulation, depends upon the gradient of mass densities of both species, see e.g. 4 ' 9 . The propagation of harmonic waves is studied and normal modes for the transverse and longitudinal velocities are considered. Since the perturbed fields blow up, we have a set of instability conditions derived by the dispersion relations. Such conditions define the constraints of the constitutive parameters of the model derived from the present analysis.
2. Preliminary and Balance Equations Let Bs and Bf be the reference configurations of the solid and of the fluid constituents, respectively. The actual configuration is denoted by B. Let x € B be the Eulerian coordinates and Xs, Xf be Lagrangian coordinates of the solid and the fluid, respectively. From the mathematical point of view, this means that there exist two functions Xs a n d Xf> called the current placements of the solid and of the fluid, such that the position x e B i n the actual configuration is given by, x = x,(X8,t) = X/(X/,t), v X a e B s , V X / e B / , V i e M .
(1)
The derivative with respect to time of (1) gives the velocity fields vs and Vf of the solid and of the fluid, respectively, i.e.,
».«•«*'> = £ - & • „-.,<*<)-£-£.
W
where ^ | and -^ denote the time derivatives keeping X s and X / constant. The derivative with respect to the particles X s and X / give the deformation gradients F 3 and Fy of the solid and of the fluid, i.e.,
F s =F s (x,*) = J | i = I + Hs,
F / = P/(x,«) = ^
L
= I + H /) (3)
where H s and H / are the displacement gradients of the solid and of the fluid, respectively. In the following, the symmetric part E s Es = i ( H s + H j )
(4)
of the displacement gradient H 5 of the solid constituent, also called the infinitesimal deformation, will be extensively used.
172 The balance equations of masses and of linear momenta of the solid and of the fluid are assumed to be the following, ~
+ V • [Q.V.] = gsTsf,
^
+ V • [gfvf] =
QfVfs,
—^r- + V • [£ s v s
(6)
We underline that Tg and TQ are intended to be Cauchy stress tensors. This means that the forces producing this kind of pre-stress are prescribed in the actual configuration and are per unit area in the actual configuration, too. Note that not all the forces fulfil this characteristic. The forces due to an hydrostatic pressure acting on the mixture is one of this kind; the forces due to the weight or to the traction in one fixed direction are not ! The characterization of the initial (reference) configuration is now complete once one assigns the partial mass densities QQ and QQ of the solid and of the fluid constituents. We will denote the fields evaluated in this (reference) configuration with the subscript 0 and we can write, £o = const., £>Q
=
const., VQ = VQ = 0., p^ = const., pi = const. (7)
Superimposed to this equilibrated and pre-stressed state, we consider an arbitrary perturbation, that is defined by the two displacement fields us and u^ and by gs and g~f, the change of partial mass densities, i.e., x = Xs+us(XS:t)
= X/+u/(X/,i);
Qa = Qo + Qs,
Qf = Qo+Qf- (8)
The constitutive characterization of the non-simple mixture considered in this paper is derived by Sciarra 5,6 in 2001 in the limit of small perturbation. Thus, it is reasonable to consider the solid in the regime of linear
173 elasticity; the fluid is modelled as a perfect and compressible medium. The objective form of the conservative part \I> of the energy per unit mass (so that $ is the conservative part of the specific energy) is defined as follows, QOV = Ts0 • H s + ^ef + lcsW • Es - ^ H s • ( T g H s T - H S T§) 2 4 0o ( H , T g - T g H s + TgH* T - H s T TS) + §fK'f • W + \QO1H~Q2V +lHs. (9) where the superscript (A) denotes the transpose of a general tensor A and the constitutive quantities have the following general form, i.e.,
mfs = -maf
= m = -Qf (W)1
— + 0s^Vgf
+ D (vs - v ' )
(10) where the classical dissipative interaction Darcy term for the interaction force m in the equation (10)i, i.e. D (v s — v-^), is added to the formulation of Sciarra 5 . Therefore, the constitutive equations of the mixture are completely characterized once one assigns the constitutive parameters D, K.3f, 7 / / and C s . We remark that such parameters are referred to the given reference configuration and therefore are related to the prescribed state of pre-stress. The scalar D and the tensor K s ^ represent the dissipative and the non dissipative interaction between the species of the mixture; 7 / / is related to the compressibility of the fluid species and C3 is the fourth order elastic tensor related to the linear elastic solid species. Furthermore, we assume (see e.g. 3 ' 6 ), K s / = -QoSo-y.fl,
C S E S = 2 M E S + Atr (E s ) I,
(11)
where jSf is a single interaction parameter and A and fi are the Lame coefficients evaluated for the given solid component and for the given state of pre-stress. If we insert the constitutive equations (10) and (9) into the balance equations (5), then the field equations take the form ^
= - r f V • [v.], 9-§ = - f i f r • [vf],
Q ^ = (2/i - pg) V • E s + ASSV (trE s ) - ^-Vgf dt QJ0 gf0^f
= -^fVgf
+ ~XsfV(tvEs) + D(vs-vf).
(12) -D(vs-
V/)
,(13) (14)
174 where the coefficients \ap depend on 7a/g and on the pre-stresses p% with a and /3 be s and/or / (see also 3 ). Moreover, the relation between the infinitesimal deformation E s and the partial mass density of the solid constituent is derived from the mass balance, i.e., QS = —^gtrE s , in the limit of small perturbation. Since we are interested in a solution of the form Qa = ^ e i ( w t " k - x ) , v a = v a e ^ - k ' x \ a = s,f,
(15)
where UJ is the frequency and k the wave vector, both possibly complex, substituting the above expressions into the balance equations yields, Ass
Asf \ / vs \
/ 0 \
_^
, / Ass
n D et
Afs Aff J ( v s J = U J ' =*
A5f
\
U S f Aff J = 0 ' (16)
where A a/ g are the suitable matrix coefficients, A s s = (( M - |pS) k2 + iuD A s f = -iuDI
QIUJ2)
I + ( A + /i + e\g3oe°Po)
(k ® k)
+ Xaf (k ® k)
A,ff = (iwD - ^ w 2 ) I + Qf0l wood
+1—
I (k ® k) (17)
The solution of the dispersion relation given in (16)2 is not an easy task. It can be simplified decomposing the amplitudes of the velocity fields v a into a solenoidal v„ and an irrotational w^k part, i.e., Va=vi+6]lk,
k-v^=0,
V-v£=0,
V x v | l k = 0,
a = s,f
(18)
corresponding to transverse and longitudinal part of the velocity fields, respectively. Assuming that k e R3 is a real vector, then if the immaginary part u>i of the frequency CJ is positive, then the perturbations in the form of plane waves expressed in (15) are attenuated. Note that this does not mean that the system is stable under all kind of perturbations. On the other hand it is possible to express the prediction of our assumptions claiming that while w* is negative, the system is unstable. For transverse waves the dispertion relation is reduced to a third degree polynomial in u, the coefficient of which are real. Thus, the negativeness of the immaginary part u>i of the frequency w can be assured if D > 0,
ps0 < 2/x.
(19)
175 If such conditions are not fulfilled, then t h e system is unstable. T h e first condition is a direct consequence of the Second Law of Thermodynamics and simply tells us t h a t the dissipation due t o t h e Darcy t e r m in the constitutive equation (10)i is not able t o produce energy. T h e second restriction expressed in (19) gives an explicit restriction on the value of the pre-stress of t h e solid. For longitudinal waves t h e dispersion relation is reduced to a fourthdegree polynomial in ui, the coefficient of which are real. We show t h a t the immaginary p a r t w, of the frequency u> is negative if, Pso < \ (A + 2/i) + \ez (eg) 2 {iff
- 27,/), (20)
f
A + 2M - 2^pg j U (g 0)\ff + 2^Pn
>2 A;
We remark t h a t conditions (19)2 and (20) are explicit or implicit restrictions on the values of the prestresses pg a n d Po • W h e t h e r pg = p ^ = 0, t h e n such non-instability conditions must be satisfied. We can represent (19)2 and (20) in terms of a graphic in which the coordinate axes are the prestresses PQ and p$. It is easy t o show t h a t they characterize regions of the values of prestress t h a t the mixture can admit. References 1. Bowen, R.M.: Compressible porous media models by use of the theory of mixtures. Int. J. Engng. Sci. 20(6): 697 - 735 (1982) 2. Bowen, R.M.: Theory of mixtures. In Eringen AC (ed.). Continuum Physics III. New York: Academic Press. 2 - 127 (1976) 3. dell'Isola, F.; Ianiro, N.; Placidi, L.; Sciarra G.: (In preparation) (2006) 4. Miiller, I. Thermodynamics. Pitman Advanced Publishing Program. Boston (1985) 5. Sciarra, G.: Modelling of a fluid flux through a solid deformable matrix. PhD Thesis (2002) 6. Sciarra, G.; dell'Isola, F.; Hutter, K.: A Solid-fluid mixture model allowing for solid dilatation under external pressure. Continuum Mech. Thermodyn. 13, 287-306 (2001) 7. Truesdell, C : Sulle basi della termomeccanica. Nota I Rendiconti Accademia dei Lincei X X I I (fasc. I): 33 - 38 (1957) 8. Truesdell, C : Sulle basi della termomeccanica. Nota II Rendiconti Accademia dei Lincei X X I I (fasc. I): 33 - 38 (1957) 9. Wilmanski, K.: Mathematical Theory of Porous Media - Lecture Notes. XXV Summer School on Mathematical Physics, Ravello, September 2000 (2000)
SOME F U R T H E R CONSIDERATIONS O N T H E GALILEAN RELATIVITY P R I N C I P L E I N E X T E N D E D THERMODYNAMICS
F . D E M O N T I S , S. P E N N I S I A N D F . R U N D O Dipartimento di Matematica ed Informatica, Universita di Cagliari, Via Ospedale 12, 09124 Cagliari, Italy; STMicroelectronics, Consumer and Microcontroller Group, Catania, Italy e-mail: [email protected];[email protected];[email protected]
Aim of this paper is to furnish further arguments on the naturalness of the work "A new method to exploit the Entropy Principle and Galilean invariance in the macroscopic approach of Extended Thermodynamics" by Pennisi and Ruggeri; in particular, it will be shown how it was potentially included in a previous work on Galileanity, by Ruggeri. Therefore, the salient points of this work will be here revised, taking care to show the above mentioned application. The notation will be useful also for the paper "The Galilean Relativity Principle as non-relativistic limit of Einstein's one in Extended Thermodynamics" by Carrisi, Pennisi and Scanu, where the same results will be obtained starting from the Einstein' s relativity principle.
1. Expositive part Recently, (see paper [1]), Ruggeri and Pennisi have found a way to overcome the difficulties arising from the galilean relativity principle in extended thermodynamics. Here we will present new considerations on this subject, which further show how natural the results obtained in [1] are. In particular we will see that it was potentially included in [2]. In fact, it is easy to recognize, in the subsequent eqs. (15), (16), (11) and (13), their counterparts (28) and (29) of [1]. Consequently, the approach there indicated is not simply a mathematical tool to obtain the results, but is what expected from the Galilean relativity principle, except for what concerns the separation of the variables into convective and non convective parts. These results will be obtained also in [3] starting from the Einstein' s relativity principle, thus furnishing further arguments which support them. Some details are not explained because they are familiar to whomever knows the book [4]. Let Fn"%n and F'H"'ln be the independent variables in two reference frames which are equivalent from the galilean point of view, and let vl be
176
177 the velocity of the points of the second frame, with respect to the first one. The law expressing the change of independent variables is Fh--in
=
^{AF/(h--iHvih+i...vin) h=0 ^
for
n =
o,...,A/\
(1)
'
The same law, for n = N + 1 gives the transformation of the dependent variable Fn""lN+1. In the sequel, we will use the relation ph--ink
__ V ^ \
u— n h=0
n
\ p'Hii'••*>!• vih+i . . . vin) _i_ pii-
(2)
V /
which can be easily proven. Now, if h is a scalar function, the galilean relativity principle implies that /i(F*1-i»)=/»(F'il-i-).
(3)
Before imposing the other conditions, it is better to first change the variables; more precisely, in the literature one usually takes as independent variables the Lagrange multipliers defined by the equations dh = gpiTa^...t.
K-tn
(4)
-
which is also the law defining the change of variables. The new ones are also called the "mean field". Let us now see how they transform, under the above change of reference frame. By using the derivation rule of composite functions and eq. (1), we obtain n=0
that is
v
n=h
\'
n
_ V^/ n=h
V
|\.
. .
'
. ,Jh+i...,Jn
IK\
J
We note that, until now, we have not imposed the entropy principle nor that h is its density! We have only imposed that if h is a scalar function of the variables F 1 '"'™, then it must satisfy the equation (3). Now, if we assume also that h is the entropy density and <j>k is its flux, we know that this last one transforms, under the above change of reference frame, under the law
(6)
Similarly, in the literature the functions h and d> are defined by N
i
i
h = -h+Y,\il...inF *~ " 71=0
k
k
N
, ]> =-cf> + YJK~^-ink, 71=0
(7)
178 which is a change from the dependent variables h and 4>k to h and
EA,...,/-'-'"^^..,/'"'-", m=0
71=0
N
N
J2 K-im (J*-*™* - F^-^vX) = ]T K ^ j ' ^ - ^ . 771=0
(8)
71=0
Let us prove the first of these
ix-in-*" 41 '" 4 "=E E (r) ^ - - w , -imvin+i -V^F*--*"=
71=0
77=0 771=71 ^
N
m
,
=E E u 771=0 7 1 = 0 ^
'
N
JV
^ ^
v l W l
1 0
• • •f*-*" - = E ^i-^^ 1 -*"-
'
771=0
where in the second passage eq. (5) has been used, in the third one the order of the 2 summations has been exchanged and symmetrization introduced, and in the fourth one eq. (1) has been used. The proof of eq. (8)2 is similar, except that in the last passage eq. (2) has to be used. Now, by using (7)i, eq. (8)1 yields h' + h' = h + h which, by using h' = h yields h! = h . The result is that h behaves like a scalar, i.e., has the same value in the 2 reference frames! Similarly, by using (7), eq. (8)2 yields cpk + 4>k - (h + h)vk = 4>'k +
( E (f) V"^ f t + i-*y h + 1 •-vjA=
HK-ih) •
(10)
This already holds in vj = 0, so that it is equivalent to its derivative with respect to v^, that is N
I 17 \ •3h33h+v-]n 3l—Jh 'ii-jh
,,
n.
n=h+l n=h+l
dh a X
h-3h
V
.,
'
lliJh
+ a . . . ,iJn
_
179 In other words, h is a scalar function if it satisfies the condition +1
5> =o
)»#^w= 0h-j
(11)
••jhj
dX
h
h
Regarding
A'B(AA)
1
4>lk
4>'kWB^A)}
|| ~4>k{\A)-vkh{\A)
(Obtaining
4>H*A)
In other words
A
Proceeding as for h, we find ^ ( / i + 1) ~ v r /i=o
j - l - i * i + ^ * = °-
(i3)
°*h-jh
Finally, we impose the galilean relativity principle for the constitutive function F3l'":'N+1, i.e., that the diagram in the next page is commutative. In other words, we must have p'ji--JN+i
N+l
,-T
I V
(n\
\n=h
^
\.
. .
.
vih+i
h=0
^
1
1
=
'
s
^ ) (-l)N+1-hF^-jh(\A)v^
J2(
• • Vin
...VJN+^
'
(Note that F^ '"^* can be obtained from eq. (1) with — vl instead of v% and N + l instead of n). This relation becomes an identity if calculated in yi = 0 , so that it is equivalent to its derivative with respect to iP, that is N 1
"
N
dF'^'"^^1 d\>.l—ih /
Y2[-hl(N+1~
N
^2
( / J (" ~~'
—=h+l j. i i
V
n
/
h X
) h-ihjih+2~inv
-»h+2 .
\
h)(-l)N+1~hFUl-jhvjh+1
• ..vi»5JN+l)j
.
180 In other words, F' :,r " : " v + 1 must satisfy the condition J > + i)———\>.^.h. + (N + i)F'^"^~+Oi
=
o.
(14)
h=0
\A
*B&A) P'jl---JN
+1
JP31--JN + 1
F'^-3"+i[x'B(XA) II N
N+l h
Eh=o ( ~l
F^-3"+^(XA)
) {-l)
j
- F^- *{\A)v^
•••v*"*
Equations (11), (13) and (14) exhaust the conditions of the galilean relativity principle. On the other hand, one could say that with the independent variables A^...^ and the dependent variables Fl1'"l™, also these last ones must satisfy the galilean relativity principle. This is true, but we can easily see that this is a consequence of the other conditions. In fact, from eq. (4) we have dh = Xil...indFH'"ln; from this and from eq. (7)i we have dh = F^-^dX; l - " » n ' pll---ln
__
dh d\x...in
(15)
After that, we can consider eq. (10), i.e., h{X%i...%k
.)-*(£ft)v
liJh + l . . . „ • ? "
Jh3h + 1---Jnl
\n=h
V
'
By taking the derivative of this relation with respect to A^...^ and taking into account that A^ -h depends on it only for h < k, we obtain dh
pii---i
d A j11 , . . . , lk -.
E
dh ^—' dX'i „• h=o n—jh.
p'(ii---ihvih+i
h i
JX
Jh
. . ,vik)
h=0
that is the relations (1); then the galilean relativity principle on Fn'"%n is satisfied as a consequence of the other conditions.
181 We note t h a t e q s . ( l l ) , (13) are those listed in paper [1]. We stress t h a t , until now, we have not imposed the entropy principle! If kk
we impose it, we also have
p11'"*71
,
=
t\§\
Then also eq. (14) is a consequence of the other equations. In fact, eq. (13) N 1
~
can be rewritten as
v+i
}_^{h + 1 ) - J h=0
K-inj
h5 N+1
j
= 0.
° h---ih
whose derivative with respect t o A'-
„• is
^>+1)-^\>—^-*»+Na^—*5 OA
+
A
OA
+ F»i-"*r+i=o,
h=0 h-ih (h-JN-x where eqs. (15) and (16) have been used. T h e result is exactly eq. (14). Moreover, we see t h a t eq. (13), by using eqs. (16) and (15), can be rewritten
h=0
OA
h-jhk
In this way, the galilean relativity principle implies conditions (eqs. (11) and (17)) only on the function h, in the form of partial differential equations. Obviously, also t h e compatibility between eqs. (15) and (16) must be imposed, which we have found from the entropy principle. We conclude by noting t h a t eqs. (10) and (12) prove the Proposition 1 of ref. [5], where they were justified in another way and also by the fact t h a t they hold in the kinetic approach t o this subject. References 1. Pennisi S., Ruggeri T., "A new method to exploit the Entropy Principle and Galilean invariance in the macroscopic approach of Extended Thermodynamics", to be published. 2. Ruggeri T., "Galilean invariance and Entropy Principle for Systems of Balance Laws. The Structure of Extended Thermodynamics" (1989), Continuum Mech. Thermodyn. ?, pp. 3-20. 3. Carrisi M.C., Pennisi S., Scanu A., "The Galilean Relativity Principle as nonrelativistic limit of Einstein's one in Extended Thermodynamics", submitted to the Proceedings of WASCOM 2005. 4. Miiller,I., Ruggeri,T.( 1998). Rational Extended Thermodynamics - Springer Tract in Natural Philosophy - Springer Verlag. New York 37. 5. Pennisi S., Scanu A., "Judicious Interpretation of the Conditions Present in Extended Thermodynamics", Proceedings of WASCOM 2003, pp. 393-399.
WAVES A N D VIBRATIONS I N A SOLID OF S E C O N D G R A D E
M. DESTRADE Laboratoire de Modelisation en Mecanique (UMR 7607), CNRS / Universite Pierre et Marie Curie, 4 place Jussieu, case 162, 75252 Paris Cedex 05, France E-mail: [email protected] G. SACCOMANDI Sezione di Ingegneria Industriale, Dipartimento di Ingegneria dell'Innovazione, Universita degli Studi di Lecce, 73100 Lecce, Italy E-mail: [email protected] We study the viscoelastic second grade solid, for which the constitutive equation consists in the sum of a purely elastic part and a viscoelastic part; this latter part is specified by two microstructural coefficients a.\ and a.2, in addition to the usual Newtonian viscosity v. We show via some exact solutions that such solids may describe some interesting dispersive effects. The solutions under investigation belong to special classes of standing waves and of circularly-polarized finite-amplitude waves.
1. Introduction Fosdick and Yu 1 studied the thermodynamics and the non-linear oscillations of a special class of of viscoelastic solids of differential type where the rate effects are characterized by two microstructural coefficients a.\ and ct2, in addition to the usual Newtonian viscosity v. This is a most promising model, which generalizes the usual dissipative solid used in many applications (for example in non-linear acoustics2) and which introduces more than one characteristic time or speed. This model is the solid-mechanic analogue of the much-discussed second grade fluid and for this reason it has been dubbed "the solid of second grade". Here we point out some important dynamic features of this solid by considering some simple, fmite-
182
183 amplitude, motions. In Section 3, we record some unexpected behaviors for the incompressible Mooney-Rivlin solid of second grade, for which the elastic part of the Cauchy stress tensor is linear in the first two invariants of the Cauchy-Green tensor. We use a finite-amplitude, rectilinear shear motion to study the influence of rate effects on some classical problems. First, we find that the creep and recovery experiments will undergo some time-dependent oscillations if those effects are strong enough (in the corresponding purely elastic case, creep and recovery have exponential time dependence.) Second, we describe a slab with one face fixed and the other oscillating: we find that, above a certain threshold, the rate effects will cause the amplitude of the resulting oscillations within the slab to be rapidly (exponentially) attenuated with distance from the moving plate, even when there is no Newtonian viscosity (in the corresponding purely elastic case, the oscillations are transmitted sinusoidally through the slab.) We argue that these effects are intimately linked to the microstructure of the solid via the parameter a\, and that they are symptomatic of the behavior of solids of second grade in general and are not restricted to these two particular problems, nor to the specialization to the Mooney-Rivlin model. This point is further pressed in Section 4, where we make a connection with the results of a previous article, for finite-amplitude, elliptically polarized, transverse plane waves. We find that the nature of generalized oscillatory shearing motions 3 might not change with the introduction of second grade effects, but that the nature of sinusoidal standing waves is affected at high frequencies.
2. Basic equations As is conventional in continuum mechanics, the motion of a body is described here by a relation x = x(X,£), where x denotes the current coordinates at time t of a point occupied by the particle of coordinates X in the reference configuration. The deformation gradient F(X,£) and the spatial velocity gradient L(X,t) associated with the motion are defined by F := <9x/SX and L := grad x, respectively (here the dot denotes the material derivative, see Chadwick4 for instance). The other geometrical and kinematical quantities of interest are the left Cauchy-Green strain tensor B := F F T and the first two Rivlin-Ericksen tensors Ai := L + L T and A 2 := Ai + A X L + L T A X . In this article we restrict our attention to incompressible materials so
184 that only isochoric motions are possible; hence in particular, det B = 1 and tr D = 0 at all times. Further, we consider incompressible materials with a special constitutive equation 1 , for which the Cauchy stress T is split in an additive manner into an elastic part T E and a dissipative part T D . For the elastic part, the constitutive equation is that of a general hyperelastic, incompressible, isotropic material, that is 4 TE = - p l + 2 | ^ B - 2 | ^ B - 1 , (1) oli dl2 where 1 is the unit tensor and p — p(x, t) is the arbitrary Lagrange multiplier associated with the constraint of incompressibility and is usually called the pressure. In Eq. (1), £ is the strain energy function, which for an incompressible material depends on I\ and i 2 , the first two principal invariants of B: ij := tr B, I2 := [if — tr (B 2 )]/2. The dissipative part is given by T D = z/Ai + a i A 2 + a2AJ,
(2)
where v is the usual classical Newtonian viscosity and ai, a2 are the microstructural coefficients. The total Cauchy stress is the sum of these two parts, namely T = -pi + 2—-B - 2 — B
dh
1
+ i/Aj + « i A 2 + a2K\.
(3)
oh
Note that for the Mooney-Rivlin model of rubber elasticity, for which 2S = C{I\ — 3) + E{I2 — 3), where C and E are material constants, the total stress specializes to T = -pl + CB-EB~1+vAl+a1A2 + a2K\. (4) For materials which such a constitutive equation, Hayes and Saccomandi5 studied the propagation of finite amplitude transverse waves superimposed onto an arbitrary homogeneous static deformation. 3. On the nature of the microstructural parameter a± Fosdick and Yu1 showed that the compatibility of the material model Eq. (3) with the laws of thermodynamics requires that a.\ + a2 = 0, and that the Newtonian viscosity v is non-negative. Moreover, they showed that the Helmholtz free energy function of the second grade solid is a minimum at equilibrium if and only if a,\ > 0. Another interesting and important result of their investigation is a remark about the evolution in time of E(t), the
185 canonical energy of a body mechanically isolated and contained in a region fit free of tractions. The canonical energy is the sum of the Helmholtz free energy and the kinetic energy and so it is defined by E(t) := [ (E + \aiAx
• Ax) dv + [ \pk • xdu,
Jilt
JVlt
(5)
(where p is the mass density). Fosdick and Yu showed that its time evolution is given by V
- I (A1-A1)dW, (6) Jnt which means that the canonical energy decreases with time. This latter equation also shows that a\ does not contribute to dissipation. This is a central point in our discussion. Now the aim of this Section is to elaborate further on the parameter a\ and to show that it is genuinely connected with the microstructure of the material (precisely, with a characteristic length). To this end we try to maintain algebraic complexity at the lower level and so we study the following simple rectilinear shear motion, E •
1
x(X,t)
= [X + u(Y,t)}ei+Ye2
+ Ze3,
(7)
where u is a yet unknown function and e i , e2, e$ form a fixed orthonormal triad. For this isochoric motion, the kinematical quantities of interest are B = 1 + Uyei ® ei + uy(ei
®e2 + e 2 ® e i ) + 2u Y t e 2 ® e 2 ,
A? = u y t ( e i ® e i +e 2
(8)
and the first two principal invariants are equal: I\ = J 2 = 3 + uY • The equations of motions, in the absence of body forces, are: div T = px in current form. Writing them in referential form, and using Eqs. (8) which, together with Eq. (3), show that the non-diagonal elements of T depend on Y and t only, we find that they reduce to dp 8Tl2 0T22 dp + =pUtu = (9) -dX -9Y -dY °' ~dZ=°According to the third of these equations, p = p(X,Y,t). Then, by differentiation of the first and second equations in Eqs. (9) with respect to X, we
186 find that pxx = PYX = 0, so that px = Pi(t), say. Finally, the first equation in Eqs. (9) is the determining equation for the shearing deformation, given by -Pl(t)
+ (QUY)Y
+ VUyyt + CXiUYYtt = PUtt,
(10)
where Q = Q{uY) = 2{dY,/dI\+dY,/dl^) is the generalized shear modulus3. To simplify further the computations in the remainder of this Section, we take the scalar p\(t) to be identically zero (pi s 0), and the constitutive equation of the material to be Eq. (4). In that case, Q = const. = //, where p, — C + E is the infinitesimal shear modulus of Mooney-Rivlin materials, and the determining equation Eq. (10) simplifies to flUYY + VUYYt + OL\UYYtt = PUtt-
(H)
We notice that this partial differential equation is exactly the same equation as that obtained by Rubin et al. 6 in the linear limit of the phenomenological theory of dispersion caused by an inherent material characteristic length; it also occurs in the framework of acoustic waves propagation in bubbly liquids 7 . If in Eq. (11) L is a characteristic length and T a characteristic time, then we may exhibit three dimensionless numbers: 7i"i = v/([iT), 7T2 = ai/{pT2) and TT3 = pL/(pT2). We now pause to examine the behavior of our Mooney-Rivlin second grade solid when it undergoes some classic experiments of viscoelasticity, namely the creep experiment and the recovery experiment. These are quasi-static experiments, so that the inertial term on the right hand-side of Eq. (11) may be neglected; in other words, w^
(12)
the constant being dictated by the type of experiment to be modeled. For creep and for recovery, the shearing deformation is considered homogeneous so that here, u is of the form u(Y, t) — K(t)Y. For recovery, K —> 0 as t —» oo and for creep, K tend to a finite value, K° say. Hence the governing equations are K + KT + eKTT = K°,
K + Kr + eKTT = 0,
(13)
for creep and for recovery, respectively. Here, Eq. (12) has been nondimensionalized by the scaling r — pijv (note that vj\i is usually called the relaxation time) and by the introduction of the dimensionless parameter e = p,ai/v2. When e — 0, we recover the classical solutions
187
K(T) = K°(l — e~T) and K(T) — K°e~T for the creep problem and for the recovery problem, respectively, where K° is the steady state amount of shear. When e < 1/4, the general solutions of Eq. (13) are also damped, but when when e > 1/4, the parameter a\ (> 4v2/p) modifies the nature of the solutions. Hence for the recovery experiment, the solution is of the form K(T) = K0e~T/2£
c o s ( y 4 7 ^ 1 r/2e),
(14)
clearly highlighting that microstructural oscillations come into play. Now turning back to the dynamic equation Eq. (11), we consider a slab of thickness L in the Y direction, and we introduce the dimensionless variables £ := Y/L and r := y/p/(pL2) t. Also, we consider the case where v = 0 (no Newtonian viscosity); then Eq. (11) is recast as Oil U
+ - 7 2 U « T T = WTT,
i(
(15)
where we note that the quantity oti/(pL2) is a dimensionless parameter. Let the slab be sandwiched between two rigid plates, one at the bottom £ = 0, oscillating with frequency ui and displacement Ucosut (say) and one at the top £ = 1, at rest. We seek an exact solution to Eq. (15) in separable form, u = £/(£) cos(u)t), say. Then <j> satisfies l - ^ V
+ ^
^
0
,
*(0) = l,
0(1) = 0.
(16)
When a\ = 0, we recover the classical elastic case solution, m
=
ainvWMMl-J),
(1?)
sin y p/p Lio When a\uj2/p < 1, the solution is essentially of the same nature: ,,^. sina(l — £) , xfpTu, Lw v 0 0 = r -, with a : = 7^===, 18 sin a y/l - (on/p.) u)2 and so, at "low" frequencies there is no fundamental difference between the purely elastic case and the Mooney-Rivlin second grade solid. When a\u2Jji > 1 however, the microstructure (via the parameter a\) dramatically alters the response of the viscoelastic slab, because then the solution is in the form
m =^ ? ^ , smh/?
with {,:= /f^ y/(a\/n)
• to2 - 1
(19)
188
Hence at "high" frequencies, the vibrations engendered in the slab by the lower plate are localized near the vibrating plate; the oscillations are "trapped" in the microstructure. Figure 1 shows the variations of the oscillations's amplitude through the thickness of the slab. Figure 1(a) represents (£) given by Eq. (18) for a = 5, 9,23; these vibrations are sinusoidal, with resonance occurring at a = nir. Figure 1(b) represents (£) given by Eq. (19) for (3 = 5,9,23; the amplitude is rapidly attenuated through the thickness; note that as a-^oj2/'// approaches 1 from above, the quantity /? tends to infinity, and the localization is greater and greater until at aiuj2/fi = 1, we have perfect isolation. These attenuation effects can exist in viscoelastic solids for which v ^ 0, c\\ — 0, but at the expense of canonical energy dissipation. Here we emphasize that the results are obtained for a Mooney-Rivlin solid of second grade with no Newtonian viscosity, v = 0, so that by Eq. (6), the canonical energy is conserved.
m
Figure 1. Influence of the microstructure on the oscillations of a slab fixed on one end. At low frequencies: oscillations; at high frequencies: attenuation.
189 4. Finite amplitude transverse plane waves In this Section we consider the following class of motions x — 7X + u(z,t),
y = 7Y + v(z,t),
z = XZ,
(20)
that is, motions describing a transverse wave, polarized in the (XY) plane, and propagating in the Z direction of a solid subject to a pure homogeneous equi-biaxial pre-stretch along the X, Y, and Z axes, with corresponding constant principal stretch ratios 7, 7, and A (72A = 1). Here u and v are yet unknown scalar functions. Then the geometrical quantities of interest are the left Cauchy-Green strain tensor _ 2
B =
7 + X2u2z X2uzvz X2uz~ 2 2 2 2 X uzvz 7 + X v X2vz 2 X uz X2vz A2 .
(21)
its inverse A 0 -Xuz 0 A -Xvz -Xuz -Xvz X(u2z + v2) +
B-1
(22) 4 7
and the kinematical quantities of interest are 0 0
Ai
0 uzt' 0 vzt .
u2zt
A? = UztVzt .
Uzt Vzt 0 _
uztvzt
0
Vzt
0
(23)
0
u2zt +
0
u
zt.
and A2 =
" 0 0
0 0
uztt vztt
(24)
J
Uztt vztt 2 («zt + "zt)J
and the first two invariants are h
= 2 T 2 + A2 + X2(u2z + v2),
h = 2X + 7 4 + X(u2z + v2).
(25)
Now the equations of motion, in the absence of body forces, are given in current form as div T = /?x, or here,
dp •Yx
+
dT13 -dz-=(mtU
dp , dT23 -dy- + -dz-=PVtU
9T 33 IT
0.
(26)
Differentiating these equations with respect to x, we find pxx = pyx = pzx = 0, so that px = pi(t), say. Similarly, by differentiating the equations with
190 respect to y, we find py = P2(t), say. Now the first two equations reduce to - Pi(t) + (Quz)z + vuzzt + aiuzztt
= putt,
- P2(t) + {Qvz)z + vvzzt + a!VzzU = pvtt, 2
(27) 2
and the third equation determines p. Here, Q = Q{u + v ) is the generalized shear modulus, now defined by Q = 2(X2dT,/dh
+ \dT,/dI2).
(28)
8
Following Destrade and Saccomandi , we take the derivative of Eqs. (27) with respect z, we introduce the notations U := uz, V :— vz and the complex function W := U+iV, and we recast Eq. (27) as the single complex equation (QW)ZZ + vWzzt + onW^a 2
= pWtt,
2
2
(29) 2
where Q is now a function of U +V alone, Q = Q(U +V ). the complex function W into modulus and argument as W(z,t)
We decompose
= n(z,t)esxp{i0(x,t)),
(30)
say, and we seek solutions in the separable forms, Sl(z,t)=a1(z)Q2(t),
0(x,t) = 01{x) + 02(t),
(31)
say. The solutions obtained from this ansatz are remarkable, first of all because they reduce the partial differential equation Eq. (29) to a system of ordinary differential equations, and also because they contain the generalization to a nonlinear setting of often encountered classes of wave solutions, such as damped and attenuated plane waves. It is interesting to note that in some cases, the solutions of Eqs. (29) are similar to the classical viscoelastic case. For example, when we restrict our attention to the following generalized oscillatory shearing motion W(z,t)
= [iP(t)+i
(32)
where & is a constant and ip, <j>, and 9 are function of time alone, we recover for our present setting the same equations (and therefore the same solutions) as those already reported by Destrade and Saccomandi8 in the context of compressible dissipative solids. Establishing a formal correspondence requires the replacement of their density po with the positive quantity p + k2a\ here; it also shows that no noticeable differences arise from the introduction of strong rate effects. For instance, such is the case when we consider, at v — 0, circularly polarized harmonic waves in the form u(z,t) = Acos(kz— wt),
v(z,t) = ±Asin(kz
- cot).
(33)
191 Now, U2+V2 = A2k2 and therefore Q is independent of z; then the equation of motion Eq. (29) leads to the following dispersion equation: Q(A2k2) = (p + k2a1)u2,
(34)
which is similar to the equation for the purely elastic case. When we consider generalized shear sinusoidal standing waves8, we obtain more noticeable results. Hence Carroll 9 took u(z,t) = (p(z) cos tot,
v(z,t) = cf>(z) sin cut,
(35)
where <> / depends on z only. Then Eq. (29), written at v = 0, reduces to {[Q($2) - aiw2]$}" + pu2$ = 0,
(36)
where $ :— 4>'. Here the main difference between this equation and the corresponding equation in the purely elastic case is the introduction of a characteristic dispersive length a\uj2. We also note that Eq. (36) is remarkable because the usual substitution for autonomous equations, * ( $ ) = $', reduces this equation to a linear first order differential equation in \E'2, so that many solutions can be found in analytical form. 5. Concluding remarks We showed in this note that dispersion and dissipation are strongly correlated for the solid of second grade. We provided several interesting exact solutions in special cases, and gave pointers on how to reduce the general dynamical equations down to ordinary differential equations. As it happens, some methods and results developed elsewhere in the literature can be applied to the present framework in a straightforward manner. Our next step will be to extend the present model for a solid of second grade to a wider nonlinear setting, by taking the microstructural parameter a i to be a function of the invariants and no longer a constant. In such a way, it might be possible to balance dispersive and nonlinear effects, and perhaps to obtain some solitons and compactons 8 . That derivation would constitute a major advancement in our understanding of the nonlinear mechanics of solids, with many important technical applications. Acknowledgments This work is supported by the French Ministere des Affaires Etrangeres under the scheme: "Sejour Scientifique de Haut Niveau".
192
References 1. R. L. Fosdick and J. H. Yu, Int. J. Nonlinear Mech. 3 1 , 495 (1996). 2. A. Norris, in Nonlinear Acoustics, M. F. Hamilton and D. T. Blackstock (eds.) Chap. 9 (Academic Press, San Diego, 1998). 3. M. M. Carroll, J. Elast. 7, 411 (1977). 4. P. Chadwick, Continuum Mechanics (Dover, New York, 1999). 5. M. A. Hayes and G. Saccomandi, J. Elast. 59, 213 (2000). 6. M. B. Rubin, P. Rosenau and O. Gottlieb, J. Appl. Phys., 77, 4054 (1995). 7. P. M. Jordan and C. Feuillade, Int. J. Engng. Sci. 42. 1119 (2004). 8. M. Destrade and G. Saccomandi, Phys. Rev. E 72, 016620 (2005). 9. M. M. Carroll, Q. J. Mech. appl. Math. 30, 223 (1977). 10. M. Destrade and G. Saccomandi, Solitary and compact-like shear waves in the bulk of solids, submitted (2006).
A B O U T T H E LARGE TIME BEHAVIOR OF DISSIPATIVE EQUATIONS W H E N A P R I O R I B O U N D S A R E SLOWLY GROWING
L. D E S V I L L E T T E S CMLA
- UMR 8536 du CNRS, ENS de Cachan, 61 Av. du Pdt. Wilson, 94235 Cachan Cedex, France. E-mail: desville&cmla.ens-cachan.fr
We are interested in obtaining explicit rates of convergence toward the equilibrium for equations having a Lyapounov functional (entropy). We assume that the dissipation of entropy dominates the entropy itself. However, instead of being independant of time, we only suppose that the rate of this domination deteriorates slightly when the time goes to infinity. Such a situation was described first by Toscani and Villani in the context of the Boltzmann equation (and its variants when grazing collisions are predominant). We show here how the estimate of convergence toward equilibrium obtained there can be used to establish new (global in time) a priori estimates for the equation under study, which, in turn, sometimes enable to precise the rate of convergence toward equilibrium. Examples of applications of thise ideas are taken from works in collaboration with Mouhot ( 1 0 ) and Fellner ( 8 ), respectively for homogeneous kinetic equations and reaction-diffusion systems.
1. Introduction 1.1. De La Salle's
Principle
We look for explicit estimates of convergence toward equilibrium for equations where dissipative effects are predominant. We describe the principle of the entropy method for an abstract equation dtf = Af,
(1)
where A is any kind of linear or nonlinear operator. We suppose that there exists a (bounded below) Lyapounov functional H = H(f) (usually called entropy (or opposite of the entropy)) and a functional D = D(f) (usually called entropy dissipation) such that (what is 193
194 called in kinetic theory the first and second part of Boltzmann's H-theorem) holds :
D(f) = 0
dtH(f)
= -D(f)
< 0,
<=•
Af = 0
^
(2) f = feq,
(3)
where feq is a given function. At this level, using what is sometimes called De La Salle's principle, it is often possible to prove that the decreasing function 11—• H(f(t)) converges toward its minimum, that 11—• D(f(t)) converges toward 0, and that lim f{t) = feg
(4)
in a convenient topology. Note that the situation described above (with a unique equilibrium feq) is usually valid only once a certain number of conserved quantities (in the evolution of eq. (1)) have been taken into account. 1.2. The entropy/entropy
dissipation
estimate
The structure described above only allows to obtain a theorem of convergence, it does not enable to estimate the rate of this convergence. In order to obtain such an estimate, one looks for functional inequalities of the form D{f) > *{H{f) - H(feq)),
(5)
where $ : M+ —> M+ is a function such that $(x) = 0 <^=> x = 0. One tries to find a function $ which increases as much as possible near 0. In the case when there are conserved quantities in the evolution of eq. (1), it is enough to prove estimate (5) when the corresponding quantities are fixed. Assuming that estimate (5) holds, we get (for a solution of (1)) thanks to estimate (2) the differential inequality dt(H(f(t))
- H(feq))
< -$(H(f(t))
- H(feq)),
(6)
and Gronwall's lemma ensures that H(f(t))-H(feg)
(7)
where R is the reciprocal of a primitive of —1/$. Then, if H has good properties of coercivity, we obtain
^(/(t)-/e,)<5(t),-
(8)
195 where S is related to R, and JVi is some norm which depends on the problem. Such an estimate is recovered from (7) thanks to inequalities like Cziszar-Kullback's (Cf. 6 , 1 8 ). When one can take <3?(a;) = Cstx, one gets R(t) < e~Cstt. Sometimes, it is however only possible to take $(x) = Cste x1+£ for some (or all) e > 0, and consequently R(t) = Cstet~x/e. Note also that generically (that is, if H behaves quadratically in a neighborhood of / e g ) , one can take R(t) = CstS(t)2. 1.3. Slowly growing a priori
bounds
It happens very often that estimate (5) is not true for all / (or even for all / for which the conserved quantities have a given value), and one only has the possibility to prove D(f) > U(N2(f))
$(tf (/) - H(feq)),
(9)
where N2 is some norm (sometimes it is a more complicated functional which depends upon / through a certain number of norms, moments, lower bounds, etc.) and U is some decreasing strictly positive function. Then, if during the evolution of eq. (1), the quantity N2(f{t)) is bounded (when t —> oo), we are in the same situation as in the previous section. However, it sometimes happens that t — i > JV2 (/(£)) is not known to be bounded (when t i-> f(t) is the solution of eq. (1)), but is only known to grow in a certain way when t —» 00. Then, eq. (6) has to be replaced by the more complicated differential inequality dt(H(f(t))
- H(feq))
< -a(t) b(H(f(t))
- H(feq)),
(10)
where a is some nonnegative decreasing function of t. As a consequence, estimate (7) still holds with R the composition of the reciprocal of a primitive of —1/6 and of a primitive of a. Whenever this estimate gives a significant result of convergence to equilibrium, one speaks of slowly growing a priori bounds (slowly means here that the growth is sufficiently moderate to preserve a large time behavior which can be controlled in a reasonable way; it has no intrinsic meaning). Here are two typical situations one can encounter in practice : (1) In the first one, we assume that a(t) = Cst (1 +t)~a (for a e ] 0 , l [ , 13 > 1). Then,
R(t) =
Cst(l+tyv3.
and b(u) = vP
196 (2) In the second one, we assume that a(t) = Cst (log(e + £)) * and b(u) = u. Then, R(t)
of the
method
We now assume that we are in a situation where estimates (9) and (10) hold, so that we have estimate (7) for a certain function R and, subsequently, estimate (8) for another function S and a norm Ni. Usually, the norm JV2 is stronger than JVi, so that estimate (8) is not enough to conclude that t H-> N2(f(t)) is bounded. However, it happens sometimes that there exists another norm N3 which enables to obtain the interpolation (for some S > 0) N2(f)<
CstN!{f)s
N^f)1-6,
and the estimate
N3(f(t))
< N2(f(t) - feg)5
- feq) +
N3(f(t)
< S(t)5 (T(t) + N3(feq)y-5
- feq))1-5 +
N2(feq) +
N2(feq)
N2(feq).
As a consequence, we see that t i-> N2(f(t)) is bounded as soon as ( H> S(t)sT(t)1-5 is bounded and AT3(/eg) is finite. We give in subsection 2.1 a typical example of this situation : all the moments of the solution of the spatially homogeneous Boltzmann equation with soft (regularized) potentials are propagated uniformly in time. We now assume that thanks to the previous method, we have been able to prove that t H-> N2(f(t)) is bounded. Then we can replace estimate (9) by estimate (5) and (sometimes) get in this way a better decay to the equilibrium. We show in subsection 2.2 how this idea works in the case of a reactiondiffusion system constituted of four equations.
197 We summarize the line of ideas described above : first, one proves that certain quantities (namely, A ^ / ) ) do not increase too much when t —> +00 along the flow of eq. (1). Secondly, thanks to some entropy/entropy dissipation estimate (that is, estimates (9) and (10)), one shows that some explicit decay toward equilibrium in a weak norm holds (estimate (8)). Thirdly, one proves that some norm stronger than JV2 (namely, A^) also do not increase too much when t -> +00 along the flow of eq. (1). Thanks to the interpolation described above, one obtains that the quantity A ^ / ) is bounded when t —» +00 along the flow of eq. (1). Finally, the explicit decay toward equilibrium can be improved because estimate (9) can be replaced by (5). 2. E x a m p l e s 2.1. The spatially homogeneous soft potentials
Boltzmann
equation
with
We now consider the case when (1) is the (spatially homogeneous) Boltzmann equation of rarefied gases. The unknown is / = fit, v) > 0, with v € RN (f(t, v) is the number density of molecules of a gas which at time t have velocity v), and the operator A in (1) is a quadratic and integral operator modeling the binary collisions between the molecules. It is defined by: Af(v) = Q{f){v) = f f
{/(„') f(vi) - f{v) / ( „ . ) }
x B ( \v — v* 1,1
\v-vJ
(11)
^7 • a I dadv*,
J
with , v + v* \v -1>* v = — 1- -1— cr,
• w* \v — v* 2 2 and B is a cross section depending on the interaction between the molecules. For more details about this kernel, we refer to 5 . The conserved quantities for this model are the mass mo — f fdv, the (i-th component of the) momentum mu = J fvidv and the energy TO2/2 — J f ^y- dv. We assume here (without restriction, since this can be
198 obtained by multiplication, translation and dilation) that at time t = 0, one has mo = 1, mu = 0 and m2 = N. Finally, we denote by M(v) = (2n)~N^2 e~^ I"2 the normalized Maxwellian. One defines the (relative) entropy by
H{f
^Lmios(M)dv-
<12)
Differentiating (1) along the flow (with A given by (11)), we get (2) with
x log ( ^ T T T T T ) B (\V ~ v*\> r ^
• A d°dv*dv-
(13)
Then (assuming that B > 0 a.e.), relation (3) holds with feq = M (and H(feq) = 0). This is exactly Boltzmann's H-theorem (Cf. for example 5 ) . In the sequel, we shall use the notation (for / : RN —» E, p > 1, s > 0),
\Hv)\p(i+\v\2rt/2dv.
/
II/HL?=
JvmN We are interested here in the case of the so-called mollified (moderately) soft potentials with angular cutoff, that is, the cross sections B satisfying Assumption 1 :
B
(i-<^f')-*(i-*i)ft(i^-'>
with Bi smooth and Bi(x) ~ x ^+oo x~a, a e]0,2[, B 2 € l 1 ( ] - l , l [ ) . We recall that (under assumption 1) for any initial datum fin e L\ (s > 2), there exists a unique solution to eq. (1, (11)) conserving the mass, momentum and energy (Cf. 3 for the existence). One can show in this situation that the L 1 moments of / grow at most linearly (Cf. 10 for example) : Proposition 2.1. Let s > 2. Then for any initial datum fi„ € L], the unique associated solution f = f(t,-) to equation (1), (11) under assumption 1 satisfies the bounds V*>0,
\\f(t,-)\\Ll
+ t).
199 Then, eq. (9) can be obtained with $(1) = Ctes xl+e (for all e > 0), so that the following proposition (proven by G. Toscani and C. Villani in 22 ) holds : Proposition 2.2. We consider an initial datum 0 < f%n G L\ and r > 0. Then there exists qo > 0 such that if moreover fin £ L 2 0 , the unique associated solution f — f(t,-) of equation (1), (11) under assumption 1 satisfies Vt>0,
\\f(t,-)-M\\Li
+ t)-T.
It is then possible to use the interpolation method of subsection 1.4 to obtain the boundedness when t —> +00 of the moments of / (this is proven in detail in 10 ) : Proposition 2.3. Let s > 2 be given, together with an initial datum 0 < fin € L\. We consider the unique solution fit, •) > 0 to equation (1), (11) under assumption 1. Then there exists qo > 0 (depending on s) such that if hn^L\s(M2qo, sup\\f(t,-)\\L]
Here, the ideas of subsection 1.4 are used with Ni(f) = | | / | | n , ^(f) H/IILJ, N3(f)
=
= \\f\\LL, 5 = 1/2, T(t) = l + t, S(t) = (1 + t)~K
We refer to 4 , u , 21 , 22 and 23 for a complete picture of the state of the art concerning the use of the entropy/entropy dissipation method for the spatially homogeneous Boltzmann equation : the logarithmic Sobolev inequality of Gross (Cf. 16 ) plays there a central role. Those results should be compared with what is obtained thanks to spectral theory and linearization, Cf. for example 2 . For the theory of moments of the solution of the (spatially homogeneous) Boltzmann equation, we refer to 17 , 12 , 7 , 19 , 24 and 25 . We conclude this example by the following remarks : First, it is also possible (Cf. 10 ) to obtain results on the uniform in time propagation of smoothness of the solution of the spatially homogeneous Boltzmann equation with soft potentials (typically, Lps norms for p > 1 or weighted Sobolev norms). Secondly, we do not present here the last step of the plan described in subsection 1.4, since it does not seem possible to improve in this case the result of "almost exponential" convergence presented in 22 .
200
2.2. A ID reaction-diffusion equations
system
of four
reversible
We now introduce a reaction-diffusion system modeling the evolution of the concentration a^ = <2j(£, x) > 0 of species Ai, i = 1, ..,4, which undergo a reversible chemical reaction Ai + A2 ^ A3 + A4. The system of PDEs satisfied by the <2j is (after rescaling) the following : dtcii - 5i dxxa,i = ( - l ) M i (01 a2 - a3 a 4 ),
(14)
where x e]0,1[, and homogeneous Neumann conditions and initial conditions hold dxai(t,0)
= dxa,i(t, 1) = 0,
ai(0,x) = ai0(x).
(15)
Here, fx\ = \i2 = —1, fj,3 = 114 = 1, and Si > 0. Then, using the conservation of the number of molecules, we define M13 = / (ai(t,x) + a3(t,x))dx= Jo
/ (aw(x) + a30{x))dx, Jo
(16)
M14 = / (ai(t,x)+a,4(t,x))dx Jo
= / (aw(x) + al0(x))dx, Jo
(17)
M23=
= / (a2o(a?) + a30(x))dx, Jo
(18)
/ (a2(t,x)+a3(t,x))dx Jo and as a consequence
M24 = Mu + M23 - M13 = / (a2(f,x) + a 4 (t,x))da; • Jo
(19)
i
1
(a2o(x) + a 40 (:r))cte. /o Finally, we introduce the total mass M1234 = M13 + M24 = Mu + M2Z .
(20)
Then, for system (14) - (15) satisfying (16) - (18) (with all M{j > 0), there exists a unique equilibrium state (aioo>a2ooj a3oo,G4oo) defined by the unique positive constants solving a l o o + a 3oo = M 1 3 , ai00 + a400 = -Wi4> &200 + a3oo — M23, and a i ^ 0200 = a3oo ^4oo : a
^
a
=^ > 0 ,
3oc = ^ f f > 0 ,
a
2oo = M
a4oo = M
2 3 1 4
- ^ ^ = ^ ^ > 0 , - ^ ^ ^ ^ f
(21)
> 0 . (22)
201 Global existence and uniqueness of classical solutions of (14)—(15) have been shown in the work of * and 20 , under weak assumption on the (nonnegative) initial data. For this system, the standard estimates on the ID heat equation yield a polynomially growing L00 bound (Cf. 8 ) : Lemma 2.1. Let ai, i = 1,2,3,4 be the solutions of the system (14)-(15) with nonnegative initial data a^o G L°°(]0,1[). Then, for T > 0 llai|U°°([0,T]x[0,l]) < Cst
(l+T^Y
(23)
Using this lemma and the ideas of subsection 1.3, one gets the following estimate of almost exponential convergence in L1 (Cf. 8 ) : Proposition 2.4. Let the initial data a,o be nonnegative functions of L°°(]0,1[) with strictly positive masses M\z, Mu, M23, and M24. Then, the unique nonnegative classical global solution to equations (14) - (15) satisfies the following almost exponential decay towards the steady state a^oo given in (21), (22) : \\ai(t,-)-aioo\\LiQ0tl[)
S
Then, we apply the program of subsection 1.4 with Ni(f) N2(f)
= II/HL~(]O,I[), N3(f)
j .
(24)
= ||/||i,i(]o i[),
= | | / | | f f l ( ] 0 , i D , 8 = 1/4,
S(t) = Cst exp I — —
J,
T(t) = polynomial in t.
We obtain in this way the uniform L°° estimate : l|ffli||L~([0,-(-oo[x[0,l]) < Cst.
(25)
The last step of the program of subsection 1.4 leads to improve the estimate (24) of decay towards equilibrium, so that it becomes truly exponential (Cf. 8 ) : Theorem 2.1. Under the assumptions of proposition 2-4, the concentrations ai, i = 1, 2,3,4 decay exponentially towards the steady state ajoo given in (21), (22) : \\a,i(t,-) -a i o o || L i(] 0 > i[)
(26)
202 For other results obtained by t h e entropy method in t h e study of t h e large time behavior of reaction-diffusion equations, we refer t o 1 5 , 1 3 , 1 4 and 9
3.
Conclusion
As we have tried t o show, the lack of uniform in time a priori estimates is not necessarily a disaster when one tries t o use the entropy/entropy dissipation estimate. In fact, t h e estimates on the large time behaviour are sometimes the key tool t o pove t h a t a priori estimates are uniform in time (this is w h a t happened in t h e two examples we presented). Once this uniformity is proven, it is sometimes possible t o precise t h e rate of decay to equilibrium (as in the second example presented in this work). Note finally t h a t a common feature of all t h e estimates given here is t h a t they are explicitly computable in t e r m s of t h e d a t a of the problem (in particular, we never used any compactness argument). References 1. AMANN, H. Global existence for semilinear parabolic problems. J. Reine Angew. Math. 360, (1985), 47-83. 2. ARKERYD, L. Stability in L1 for the spatially homogeneous Boltzmann equation. Arch. Rational Mech. Anal. 103, 2 (1988), 151-167. 3. ARKERYD, L. Intermolecular forces of infinite range and the Boltzmann equation. Arch. Rational Mech. Anal. 77, (1981), 11-21. 4. CARLEN, E., CARVALHO, M. Entropy production estimates for Boltzmann equations with physically realistic collision kernels. J. Stat. Phys. 74, 3-4 (1994), 743-782. 5. CERCIGNANI, C. The Boltzmann Equation and its Applications. Springer, 1988. 6. CSISZAR, I. Information-type measures of difference of probability distributions and indirect observations. Stud. Sci. Math. Hung. 2, (1967), 299-318. 7. DESVILLETTES, L. Some applications of the method of moments for the homogeneous Boltzmann equation. Arch. Rational Mech. Anal. 183(1993), 387-395. 8. DESVILLETTES, L., FELLNER, K. Entropy methods for reaction-diffusion equations : degenerate diffusion and slowly growing a priori bounds. Preprint 200519 of the CMLA, ENS de Cachan. 9. DESVILLETTES, L., FELLNER, K. Exponential Decay toward Equilibrium via Entropy Methods for Reaction-Diffusion Equations. Accepted for J. Math. Anal. Appl. 10. DESVILLETTES, L., MOUHOT, C. Large time behavior of the a priori bounds for the solutions to the spatially homogeneous Boltzmann equations with soft potentials. Preprint 2005-2 of the CMLA, ENS de Cachan.
203 11. DESVILLETTES, L., VILLANI, C. On the spatially homogeneous Landau equation for hard potentials. Part II : .ff-theorem and applications. Comm. Partial Differential Equations 25, 1-2 (2000), 261-298. 12. ELMROTH, T. Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range. Arch. Rational Mech. Anal. 82, 1 (1983), 1-12. 13. GLITZKY, A., GROGER, K., HUNLICH, R. Free energy and dissipation rate
for reaction-diffusion processes of electrically charged species. Appl. Anal. 60, 3-4 (1996), 201-217. 14. GLITZKY, A., HUNLICH, R. Energetic estimates and asymptotics for electroreaction-diffusion systems. Z. Angew. Math. Mech. 77, (1997), 823-832. 15. GROGER, K. Free energy estimates and asymptotic behaviour of reactiondiffusion processes. Preprint 20, Institut fur Angewandte Analysis und Stochastik, Berlin, 1992. 16. GROSS, L. Logarithmic Sobolev inequalities and contractivity properties of semigroups. In Dirichlet Forms (1992), P. et al., Ed., vol. 1563, Led. Notes in Math., Varenna, Springer-Verlag, 54-88. 17. IKENBERRY, E., TRUESDELL, C. On the pressures and the flux of energy in a gas according to Maxwell's kinetic theory. I. J. Rat. Mech. Anal. 5 (1956), 1-54. 18. KULLBACK, S. A lower bound for discrimination information in terms of variation. IEEE Trans. Inf. The. 4, (1967), 126-127. 19. MISCHLER, S., AND WENNBERG, B. On the spatially homogeneous Boltzmann equation. Ann. Inst. H. Poincare Anal. Non Lineaire 16 (1999), 467501. 20. MORGAN, J. Global existence for semilinear parabolic systems. SI AM J. Math. Ana. 20, (1989), 1128-1144. 21. TOSCANI, G., VILLANI, C. Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Comm. Math. Phys. 203, 3 (1999), 667-706. 22. TOSCANI, G., VILLANI, C. On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds. J. Statist. Phys. 98, 5-6 (2000), 1279-1309. 23. VILLANI, C. Cercignani's conjecture is sometimes true and always almost true. Comm. Math. Phys. 234, 3 (2003), 455-490. 24. WENNBERG, B. On moments and uniqueness for solutions to the space homogeneous Boltzmann equation. Transport Theory Statist. Phys. 23 (1994), 533-539. 25. WENNBERG, B. Entropy dissipation and moment production for the Boltzmann equation. J. Statist. Phys. 86(1997), 1053-1066.
DEFORMATION WAVES IN M I C R O S T R U C T U R E D SOLIDS A N D DISPERSION
J. E N G E L B R E C H T , A. B E R E Z O V S K I Centre for Nonlinear Studies, Institute of Cybernetics at Tallinn TV, Akadeemia 21, 12618 Tallinn, Estonia E-mail: [email protected], arkadi. [email protected]. ee F. PASTRONE Department of Mathematics, University of Turin, Via Carlo Alberto 10, 10123 Turin, Italy E-mail: [email protected] M. B R A U N Institute
of Mechatronics and System Dynamics, University of Duisburg-Essen, 4704S Duisburg, Germany E-mail: [email protected]
The governing equations for a ID case of the Mindlin model for microstructured materials are derived and analysed. These equations exhibit hierarchical properties assigning the wave operators to internal scales. The dispersion of waves is characterized by higher-order derivatives including also the mixed derivatives with respect to coordinate and time.
1. Introduction Materials used in contemporary high technology are characterized often by their complex structure in order to satisfy many requirements in practice. This concerns polycrystalline solids, ceramic composites, alloys, functionally graded materials, granular materials, etc. Often one should also account for the damage effects when materials have microcracks. All that shows the existence of intrinsic space-scales in matter, like the lattice period, the size of a crystallite or a grain, and the distance between microcracks. This scale-dependence should also be taken into account in governing equa-
204
205
tions. The classical theory of the continuous media is built up using the assumption of the smoothness of continua. The microstructured materials, i.e. materials with irregularities have one or more internal scales and their complex dynamic behaviour cannot be explained by the classical theories. Within the theories of continua the problems of irregularities of media have been foreseen long time ago by the Cosserats and Voigt, and more recently by Mindlin [1] and Eringen [2]. The elegant mathematical theories of continua with voids or with vector microstructure, or continua with spins of Cosserat continuum or micromorphic continuum, etc. have been elaborated since then, see overviews by Capriz [3] and Eringen [4]. The straight-forward modelling of microstructured solids leads to assigning all the physical properties to every volume element in a solid introducing so the dependence on material coordinates. This leads to an extremely complex system. Another probably much more effective way is to separate macroand microstructure in continua. Then the conservation laws for both structures should be separately formulated (Mindlin [1], Eringen [2, 4]), or the microstructural quantities are separately taken into account in one set of conservation laws (Maugin [5]). Here we proceed according to the ideas of separating macro- and microstructures.
2. Governing equations The governing equations are derived following Mindlin [1] who has interpreted the microstructure "as a molecule of a polymer, a crystallite of a polycrystal or a grain of a granular material". This microelement is taken as a deformable cell. Note that if this cell is rigid, then the Cosserat model follows. The displacement u of a material particle in terms of macrostructure is defined by its components «; = Xi—Xi, where Xi,Xi(i = 1,2,3) are the components of the spatial and material position vectors, respectively. Within each material volume (particle) there is a microvolume and the microdisplacement u' is defined by its components u^ = x[ — X't, where the origin of the coordinates x\ moves with the displacement u. The displacement gradient is assumed to be small. This leads to the basic assumption of Mindlin [1] that "the microdisplacement can be expressed as a sum of products of specified functions of x\ and arbitrary functions of the Xi and t". The first approximation is then
u'j =
x'k(pkj(xi,t).
(1)
206
The microdeformation is
d£i=Wi
= <ev-
(2)
Further we consider the simplest ID case and drop the indices i,j dealing with u and ip only. The indices t and x used in the sequel denote differentiation. The fundamental balance laws for microstructured materials can be formulated separately for macroscopic and microscopic scales using the Lagrangian and the Euler-Lagrange equations (see Engelbrecht et al. [6]). For the basic single-scaled model we take the potential energy W in the form of a quadratic function
W = l (au2x + 2A
(3)
with a, A, B, C denoting material constants. Then the governing equations take the form
putt = auxx + Aipx,
(4)
Iipu = Ccpxx - Aux - B
(5)
where p is the density and / the microinertia. In the two-scale situation (scale within the scale), it is assumed that every deformable cell of the microstructure includes new deformable cells at a smaller scale. The displacements at the different scales are then (cf. (1)): Uj = Uj {Xi, t),
u'j = X'k ipkj
(Xi,t),
u'- = x'klpkj (x'i,t),
(6)
respectively, where x'k,x'l correspond to the local coordinate within respective cells. As we are interested in motion on the macrolevel, it is assumed that u'j = x'k^kj(xi,t).
(7)
Then we get du
'i
du
'i
^ T ^ ' a<=^-
(8)
207
As before, we drop the indices i,j,k energy function is taken w
= g iaul + 2Aiuxy
for the ID case. Now the potential
+ Bnp2 + Cwl + 2A2uxip + B2ip2 + C2ip2x), (9)
where a and Ai,Bi,d (i = 1,2) denote material constants. The governing equations are then the following: puu = auxx + Aiipx, hftt
(10)
=Ci
(11)
hiptt - C2ipxx - A2
(12)
generalizing the system (4), (5). Here 7j (i = 1,2) denote microinertia of corresponding microstructures. 3. Hierarchies of waves Whitham [7] has described certain complicated wave systems where a scale parameter S plays a crucial role. Depending on its limit values, 5 —> oo or 6 —> 0, one or another wave operator governs the process asymptotically. Thus, the full system includes a hierarchy of waves with certain stability conditions [7]. Here we show that waves in microstructured materials exhibit the hierarchical behaviour governed by a parameter which is the ratio of the characteristic scale of a microstructure and the wave length of the excitation. First, the single scale. Let the scale of the microstructure be I and the excitation characterised by its amplitude UQ and wavelength L. Then we can introduce the following dimensionless variables and parameters U = u/U0,
X = x/L,
T = c0t/L,
8 = l2/L2,
e = U0/L,
(13)
where CQ = a/p. We also suppose that I = pl2I*, C = l2C*, where I* is dimensionless and C* has the dimension of stress. The difference of densities is embedded in I*. By means of series representation and the slaving principle (Christiansen et al. [8]) we get finally UTT = (1 - *) UXX + 51*m (UTT ~ nUxx)xx where k = c2A/c2, m = A2/B2, n = c2/c20, c2=C/I, In case of multiple scales we have to introduce h = l2IL2,
82 = l2/L2,
,
(14) c2A=A2/pB. (15)
208
where h and l2 are the scales of microstructures
+
~ <^2 TO 2 (UTT - n2UXx)Xxxx nx = Cf/alf,
m2 =
(16)
A\{Alf/B\Bl,
4. Discussion Equations (14) and (16) reflect clearly the hierarchical character of wave propagation in microstructured materials as indicated by Whitham [7] within general wave theory. Indeed, in case of eq (14): (i) if S is small, then the last two terms are negligible, if 5 is large, then the first two terms are negligible and the properties are governed by properties of microstructure; (ii) the wave speed in the compound material is affected by the microstructure (1 versus k — c\/c20) and only A = 0 (no coupling) excludes this dependence; (iii) the influence of the microstructure is, as expected, characterized by dispersive terms; however, contrary to the idealized models, the double dispersion (different terms UTTXX and Uxxxx) is of importance. The multi-scale model (16) actually prolongs the hierarchical properties of the single-scale model (14). Indeed, the wave operators macro versus micro 1 and micro 1 versus micro 2 are related by similar sign convention, and the wave velocity in microstructure 1 is affected by the properties of microstructure 2 in a similar way as the wave velocity in macrostructure is affected by properties of microstructure 1. It is seen that higher-order dispersive terms Uxxxx, Uxxxxxx, • • • coincide with those derived from the lattice theory [9] but again mixed derivatives UTTXX , UTTXXXX , • • • reflecting the role of microinertias also enter the equations. The dispersion analysis [6] shows explicitly how the dispersion curves tend asymptotically from one velocity (macrostructure) to another (microstructure). Although the eqs (14) and (16) are asymptotical, the corresponding dispersion curves are close to the exact ones derived from the system (4), (5) and (10), (11), (12), respectively.
209 In case of nonlinear waves, the potential energy W should include also the cubic terms. Such problems are discussed in [10, 11]. T h e balance of nonlinear and dispersive effects in microstructured solids may lead to the emergence of solitary waves. T h e nonlinearity at the macroscale together with dispersive effects results in a symmetric solitary wave while the nonlinearity at the microscale [10] causes t h e emergence of an asymmetric solitary wave [11]. T h e prospects of using special characteristics of waves in microstructured materials briefly described above for Nondestructive Testing (NDT) of materials are now extensively studied (cf also [11]).
Acknowledgements Support from the E S F - P E S C programme NATEMIS and t h e Estonian Science Foundation is gratefully acknowledged. References 1. 2. 3. 4. 5. 6.
7. 8. 9. 10. 11.
R.D. Mindlin, Arch. Rat. Mech. Anal., 16, 51 (1964). A.C. Eringen, J. Math. Mech., 15, 909 (1966). G. Capriz, Continua with Microstructure, Springer, New York (1989). A.C. Eringen, Microcontinuum Field Theories. I Foundations and Solids, Springer, New York (1999). G.A. Maugin, Material Inhomogeneities in Elasticity, Chapman & Hall, London (1993). J. Engelbrecht, F. Pastrone, M. Braun, A. Berezovski, Hierarchies of waves in nonclassical materials. In: P.-P. Delsanto (ed), The Universality of Nonclassical Elasticity with Application to NDE and Ultrasonics, Springer, New York (in press). G.B. Whitham, Linear and Nonlinear Waves, J. Wiley, New York (1974). P.L. Christiansen, V. Muto, and S. Rionero, Chaos Solitons Fractals, 2, 45 (1992). G.A. Maugin, Nonlinear Waves in Elastic Crystals, Oxford University Press, Oxford (1999). J. Engelbrecht and F. Pastrone, Proc. Estonian Acad. Sci. Phys. Math., 52, 12 (2003). J. Janno and J. Engelbrecht, J. Phys. A: Math. Gen. 38, 5159 (2005).
A N ALLELOPATHIC COMPETITION W I T H A N EXTERNAL TOXICANT INPUT
P. FERGOLA AND M. CERASUOLO Dipartimento di Matematica e Applicazioni "R. Caccioppoli" Universita degli Studi di Napoli Federico II Via Cintia, 80126 Napoli (Italy) E-mail: [email protected], [email protected]
An allelopathic competition between two populations of microorganisms, taking place in a chemostat-like environment, is analyzed. The allelochemicals production by one of the two species is supposed delayed. The same allelochemical compound is also introduced as an external input concentration. Meaningful steady-state solutions and their stability properties are analyzed. The survival of the producing species is, in particular, studied in the special case of an exponential delay kernel.
1. Introduction A mathematical model for an allelopathic 1-4 competition between two populations of microorganisms is analyzed when an external input of the same internal allelochemicals, produced by one of the two species, is introduced in the chemostat. Analogous biological scenarios have been considered in some recent works 5 - 7 . In this paper, in particular, we study this problem by assuming that the environment is of chemostat type and, furthermore, that the production process of the internal allelochemicals is delayed. More precisely, by supposing that the competition takes place in an open culture of a biochemical laboratory, we model the delayed production process by means of a linear delayed model of quorum sensing, 2 ' 4 represented by a distributed delay term which takes into account the previous production history 5 . Furthermore, by assuming that the allelochemicals have an inhibitory effect on the susceptible species, we take advantage from the wide laboratory's chances of keeping constant during the experiments temperature, light, pH and the solution density. We represent this allelopathic competition by means of a mathematical model of 4 non-linear functional differential equations. The plan of paper is as follows. In Section 2, basic notations and
210
211 the mathematical model are introduced. Meaningful steady-state solutions and their stability properties are analyzed in Sections 3 and 4 respectively. Finally, Section 5 is devoted to the analysis of an equilibrium ensuring the survival of the producing species in the special case of an exponential delay kernel.
2. The delayed quorum sensing model Let us denote by S(t) the nutrient concentration at time t in the culture vessel; x(t), y{t), the concentration of the sensitive and toxicant producing population respectively, and pit) the concentration of toxicant. The mathematical model is the following one: \S = (S°- S)D - h(S)e~^fx ± = a:[/i(S)e-»,-D] y = y
1-0
p=(p°-P)D
-
j _ <»i{Z)v(t + + 0f2(S)y
f
f2{S)% Z)d£\h{S)-D
Ul($)y(t
(1)
+ Qd£,
J-T
where fi(S) = ™^s are the Michaelis-Menten functional responses of the sensitive and producing populations respectively (i = 1,2), and all the parameters are supposed constant and positive. Moreover, 5° is the constant input concentration of the limiting nutrient; D is the constant washout rate; a» is the half saturation constant, i=l,2; rrii are the maximal specific growth rate of the two populations, respectively, i=l,2; 7 is a measure of the inhibiting effect of the allelochemicals on the sensitive population. Finally, J ° T Ou^yit + £ ) C 0 < f_T 6y{t + 0 ^ i ( £ K < 1, indicates the fraction of potential growth devoted to produce allelochemicals, and wi(£) is the kernel function which weights the past values of this production. In order to reduce the number of parameters the equations are scaled, with the usual scaling for chemostat 8 equations. Specifically, let us set S = SS°, i = 75,
X = XT]IS°, ai = aiS°,
y = yr]2S° p = pS°, a2 = a2S°,
SL
mi = rhiD,m2 = fh2D, 0i= 0m
02 =
^
By substituting the new dimensionless variables and dropping the bars,
212
system (1) can be written as follows:
'S = l-S-h{S)e-^x-f2(S)y x = x[f1(S)e-^ - 1} y = y (l-Oif p= 1-
p
(2)
W!(eM* + 0 ^ ) / a ( 5 ) - l
+ e2f2(S)y
f
u>i(£)y(i + 0 #
We observe that by defining z(t):=(S(t),x(t):y(t),p(t))eU4 and zt{0 = z{t + 0 , i G [-r,0],
for allt > 0,
system (2) assumes the form
m = F(Zt)
(3) 4
with initial conditions at t — 0 given by 0 G C([—T, 0], SR ), where C([—r, 0], 5ft4) is the Banach space of continuous functions mapping the interval [—r, 0] into 5ft4 with the norm
||0||= sup 10(01, Ce[-r,o]
4
and | • | is any norm in 5ft . Moreover, a solution of (3) can be denoted by z(t) = z(<j>,t)
£G[-r,0]
(4)
where 0 is any non-negative initial condition to equation (3) 0(0 >0, £G[-r,0),
with 0(0) > 0 .
By standard procedures qualitative properties of the solutions of system (3) can be proved like positiveness, boundedness, global existence in the future, uniqueness 6 . For sake of brevity we omit the proofs. 3. Equilibria It is easy to prove the following: Theorem 3.1. Let us suppose mi > 1, i — 1,2. System (2) admits the following steady state solutions i. EQ = (1,0,0,1) always;
213
ii. £ i = ( ^ , 1 - ^ , 0 , 1 ) with\1 =
mi^_1,iff1(l)>e';
3 ( Hi. E22 =[S,0, , i?7,V. 1 + 9>if? A2 A!l"i = [",»,a {eiA(T)f ( T ) / ( S ) ) with 2(S)' 2
S =
1 20IA(T)
[l - m 2 + 0iA(T) - 6iA(r)a2
+^461A{T)a2{l
+
+ ^iA(r)) + (-1 + m 2 - 0 I A ( T ) + 0iA(r)a 2 ) 2 1,
A(r>-£ wi(Ode «/
A2 < 1,
with 0-2
(5)
m2 - I Proof. Let us consider the system (l-S-/i(S)e-™c-/2(S)!/ = 0 x[/i(S)e-^-l]=0 2/[(l-0iA(r)j,)/2(S)-l]=O L l - p + 02A(r)/2(5)y2=O
(6)
and the following four different cases linked to different invariant subspaces: i. x = y = 0. In this case we find the equilibrium EQ = (1,0,0,1) which always exists, ii. y = 0. In this case system (6) reduces to 1 - S - fi(S)e-fPx
/!(S)e-™-l = 0 l-p=0
=0
( l - S - x =0
=U /!(
S = (l-Ai) p= l
So we find the equilibrium E\ = (A1; (1 - Ai),0,1 J with Ax that exists if m\e ~*~ 1 ' Ai < 1 that is /i(l)>e
7
214 iii. x = 0. In this case system (6) reduces t o
l-S-f2(S)y
=0
(1 - 0 I A ( T ) S / ) / 2 ( S )
l-p
+ e2A(r)f2(S)y2
»IA(T)
,",=
- 1 = 0
=0
f}= ^
MS)-1
(7)
1 I e2(h(S)-l)2 "•" 0 ? A ( T ) / 2 ( S )
By considering t h e first equation we find ( 0 I A ( T ) - S 0 i A ( T ) ) ( a 2 + S) - m 2 ( S ) + a 2 + S = 0
i.e. 0i A ( r ) S 2 - ( ^ i A ( r ) - a 2 ^ A ( r ) - m 2 + 1 ) 5 - o 2 ( l + 0 I A ( T ) ) = 0. (8) Since ((9iA(r) - a 2 6>iA(r) - m 2 + l ) 2 + 46>i A ( r ) a 2 ( l + ^ ( T ) ) > 0 then, whatever is t h e sign of t h e S coefficient, there exist just one positive solution of (8) which is given by l
20i Mr) A(T) i
m 2 + 0 I A ( T ) - 6>iA(r)a 2 +
+y / 40iA(-r)a 2 (l + 6>iA(r)) + ( - 1 + m 2 - 0 I A ( T ) + e1A(T)a2)2
].
If A2 < 1, it is easy t o prove, by using (5), t h a t S < 1. This implies, due t o t h e first equation of (7) 2 , t h a t S > A 2 . Hence, f2(S) > 1 a n d y > 0. Therefore, we find t h a t E2 there exists. • 4. L o c a l s t a b i l i t y p r o p e r t i e s By means of t h e following change of variables involving t h e generic equilibrium E = (S,x,y,p) where all t h e components are constant and, in particular, S,p are positive, a n d x,y are non-negative
xi = S-S,
x2=x-x,
x3 = y-y,
x4=p-p,
system (2) is transformed into t h e system 'x1 = l-S-x1-fi(x1
+ S)(x + x2)e-^p+Xi)
x2 = (x2 + x) f/i(xi + S)e"l{p+Xi)
- f2(Xl
+ S)(y + x3)
- ll
x3 = (x3 + y) (1 - 0iyA(r)) / 2 ( x i + S) -J
Qiui{0x3(t
- f° X4 = (x3 + y)h (xi + S)J 02Ui{Z)(x3{t + 0 + y)d£
+ £ R / 2 ( * 1 + S)-\ +l-(x4+p). (9)
215 The stability analysis of E can be performed by analyzing the characteristic equation associated to the linearized system of (9) in E: '±1 = {jl_-fi(S)xK-f2(S)y)x1-h(S)KX2-f2(8)x3+1f1(S)xKx4 £2 = fi(S)xnxi + [fi(S)K - l]x2 - 'yxfi(S)nx4 ±3 = f2{S)y{l - 0iyA(r)) xx + [(1 - 0 I S A ( T ) ) / 2 ( 5 ) - l]x 3 +
-vh(S)0iJ
wi(0*3(* + 0«^
±4 = MS)v teMr)*! + f2(S)y92A(T)x3
-x4 + yh{S)e2 !/_r
l ( 0
x3(i + £ K (10)
where K = e 7P. If we represent the right hand side of system (10) as a sum of two terms as follows /•O
x = Lx + J K(Z)x(t + £)d£ where -l-f[(S)xK-fi(5)y ~fi(S)K }[{S)XK h(S)K-l /^(5)S(l-(9iyA(T)) 0
L=
0
/2(S)^02A(T)
-f2(S) 0 (1 - e1yA(r))f2(S) h{S)ye2A{r)
7fi(S)xn -7*/i(S)" - 1 0 (11)
and "0 0 0 0 00 0 0 K(£) = 0 0 - y / ( 5 ) t f ( e ) 0 2 lWl L0 0 yf2(S)62Vi{Z) °. the characteristic equation is given by
pI-L-J
K{0e*d£
= 0
(12)
(13)
where / € 3?4 is the identity matrix. The explicit expression of (13) is -/|(S)i« -/2<S)S(1- «ISA(T)) 2
-/£(S)S 02A(T)
p - ( / I ( S ) K - 1) 0 7*/l(S)K 0 p - [(1 - fliBA(T))/2(S) - 1 - *i(p)] 0 0 -/2(S)5«2 A ( T ) -*2
where
HP) = j
y/2(S)6Wi(£K^
i = l,2.
Local stability properties of boundary equilibria are proved in the following:
216 Theorem 4.1. The following statements hold true i. If Aj > 1 and A2 > 1 then the equilibrium EQ is asymptotically stable; ii. If E\ exists and \\ < A2 then E\ is asymptotically stable. Proof, i. In E0 = (1,0,0,1), K(£) is represented by the null matrix, therefore (14) becomes 1+p +/i(l)e-T +/ 2 (1) 0 p-LAQOe-T-l] 0 0 0 p_[/2(i)_i] 0 0 0
0 0 0 1+p
By computing the roots of the characteristic equation we obtain Pl=~h
f>2 = fl{l)e~T - 1,
P3 = / 2 ( l ) - l ,
P4 = ~l-
Therefore the steady state EQ turns out asymptotically stable if P2<0=S>l>/i(l)e-T p3<0=M>/2(l) i.e. Ai > 1 and A2 > 1. ii. In Ei = ( A I , ( 1 — Ai),0,11, if(£) is represented by the null matrix, therefore (14) becomes l + f'i{\i)xe-i + p /i(A!)e-T +/ 2 (A!) -jfi(Xi)xe^ -/i(Ai)Se"T p 0 + 7 5/i(A 1 )e-T 0 0 -/2(Ai) + l + p 0 0 0 0 1+p
=
With few calculations it is easy to find that the characteristic equation can be written as follows (l + p)(p-(f2C\i)-l))[p2
+ (l + f[(\i)xe-'<)
p + f{CXi)xe^}
= 0 (15)
By computing the roots of (15) we obtain Pi=/2(Ai)-l,
P2 = - 1 ,
P3 = -/i(Ai):re~ T ,
p4 = - 1 .
Therefore, if E\ exists, it is asymptotically stable if / 2 (Ai) — 1 < 0 i.e. if Ai < A2.
D
We note that the stability conditions for both equilibria EQ, Ei are independent of delay.
217 5. Survival of the producing population As we can see from equation (14), in order to study the stability properties of the equilibria ensuring the survival of the producing species, it is necessary to specify the function wi(£)- In this Section, we limit ourself to_ analyze the stability of the boundary equilibrium E2 here denoted by (S,0,y,p). Equation (14) in E2 becomes p + l + f2(S)y +fi(S)K 0 p-(fl(S)K-l) -/£(3)j7(l-0ii7A(T)) 0 -f2(S)f92A(T) 0
+f2(S) 0 P + $I(P) -/2(5)^2A(T) -
$2(P)
0 0 0 p+1
With few calculations it is easy to find that the characteristic equation can be written as follows (1 + p)(p-
(fi(S)K - 1)) [(p+l + ti(S)y)(p + *i(p))+ +/2(S)/2(S)y(l-01yA(r))]=O
that is, (1 + p)(p-
(h(S)K - 1)) [p2 + {f'2{S)y + l)p + &(S)y+ +*i(j>){ft(S)y + l + p)]=0.
(16)
We assume that the delay kernel is given by the following exponential function: Wl(0
= /3eae,
a, /? e R+, £ G [-r, 0].
(17)
Then, it is easy to check that 1 _
e-(a+p)r
*i(p) = vf2(S)01/3
(18) a+p
and A(r) = ^ ( l - e — ) . By substituting in (16) we obtain 3 2 (1 +p)(p( / I ( S ) K - 1)) [p + (f2(S)y + 1 + a)p + {f'2{S)ya + a+ +fi(S) + MS)y9^)p + m)ya + f2{S)y61(3 + f2,(S)f2(S)y2e1(3+ a -e-"T (e~ -•f2{S)y9il3(ti(S)y + 1 + p))] = 0 (19) Prom (19) it can be immediately checked that E2 is unstable if JI(S)K > 1, that is if S > \\. The general analysis of equation (19) is quite complex. Analogously, it
218 appears difficult to study the possible stability switches of this equilibrium. In particular, the Beretta-Kuang method 9 seems to be unsuccessful in this case. In fact, by analyzing the third factor in the left hand side of equation (19) we note that it can be put in the form P(p,T) + Q ( p , T ) e - ^ = 0 where P is the third order degree polynomial P{P,T)
= ps +p2(r)p2
+PI(T)P
+ P0(T)
with delay dependent coefficients PO(T) = ft(S)ya + / 2 (S)yl?i/J + K(S)f2(S)y291, Pi(r) = &(S)ya + a + f^(S) + f2{S)y9xp P2(r) = fi,(S)y + l + a0
(20)
and Q is the first order polynomial Q(P,T)
=qi(T)p + qo(T)
where qo(T) = e-™ h(S)yOM{S)y
+ 1)
,
,
In this case the characteristic equation has time delay dependent coefficients, therefore by using the algorithm shown in 9 , the function F(oj,r):=\P(iu,,r)\2-\Q(iuj,T)\2 with P{IUJ,T)
= (PO(T) - W2P2{T))
Q(iw,
= q0(r) + iqx
T)
+ I{PI{T)UJ -
w3)
(T)U
is given by F(U,T)
= w6 + (P2(r)2 -go(r)2+po(r)2
2PI(T))LU4
+
(2P2(T)PO(T)+PI(T)2
-
q^r)2)^2
(22) Due to the complexity of coefficients (20) and (21) of (22) it seems very difficult, in the general case, to find positive roots of the equation F(w,r)=0.
(23)
Numerical simulations performed with MATLAB, by using numerical values for the parameters deduced by the current literature, have shown that the existence of real positive roots of (23) does not imply any stability switch. Further investigations based on different procedures are still in progress.
219 References 1. S.B.Hsu, P.Waltman, Competition in the Chemostat when One Competitor Produces a Toxin. Jpn.J.Ind.Appl.Math., 15 (1998), 471-490. 2. J.P.Braselton, P.Waltman, A Competition model with dynamically allocated inhibitor production. Mathematical Biosciences, 173 (2001), 55-84. 3. E.M.Gross, Allelopathy of aquatic autotrophs (invited review). Critical Reviews in Plant Science 22, 313-339, 2003. 4. P.Fergola, F.Aurelio, M.Cerasuolo, A.Noviello, Influence of mathematical modelling of nutrient uptake and quorum sensing on the allelopathic competitions. Proocedings "WASCOM 2003" 13th Conference on Waves and Stability in Continuous Media, 2004. 5. P.Fergola, M.Cerasuolo e E.Beretta An Allelopathic Competition Model with Quorum Sensing and Delayed Toxicant Production, Math. Biosci. Engin. Vol. 3, n.l, (2006), 37 50. 6. P.Fergola, E.Beretta e M.Cerasuolo Some New Results on an Allelopathic competition models with quorum sensing and delayed toxicant production, to appear on Nonlinear Anal. Real World Appl. 7. P.Fergola, M.Cerasuolo, A.Pollio, G.Pinto and M. Delia Greca, Algal Allelopathic Competitions: Experiments and Models, to appear. 8. H.L.Smith, P.Waltman, The Theory of the Chemostat - Dynamics of Microbial Competition. Cambridge Studies in Mathematical Biology, 1995. 9. E.Beretta, Y.Kuang, Geometric Stability Switch Criteria in Delay Differential Systems with Delay Dependent Parameters. SIAM J.Math.Analysis, 33, No5, 1144-1165, 2002.
STABILITY AND O T H E R CONSIDERATIONS FOR A NONLINEAR DIFFUSION SYSTEM*
J.N. F L A V I N A N D M . F . M C C A R T H Y Department of Mathematical Physics, National University of Ireland, Galway, Ireland E-mail: [email protected] S. R I O N E R O Departimento di Matematica e Applicazioni Universita degli Studi di Napoli "Federico
"R. Caccioppoli", II", Napoli, Italy.
A nonlinear double diffusive system is the subject of this article: we consider two simultaneous p.d.e.'s in two dependent variables, first order in time and second order in the spatial variables. Dirichlet boundary conditions, independent of time, are assumed. The principal concern of the article is the stability of the steady states together with the convergence of the unsteady states thereto. Novel Liapunov functionals are used to this end. Both linear and nonlinear stability are discussed together with some aspects of (linear) instability. Prior to these considerations, some relevant remarks are made concerning uniqueness and nonuniqueness of the steady states.
1. I n t r o d u c t i o n A nonlinear double diffusive system is the subject of this article: we consider two simultaneous p.d.e.'s in two dependent variables, first order in time and second order in the spatial variables. Dirichlet boundary conditions, independent of time, are assumed. "The authors gratefully acknowledge the work of Dr. P. O'Leary (N.U.I. Galway) in computing the approximations to eigenvalues given in the Appendix. This work has been performed under the auspices of G.N.F.M. of I.N.D.A.M. and M.I.U.R. (P.R.I.N.) "Nonlinear wave propagation and stability in the thermodynamical processes of continuous media". The author J.N.F. wishes to thank G.N.F.M. of I.N.D.A.M. for a visiting Professorship held in Italy in September 2004. The author S.R. gratefully acknowledges the warm hospitality extended to him in 2004 by the National University of Ireland, Galway.
220
221 The principal concern of the article is the stability of the steady states together with the convergence of the unsteady states thereto. (The term stability etc. used in this paper, everywhere relates to perturbations/changes in the initial conditions, with one exception occurring in Section 4.) To this end, we use a Liapunov functional analogous to one used previously in the context of single, nonlinear diffusion equations and allied systems ([1],[2] and [3],[4]). Prior to these considerations, it is deemed appropriate to make some remarks concerning uniqueness and nonuniqueness of the steady states. The organization of the paper is now discussed. Section 2 describes the initial boundary value problem, the steady state problem, and the perturbation (of the steady state) problem. Section 3 focusses on steady states: it contains a uniqueness criterion and discusses some examples of non-uniqueness. Section 4 discusses matters concerning linear stability and instability of steady states: a stability criterion based on a Liapunov fuctional is derived; and, concerning an example of two steady states identified in Section 3, we discuss evidence for the stability of one and the instability of the other. The section concludes with a discussion of constant steady states (corresponding to constant boundary conditions). Section 5 discusses nonlinear stability of the steady states using a functional involving the first spatial derivatives. This functional is an analogue of the functional of the "second kind" arising in a similar context for a single nonlinear diffusion equation [1]. A more extensive and more detailed treatment of the matters dealt with in this paper are available in [5], including proofs of most of the results. The principal results relating to stability etc. are recorded as Theorems and the principal results on other matters as Propositions. General discussions on stability relevant to this article may be obtained in [6],[7]. 2. Formulation, Preliminaries We shall, in general, be concerned with a three-dimensional region V with smooth boundary dV. We shall be concerned with the double diffusive system: T(x,t),S(x,t) are smooth functions satisfying ~
= V • {k(T)VT
+
m(S)VS} in V,
^ = V - { n ( T ) V T + /(5)VS}
(1)
222
where the diffusivities k(T) > 0,£(S) > 0,m(5),n(T) are specified, continuously differentiable functions of their arguments, subject to the boundary conditions T = T ( x ) , 5 = 5(x) on dV
(2)
and subject to the initial conditions T(x,0) = / ( x )
5(x,0)=5(x),
(3)
where T, S, / , g are specified functions. We shall be concerned also with the corresponding steady state system: Denoting the steady states, corresponding to T, S respectively by (7(x), V(x), we have V • {k{U)VU + m(V) W } = 0 mV, V • {n{U)WU + £(V)VV}
(4)
=0
or, equivalently and more conveniently,
f fc(OdC + I o
Jo
rn(V)dV in V,
/
Jo
n(Od£ + / e(V)dr, Jo
(5)
o,
subject to the boundary conditions U = f (x) , V = §{x) on dV.
(6)
(The use of the symbol V to denote both the region and a steady state is not liable to cause confusion.) One expects the unsteady state to tend to a steady state as the time t —> oo in appropriate circumstances. Indeed, our principal concern will be the stability and instability of the steady states (with respect to perturbations in the initial state). To this end, we consider the perturbations u,v defined by u = T-U,
v=
S-V
(7)
and one finds that u, v satisfy «t = V 2 [J0Ufc(£+ U)d£ + / ; m ^ + V)dn] vt = V 2 [/0U n(C + U)d£ + / ; £(V + V)dr1] ,
(8)
223
(where subscript t denotes partial differentiation with respect to this variable) subject to u = v = 0 on dV
(9)
and u(x,0) = / ( x ) - 1 7 ( x )
Xx,O)=0(x)-V(x).
(10)
One context to which the above system (l)-(3) etc. may be relevant is a predator-prey system where the advection velocities are neglected; see [7], for example. The quantities T, S denote the concentrations of predator and prey respectively, and m{S){> 0) corresponds to diffusion of predator towards increasing population density of prey, while n(T)(< 0) corresponds to diffusion of prey away from increasing concentrations of predator. 3. R e m a r k s on U n i q u e n e s s and Non-Uniqueness of S t e a d y States One may prove the following uniqueness criteria (see [5]) Proposition 1. If k(-) > 0 and (.(•) > 0 and either m(-) is uniformly non-negative and n(-) is uniformly non-positive or vice versa, then there is at most one steady state. Proposition 2. Suppose that there exist (integrable) quantities &o(x)(> 0), ^o(x)(> 0),m o (x)(> 0),n o (x)(> 0) such that the following five inequalities hold a.e. H-) > koJ(-) > to, |m(-)| < mo, |n(-)| < n 0 where 4.k0£0 > (m 0 + n0)2, then there is at most one steady state. We now give an example of non-uniqueness: we give a non-trivial, explicit example of two steady states,which can be anticipated from symmetry considerations. Suppose, for the sake of simplicity, that the diffusivities are given by jfc(r) = 1 + r = m{r), n{r) = £{T) = 1.
(11)
224
Suppose, also for simplicity, that we are dealing with a one-dimensional region 0 < x < 1, x being a rectangular cartesian coordinate, and suppose that the boundary conditions are {/ = V = 0,x = 0; U = V=l,x
= l.
(12)
One easily proves that there are two steady states: U = x + v x — x2, V = x — v x — x2 and
(13) 2
2
U = x — \Jx — x , V = x + v x — x . We summarize as follows: Proposition 3. In the context of diffusivities given by (11), and that of the one dimensional rectilineal region 0 < x < 1, with boundary conditions given by (12), there are two steady state solutions [o/(4)] given by (13).
4. Linear Stability, Instability One may readily verify that the linearised version of the perturbation equations (8) are ut = V 2 [k(U)u + m(V)v],
vt = V 2 [n(U)u + £{V)v].
(14)
In order to consider the linear stability etc.of the steady state solution, consider the (Liapunov) functional F
(t) = \f
{{v(HU)u
+ m(V)v)}f
+ {v(n(U)u
+ l{V)v)f]dV.
(15)
Under appropriate conditions, the integrand in (15) is positive-definite in u, v, as may be seen from the following considerations: Since the integrand is plainly non-negative, it remains to prove that the vanishing of the integrand implies that u = v — 0. The vanishing of the integrand together with the vanishing of u, v on the boundary imply k(U)u + m(V)v = 0, n(U)u + £{V)v = 0.
(16)
Assuming k(U)£(V) - m(V)n(U)
+0
(17)
except possibly on the boundary dV (where u,v are, in any case, both zero), then (16) implies that u = v = 0.
225
We summarize: Proposition 4. Assuming the condition (17), the integrand in the functional (15) is positive definite in u, v. One may readily verify that (see [5]) dF_ dt
I
" k(U){\72(ku + mv)}2 + £{V){\j2(nu + £v)}2 dV. + {m(V) + n(U)} V 2 (ku + mv) \/2 (nu + £v)
(18)
Suppose now that k(U) > 0,£(V) > 0, 4k(U)£(V) > {m(V) + n(U)}2,
(19)
(Note, en passant, that (19) implies (17)). The conditions (19) imply that the integrand is positive-definite in \/2(ku + nv) and \j2{nu + Iv). The vanishing of these latter quantities plainly imply that u = v = 0. Hence dF — <0 dt ~ the equality sign occurring if u — v = 0. This may be verbalized as follows: Theorem 1. Assuming the condition (19) ; the steady state U,V is linearly stable with respect to the Liapunov functional F (positive-definite in u,v) in the sense that F(t) is a non-increasing function of t. Remark 1. Let us consider the two steady states given by (13). From the point of view of linear stability etc., the overwhelming likelihood is that the first is linearly stable and that the second is linearly unstable. In the case of the solution (13i) the (sufficient) condition for linear stability (19) is very nearly satisfied everywhere: one finds numerically that the quantity 4k{U)£{V) - {m(V) + n(U)f
= 4 j l + x + {x - x 2 ) 1 / 2 }
-{2
+
(20)
x-ix-x2)1'2)2
is positive everywhere in 0 < x < .9905 (approx.) while it is negative in the remaining, tiny subinterval, apart from at x = 0 (where it is zero). The proof of the theorem strongly suggests that one has linear stability here. Moreover, a Galerkin approximation, up to the third order, exhibits negative eigenvalues in all cases (see Appendix).
226
On the other hand, for the other steady state solution (132) one has the reverse inequality 4k(U)£(V) - (m(V) + n{U)}2 < 0
(21)
everywhere, except at x = 0 (where it is zero). Linear instability is extremely likely here for two reasons: (a) A Galerkin approximation, up to the third order, exhibits at least one positive eigenvalue at each order of approximation (see Appendix). (b) One may choose initial conditions in the circumstances (21) such that dE/dt > 0 in the limit t -» 0. Remark 2. The second state solution (132) of the relevant problem is also unstable in another sense. Suppose the boundary condition (122) is altered to read U = 1 + 5, V = 1 at x = 1, where 5 is a constant, all other conditions remaining the same. A simple calculation gives U = {1 + 5/2)x ± ^ { ( 1 + 5)2 + 1} x/2 - (1 + 8/2)2x2.
(22)
If the - sign is taken it is easily seen that U(l) = 1, i.e. this does not give a valid solution. That is to say, the solution (132) is unstable to perturbations in the boundary condition at x — 1. On the other hand, the solution (22) with a + sign is seen to be a valid solution. With a view to obtaining a linear instability result, and for other purposes, essentially we consider a special case of (14): we consider the system M(X, t), v(x, t) satisfying ut = \j2(DllU
+ D12v),
vt = \72(D21u + D22v),
(23)
subject to u = v = 0 on dV,
(24)
where Dxj are constants.Let Ai be the lowest "fixed-membrane" eigenvalue for the region and X the corresponding eigenfunction (i.e. X satisfies V 2 ^ + XiX = 0 in V, X = 0 on dV).
227
Seek harmonic solutions of (23) of the form (u, v) = (u°, v°)eptX
(25)
where u°,v°,p are constants. Substitution of (25) in (23)-(24) leads to an equation for p of the form: 2p/Ai = - ( A i + D22) ± V(Dn + D22)2 - 4(Z?ni?22 -
D21D12).
Let us now suppose that the linear instability of the steady state problem (4)-(6) is considered, with constant boundary conditions: T, S are both constants. Considering the obvious steady solution U = f,
V = S,
(26)
the linearized perturbation equations (14) have the form (23) with Dn = fc(T), D12 = m(S), D21 = n(f),
D22 = £(§).
Since k > 0,£ > 0 have been assumed, the following linear instability result follows from the result of the previous paragraph: Theorem 2. The constant steady solution U,V (defined by (26)), corresponding to constant boundary conditions (U = T,V = S) is linearly unstable provided that k{T)l(S)
<
m{S)n{T).
R e m a r k 3. One may show (see [5]) that the constant steady solution, corresponding to constant boundary conditions, is linearly asymptotically exponentially stable (with respect to a measure of the L2 type) provided that k{T) > QJ(S) > 0 , m ( f )n(5) < 0. 5. Nonlinear Stability In this section we obtain a nonlinear stability result based on a functional which is an analogue of the Liapunov functional of the second kind used previously (e.g. [1],[2]) in the context of a single nonlinear diffusion equation, inter alia. We consider the non-linear perturbation equations (8) : consider the functional (which is obviously non-negative): E(t) = \ j
[iv(K(u,
U) + M(v, V))}2 + {V(N(u,
U) + L(v, V))}' dV, (27)
228
where K{u,U)=
k{T + U)dr, M(v,V)= Jo
m(Z + V)d£,
(28)
Jo
[ n(T + U)dr, L{v,V)= f e{£ + V)d£. Jo Jo The issue of positive definiteness in u, v of the functional (27) requires one to examine when u = v = 0. Propositions 1,2 are applicable mutatis mutandis. Thus if the conditions on k, I, m, n specified in Proposition 1 or in Proposition 2 hold, then the functional given by (27) is positive -definite in u, v, i.e. in common terminology, the functional is a Liapunov functional. One readily proves that (see [5]) N{u,U)
k(u + U) {sy2(K + M)}2 + g(v + V) {S72{N + L)}2 dV. + {m(v + V) + n{u + U)} y 2 (K + M) y 2 (N + L) (29) One may easily verify that the integrand in (29) is non-negative provided that the conditions on k,l,m,n, specified in Proposition 2 hold (i.e. dE/dt < 0). We thus have a nonlinear analogue of Theorem 1: dE_ dt
Theorem 3. The steady state (specified by (4)-(6)) is nonlinearly stable in the measure (27) provided that the conditions on k, I, m, n specified in Proposition 2 hold, in the sense that E{t) is a non-increasing function of t. Suppose now that, for all values of the relevant arguments, *(0>fci,*(•)>*!.
(30)
|(m(-)+n(-)|<2Cl
where ki,£\ are positive constants and c\ is a non-negative constant, such that hh
> c2;
(31)
and that, in addition, the constant p is defined as follows: P= \
(ki+ *i) - y/{h-*i)2
+ 4ei
for
Cl
^ 0,
(32)
p = minf/a!,^} for C\ = 0. One may prove (see[5]) that the quantity E{t) satisfies E(t) < E(0)exp(-2p\!t)
(33)
229 where Ai is the lowest (positive) "fixed membrane" eigenvalue of the region V; and, moreover, that the foregoing inequality is optimal (i.e. equality is attained therein) when k = kxJ = li and either
(34) m = n = ci or m = n = —c\,
and when the initial values of the perturbation (u, v) coincide with the eigenfunction corresponding to the lowest eigenvalue of the eigenvalue problem arising from the solution form (25). We summarize in the following Theorem 4. In the context of the perturbations (u, v) (specified by (7)(9)), about a steady state, defined by (4)-(6), the functional defined thereon by (27) decays in accordance with (33), where (i) p is defined by (32), (ii) Ai is the lowest (positive) "fixed membrane" eigenvalue of the region, provided (30),(31) hold. Moreover, the inequality (17) is optimal (i.e. equality is attained therein) if (34) holds and if the initial value of the perturbation coincides with the eigenfunction corresponding to the lowest eigenvalue of the eigenvalue problem arising from the solution form (25). Corollary. The steady state U, V is nonlinearly, exponentially asymptotically stable with respect to the Liapunov functional (27) (in the sense that (33) holds), provided that k(-),i(-),m(-),n(-) satisfy the conditions k>
h,e>£i
where k\,£i are positive constants, and where either (a) m(-) is uniformly non-negative and n(-) is uniformly non-positive or vice versa, and \m + n\ < 2c\ where c\ is a non-negative constant, together with (31), or (b) \m\ < mi,jn| < nowhere m\,ni are non-negative constants and mi + rii <• 2ci, together with (31). (The cases (a), (b) are related to Propositions 1,2 respectively and ensure inter alia that the functional (27) is positive-definite in u,v : in common terminology it is a Liapunov functional).
230
Remark 4. Theorem 4 says essentially that (under the conditions stated) a perturbed state of a given steady state evolves "rapidly" to this,or possibly, to some other steady state (if such exists). On the other hand, conditions are given in the Corollary that imply positive-definiteness in u, v of the functional E(t), and nonlinear exponential asymptotic stability of the (unique) steady state, i.e. a perturbed state returns "rapidly" to the (only) steady state.
5.1.
Appendix
The "normal mode solutions" of (14) subject to zero boundary conditions are such that (u,v) = (u(x),v(x))ext,
(Al)
where A is a constant, and where attention is confined, in the present context, to a one-dimensional rectilineal region 0 < x < 1, and where £, [k(U)u + m{V)v\ = \u, (A2) £ i [n(U)u + £(V)u] = Xv, subject to u = v = 0 on x = 0,1.
(A3)
We confine attention to diffusivities given by (11) - where one has two steady states given by (13). One may obtain Galerkin approximations to the eigenvalues of (A2),(A3) by writing n
n
u = y^ Ai sin(J7r;r), v = ^ , ®i sin(i7rx), where Ai, Bi are constants, corresponding to the nth order approximation. One multiplies each of the equations (A2i),(A22) by sin(J7rx) and integrates between x = 0, x = 1, and thereby obtains a 2n x 2n determinantal equation determining the nth approximation to the eigenvalues, together with the associated amplitude terms A.,B.. Linear stability is characterized by uniformly negative eigenvalues, or eigenvalues with uniformly negative real parts; on the other hand, linear
231 instability is characterized by at (least) one positive eigenvalue, or (at least) one eigenvalue with positive real part. T h e steady state (13i) gives rise t o the following approximate eigenvalues (all eigenvalues recorded exclude a multiplicative factor 2n2): 1st approximation: —2.57, —.250; 2nd approximation: - 1 0 . 3 6 , - 2 . 5 2 , - 1 . 1 6 , - . 2 4 9 ; 3rd approximation: - 2 3 . 4 2 , - 1 0 . 1 8 , - 2 . 8 3 , - 2 . 3 8 , - 1 . 1 6 , - . 2 2 9 , all associated with linear stability. On the other hand, the steady state (132) gives rise to the following approximate eigenvalues: 1st approximation: —2.44, .263; 2nd approximation: - 9 . 7 6 , - 2 . 4 3 , - 2 6 3 , 1 . 2 1 ; 3rd approximation: - 2 2 . 0 1 , - 9 . 6 7 , - 2 . 4 3 , .240,1.21, 2.84, these being associated with linear instability because of the presence of at least one positive eigenvalue. All the values given above have been rounded off to three significant figures. References 1. Flavin J.N. and Rionero S., Asymptotic and other properties for a nonlinear diffusion model. J. Math. Anal. Appl. 1998; 228:119-140. 2. Flavin J.N. and Rionero S., Some Liapunov functionals for nonlinear diffusion and nonlinear stability. Proceedings "WASCOM 99" 10th Conference on Waves and Stability in Continuous Media, World Scientific, Singapore etc. 2001. 3. Flavin J.N. and Rionero S., The Benard problem for nonlinear heat conduction: unconditional nonlinear stability, Quart. J. Mech. Appl. Math. 1999; 52: 441-452. 4. Flavin J.N. and Rionero S., Nonlinear stability for a thermofluid in a vertical porous medium. Contin. Mech. Thermodyn., 1999; 11:173-179. 5. Flavin J.N., McCarthy, M.F. and Rionero S., Stability and other Considerations for a Nonlinear Double Diffusive System. (To appear) 6. Flavin J.N.and Rionero S, Qualitative Estimates for Partial Differential Equations: An Introduction. CRC Press, Boca Raton, Florida, 1996. 7. Straughan B, The Energy Method, Stability, and Nonlinear Convection. Springer, New York, Second edition, 2004. 8. Okubo A. and Levin S.A., Diffusion and Ecological Problems, Modern Perspectives, Second Edition, Springer, 2001.
GENERALIZED RAREFACTION WAVES A N D R I E M A N N PROBLEM FOR A CLASS OF DISSIPATIVE H Y P E R B O L I C MODELS
D. FUSCO AND N. MANGANARO Department of Mathematics University of Messina Salita Sperone 31 98166 Messina, Italy. E-mail: [email protected] E-mail: [email protected]
Riemann problems for a class of 2 x 2 hyperbolic systems involving source-like terms are considered. By means of reduction techniques it is shown that the searched rarefaction wave is given by an exact solution which, in fact, represents a deformation of the rarefaction wave admitted by the corresponding homogeneous governing model. Hence the solution of the Riemann problem of interest is provided.
1. I n t r o d u c t i o n Within the theoretical framework of shock waves, the Riemann problem represents a subject of prominent interest. The main contributions given to this topic concern one dimensional wave motions governed by hyperbolic homogeneous systems of conservation laws of form
where U £ RN, F(U) £ RN, while t and x denote time and space coordinates, respectively. As well-known (see Refs. 1, 2 and references therein quoted) for the model (1), under the assumption of not large initial jumps, the Riemann problem admits an unique solution consisting in a superposition of shocks, contact discontinuities, rarefaction waves and constant states. On that concern it is noteworthy that the structure of model (1) permits to determine the rarefaction wave under the form of a self-similar simple wave solution depending upon the variable z — f.
232
233
Denoting by d the right characteristic eigenvector of the matrix VF, (V — g|j), associated to the eigenvalue A(u), the physical shocks are admissible (see Ref. 3): if
VA-d^O,
if
V A - d = 0,
A(u0) < s < A(ui) <^=>- 7] > 0 A(u0) = s = A(ui) ^=j> r\ = 0
(Shock),
(Characteristic Shock),
while if X(VLI) < A(u0) we have a rarefaction wave. Here s is the shock velocity, 77 is the entropy production across the shock wave while uo and ui denote, respectively, the states behind and ahead the shock front. These results are no longer valid if the local exceptionality condition VA • d = 0 holds for some u. In this case the stability of the shock is guaranteed by means of the Liu 4 " 5 conditions; that implies the generalized Lax condition A(uo) < s < A(ui). Nevertheless the entropy growth is not sufficient for the admissibility of the shock: it is necessary to add a new superposition principle for the shocks (Liu & Ruggeri 6 ). In a different way from the model (1) there is not a general method to solve analytically the Riemann problem for a hyperbolic system of balance laws of form dU 0F(U) „/TT, Actually, because of the source-like term B(U), rarefaction waves given in terms of a self-similar solution and constant states are not in general admitted by the model (2). The main contributions to the problem under interest were based mainly on numerical approaches (see Refs. 7, 8, 9, 10). Some analytical results were obtained for scalar equations (see Refs. 11, 12, 13) or for a 2 x 2 system of balance laws with a Dirac measure sink term (see Ref. 14). The main aim of this work is to give a contribution in the understanding of Riemann problem for non homogeneous models. First we are concerned with a simple example of balance laws for which a suitable reduction procedure allows to transform a given Riemann problem for the original governing non homogeneous system to a Riemann problem for an associated homogeneous model. A naive analysis of the dissipative wave process considered in section 2 shows that the shock amplitude decays in time along with an exact solution which represents a deformed rarefaction wave of the corresponding non homogeneous governing model. In line with these results and according to the investigation worked out in Ref. 15, later in section 3 we outline a possible approach to determining
234
the analytical solution of the Riemann problem for non homogeneous models. The leading idea of the present investigation is that in presence of a source term one would expect the searched solution to be a "deformation" of the corresponding solution of the associated source free case. 2. An example of 2 X 2 system Let us consider the following toy model of a p-system with source terms: vt-ux
= -k0av
(3)
ut + p'{v)vx = -a (fc0 - 1) u
(4)
where p(v) = CQV 1 and fco, a, Co, 7 are constants. Hereafter the prime denotes ordinary differentiation. The model (3), (4) is hyperbolic if c 0 7 > 0. Here our aim is to solve the Riemann problem
{
UR
for x > 0
(
;
v(x,0)=
for x > 0
VR
I
(5)
for x < 0 ( vL for x < 0 Provided that 7 ^ —1, it is simple matter to ascertain that under the variable transformation v = k1vekoat u = kxue^0-1^ (6) UL
r = - (eat - 1) the equations (3), (4) adopt the homogeneous form vT - ux = 0 uT+p'{v)vx=Q
(7) (8)
where p(v) = Civ~"y, c\ being a constant, and, in turn, the initial conditions (5) reduce to
{ {
U
R — k\UR
for x > 0 ;
= k\UL
for x < 0
VR = kiVR
for x > 0
vi — kiVL
for x < 0
UL
(9)
(10)
235
along with the relations c\ \ ""+1 (11) 7+ 1 \c0, Moreover the hyperbolicity of (7) and (8) requires C17 > 0. The pair of equations (7) and (8) is a homogeneous p-system for which the solution of the Riemann problem (9), (10) is well known (see Refs. 2, 16). In the following we assume p"{v) > 0, (i.e. 7 > —1) whereupon the solution is obtained by means of the rarefaction curves kQ
ki=
••
Ri :
u = uL+
R2 :
U — UL
yj-tfiy) I
dy;
V-P'(V) dy;
v > vL;
(12)
V
(13)
L > v,
JVT.
associated, respectively, to the eigenvalues A^) = — ^/—p'(v), \/—p'(v) and of the shock curves
X^
Si:
u = uL-
\/(v-vL)(p(vL)
-p(v);
vL > v;
(14)
S2:
u = uL-
\/(v-vL)(p(vL)
-p(v);
v>vL;
(15)
where the shock velocity is given by u •
UL
V
-VL
(16)
It can be proved (see Ref. 2) that the curves Ri and Si have a second-order contact at the point {VL,UL) as well as do the curves R2 and 52, so that they divide the (v,u) plane in four regions (see figure 1). Thus, depending
Si
/
S2
2.6
0.5
1
1.5
2
2.5
3
V
Figure 1.
Rarefaction and shock curves of (7) and (8).
on which region of the plane {v,u) the point
(VR,UR)
lies, the solution of
236
the Riemann problem (9), (10) is determined as superposition of rarefaction and/or shock waves (see Refs. 2, 16). In order to solve the problem (5) we are dealing with, we first notice that to the constant state (VL,UL) of the homogeneous system (7), (8) there corresponds, through the variable transformation (6), the non constant state vi(t) = vLe-koat;
n{t) = uLe{l-ko)at
(17)
of the non homogeneous model (3), (4). Next, it simple to verify that by using (6), the rarefaction curves (12), (13) transform, respectively, to i?i :
u = ui+
I y/-p'(y)dy;
v > vf,
(18)
vt > v;
(19)
Jvi
R2:
u = ui~
\J-p'(y)
dy;
Jvi
while the shock curves (14), (15) assume the form Si :
u = ui-
\/(v-vi){p(vi)
-p(v);
S2 :
u = ui - y/(v - vi)(p(vi) - p(v);
vi > v;
(20)
v > Vi;
(21)
where the shock velocity is given by u-ui
(22)
v - vi A straightforward calculation shows that also the curves i?i and Si as well as the curves R2 and S2 have a second order contact at the point (vi,ui), so that at t kept fixed they divide the (v, u) plane in four regions and the same considerations given for the homogeneous case also hold in the present non homogeneous case. By way of illustration, let us consider the initial values (VR,UR) such that the corresponding initial conditions (VR,UR) belong to the part I of the (v, u) plane (see figure 1). In such a case the solution of the Riemann problem is given by a superposition of the rarefaction wave associated to (18) and of the shock wave related to (21) (see Ref. 2). As far as the non homogeneous model at hand is concerned, use of the variable transformation (6) and of relationships (18) and (21) provide the corresponding exact solution to the Riemann problem (5): ' u(t) =
uLe^~k^at
v(t) =
koat
if vLe~
j ^ i < -v71^7^)
(23)
237
' u(v,t)^TiLe(i-ko)at
jvr^-)dy
+
if -
<
V-P'&L)
^ T
< -VW&j
(24)
< se~at
(25)
^°*eatY&
V(x t) = (
u(t) =
uie^-k^at
v(t) =
koat
if
- V-P'&i)
< ^
v1e-
u(t) = uRe(-1~k°^at ;f
if oT o~ knat
-B0-at
se-'tK^^
e«'-l
(26)
. v(t) = vRe
where the constants u l 7 v\ are defined in terms of UR, VR, UL, VL through the relations ui=uL+
I
y-p'{y)
dy;
vL < vi
(27)
Jvj,
and UR = Ui- y/(vR - vi)ip{vi) -p(vR);
VI < vR
(28)
while the shock velocity s is given by s= - ^ ^ e
a t
> 0 .
(29)
VR-Vi
The regions of the (t,x) plane associated to the solution (23), (24), (25), (26) are indicated in figure 2. Owing to (25) and (26), the shock tickness associated to the field variables u and v is given, respectively, by {u1-uR)e^-ko^at
(30)
{vi-vR)e-k°ai
(31)
and by
238
Figure 2.
Characteristic funs and shock lines of (3) and (4).)
A direct inspection shows that both the shock ticknesses (30) and (31) decay in time if the structural parameters a and fco fulfil the following conditions a > 0;
fc0
> 1
(32)
a < 0;
fc0
< 0.
(33)
or, alternatively,
It is straightforward to ascertain that if the source—like term involved in (3), (4) is subjected to the conditions (32) or (33), then the resulting governing hyperbolic model is dissipative. Since in the case under investigation p"(v) > 0, owing to the relationships (11) the conditions (33) cannot hold. In a similar way, taking into account relations (23) and (25) along with (32) it is possible to point out a time decay of the rarefaction wave amplitude. In passing we remark that being a > 0, the shock speed given by (29) grows in time. 3. Generalized rarefaction waves A crucial point in investigating Riemann-like problems for dissipative hyperbolic models is to determine an exact solution playing the same role of the rarefaction wave as in the source—free case. The analysis worked out in section 2 makes evident that the rarefaction wave-like solution of the dissipative governing model in point results to be a deformation of the rarefaction wave of the corresponding homogeneous case, as one would expect because of the role played by source-like terms in wave processes. That was in fact the leading idea of the reduction approach developed in Ref. 15.
239 In order to illustrate the main steps of the proposed strategy we limit ourselves to consider the system 3FW(U)
3F(U)
|
= B ( U )
(34)
dx
at where U =
; F°(U) =
Lu h(v)
;B(U)
F(U) =
—ru
(35)
f(v)
In (34), (35) the constants L and r mean for inductance and resistance, respectively; u(x,t) indicates the electric current while v(x,t) is the voltage. Moreover the material response functions g(v) = ^ and f(v) represent capacitance and conductance, respectively. The system (34) and (35), which is widely used for describing nonlinear transmission lines (see Ref. 17), with Lg(v) > 0 turns out to be strictly hyperbolic. Here the characteristic speeds and the corresponding right eigenvectors of matrix coefficient V F with respect to VF^ 0 ' involved in the system (34), (35), are given by A
y/Lirfv)'
dW,
=UW-
(36)
In the source free case (r = f(v) 0), the equations (34) and (35) specialize to the p-system. In order to construct suitable nonconstant solutions of (34), (35) which satisfy a Riemann problem, we modify the standard simple wave ansatz U = U(z(x,i)), (z(x,t) wave variable), for homogeneous system as follows U=U(t,z(x,t)).
(37)
The insertion of (37) into (34) yields: L v F + ZiVF^ju^B-VFC'Ut.
(38)
Along the lines of the approach developed in Ref. 18, for determining generalized simple waves to nonhomogeneous models, in order to satisfy (38) we assume: VF(0)Ut = B
(39)
Uz = ad(U)
(40)
zt + X(U)zx = 0
(41)
where A = A^) and d = d ' ^ , with a being an arbitrary factor. In the process the wave function z(x,t) is determined by the equation (41).
240 A direct inspection shows that the consistency of the equations (39) and (40) requires the following stuctural condition to be fulfilled
The equation (42) (or its integrated form) must be obeyed by the response functions g(v) and f(v) in order that the present approach holds. Hence it represents a mathematical tool for selecting suitable forms of capacitance and conductance functions of interest in modeling shock wave problems. Owing to (39) and (42), the integration of (40) yields + c0e"*
L\{v)
/
(43)
where CQ is an arbitrary constant. In order to get (43) use of the first component equation in (39) has been done, while the second component of (39) gives -
= ^
Ot
(44)
g(v)
^
Once f(v) and g(v) are specified according to (42), the integration of (44) and later of (41) determines v(t,z(x,t)) and z(x,t). In view of studying rarefaction or compression waves within the present framework, we require the solution at hand to satisfy the Riemann problem
{
ui
for x < 0
(
;
v(x,0)=
I
VL
for x < 0
(45)
UR for x > 0 [ VR for x > 0 where UL, UR, VL and VR are constants. Because of (45), the integration of (41) and (44) by means of the method of characteristics, with u given later by (43), provides the solution in the different regions indicated below: i) when x < \(vi{a))da (46) Jo
thus
(47)
241 with CL
=uL
\J
LX(V))^VL
(48)
and vi(t) defined (implicitly) by (47)2 ii) when /" X(vi{(i))da < x < f \{vr{a))dcr Jo Jo
(49)
thus
f« = / z f t l + <*«-** (50) with vo constant; Hi) when / Jo
X(vr(a))da <x
(51)
thus u = ur(t) = (f jgfa)
_
+cRe
I' = uRe- f t . (52)
, JJVR OB
'
/n<x) (<J)
with
c
*^-(/zx^Ls-
(53)
and vr(t) defined implicitly by (52)2Next by expecting the solution at hand to vary smoothly from region (46) to region (51) so that it is continuous in all the fan, taking (47)2, (50)2, (52)2 into account, we require UR=UL
+
rv*
J
dv
(54)
We remark that the state defined by (52) lies on the curve u = ui +
fv
dv
(55)
To the curve (55) belong all the states (u,v) which can be connected to (ui,vi) by means of a generalized rarefaction wave. Of course, different
242
rarefaction curves are related to the caracteristic speeds A^ ) and X^+\ respectively. Because of (49), when A = A^+) the condition X(vi) < X(vr)
(56)
g{v{) > g{vr).
(57)
must hold so that we have
In the same way for A — \ ^ there follows g{v{)
(58)
The model solution at hand represents a generalized rarefaction wave connected to two shock states. 4. Conclusions and final remarks The main difficulty in developing any analytical approach to solve Riemann-like problems for a governing model of form (2) is that the source term B(U) in general does not permit to represent a rarefaction wave by means of a self-similar simple wave solution depending upon z = | as in the source free case. By means of a strict approach based upon the use of an appropriate variable transformation in section 2 a Riemann problem for a dissipative p-system was solved. In line with the results therein obtained, later in section 3 there was outlined a reduction method for determining an exact solution of a 2 x 2 system representing generalized rarefaction wave whereupon a given Riemann problem was solved. Although the analysis was limited to a model of wide use in several fields of application, nevertheless the leading lines of our approach can also define a possible theoretical framework to develop a strategy to investigating Riemann problems for quasilinear nonhomogeneous hyperbolic systems of balance laws. Acknowledgments This work was partially supported by MURST, Progetto di Cofinanziamento 2003 "Nonlinear Mathematical Problems of Wave Propagation and Stability in Models of Continuous Media" and by Fondi del Programma di Ricerca Ordinario 2002 (PRA 2002) "Metodologie di Riduzione per Modem" Non Lineari di Evoluzione" of University of Mussina.
243
References 1. C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Springer Verlag, Berlin (2000). 2. J. Smoller, Shock waves and reaction-diffusion equation. Spinger-Verlag (A Series of Comprehensive Studies in Mathematics), 258, Berlin (1983). 3. P.D. Lax, Shock Waves and Entropy, Contributions to Functional Analysis, 603-634, Ed. E.A. Zarantonello. New York. Academic Press, (1971). 4. T.-P. Liu, J. Differential Equations, 18, 218-234 (1975). 5. T.-P. Liu, The Admissible Solutions of Hyperbolic Conservation Laws, Memoir of AMS, 240 (1981). 6. T.-P. Liu and T. Ruggeri Entropy Production and Admissibility of Shocks, Acta Math. Appl. Sinica (to appear). 7. P. G. LeFloch, An introduction to nonclassical shocks of systems of conservation laws, in An introduction to recent developments in theory and numerics for conservation laws, Springer, Berlin, 28-72, (1999). 8. G. Russo, Central schemes and systems of balance laws, Hyperbolic partial differential equations (Hamburgh, 2001), Vieweg, Braunschweig, 59-114, (2002). 9. E. F. Toro, Riemann solvers and numerical methods for fluid dynamics, Springer-Verlag (1999). 10. E. F. Toro, Proc. R. Soc. London A 458, 271-281, (2002). 11. T. P. Liu, Proc. Amer. Math. Soc.,71, 227-231, (1987). 12. C. Dafermos, Characteristic in hyperbolic conservation laws, Nonlinear analysis and mechanics, 1, Knops R. J. ed., Research Notes in Math. No. 17, Pitman, Boston (1977). 13. C. Sinistrari, SIAM J. Math. Anal, 28, 1, 109-135, (1997). 14. H. Holden and N. H. Risebro, Siam J. Math. Anal. 30 (3), 497-515, (1999). 15. D. Fusco and N. Manganaro, Riemann problem for hyperbolic non homogeneous models, preprint. 16. N. N. Janenko and B. L. Rozdestvenski, Systems of quasilinear equations and their Applications to Gasdynamics, American Mathematical Society, Providence, Rhode Island, 55, (1983). 17. H. M. Leiberstein H. M., Theory of Partial Differential Equations, Academic Press, New York (1972). 18. D. Fusco and N. Manganaro, A reduction approach for determining generalized simple waves, preprint.
A N EQUILIBRIUM P O I N T REGULARIZATION FOR T H E CHEN SYSTEM *
G. G A M B I N O , M. C. L O M B A R D O , M. S A M M A R T I N O Dept. of Mathematics, University of Palermo Via Archirafi 34, 90123 Palermo, Italy E-mail: {gaetana} ,{lombardo} ,{marco} @math.unipa.it
This paper addresses the control of the chaotic Chen system via a feedback technique. We first present a nonlinear feedback controller which drives the trajectories of the Chen system to a given point for any initial conditions. Then, we design a linear feedback controller which still assures the global stability of the Chen system. We moreover achieve the tracking of a reference signal. Numerical simulations are provided to show the effectiveness of the developed controllers.
1. Introduction Over the last two decades, the scientific interest in the study of chaos essentially focused onto two topics: the use and the control of chaotic systems 9 . Then, a wide class of control techniques has been developed to eliminate chaos in dynamical systems 2 , n . Moreover, the procedure of chaotification or anticontrol of chaos 3 has been applied to make chaotic a nonchaotic system and utilize the created chaos in engineering applications. Through the chaotification of the Lorenz system in its nonchaotic regime, a new chaotic system arises, the Chen system 4 . The mathematical structure of these two systems appears similar, but they are not topologically equivalent as the Chen system shows a more complex dynamical behavior 5 . In this paper the chaotic behavior of the Chen system is controlled via a feedback technique. A nonlinear feedback controller is designed such that the trajectories of the corresponding closed-loop system are globally driven to any given point of the form (xr,xr,b~1x^), where b is a parameter of 7 the system. This controller, proposed in for the Lorenz system, is robust "This paper is supported by the MURST under the grant PRIN: "Nonlinear Mathematical Problems of Wave Propagation and Stability in Models of Continuous Media 2003-2005"
244
245
with respect to changes of the parameters system, but it is a nonlinear function, which increases feedback control implementation costs. Then, in the same spirit of 1, we propose a strategy which makes the control linear. The obtained linear control preserves the same robustness and global stability properties of the nonlinear one. By using a similar technique, a state feedback controller is designed, such that one of the state variables of the Chen system can track a given signal. The same approach can be successfully applied also to the whole family of the generalized Lorenz system 6 . The plan of the paper is the following: in Section 2 we briefly introduce the Chen system and its properties and design the global nonlinear control. In Section 3 the linear control is proposed and the proof of the global stability is obtained by defining a Lyapunov function. Finally, in Section 4 the tracking of a given signal is achieved.
2. Nonlinear feedback control The Chen system is 4 :
. . . . ~ -- a{y - x) - (c — a)x — xz + cy (1) - xy — bz where a, b, c are positive real constants. It is obtained by the chaotification of the Lorenz system through a linear state feedback anticontroller 5 . If b (2c — a) > 0, the system (1) has three equilibrium points EQ = (0,0,0); E± = (±y/b(2c — a),±y/b(2c — a), 2c —a), which are all unstable for a = 35, 6 = 3, c = 28 and a chaotic attractor arises 4 . We use the feedback technique proposed in 7 for the Lorenz system, to regulate globally the trajectories of the Chen system to a point. By applying the control input u = (a — c)x + xz — cy — a(y — xr) in the equation for the state variable y, the Chen system is transformed into the closed-loop system: , H, . , . . . | 2i{x - xr) = -a(x — xT) + a{y - xr) •xr) = -a(y-xr) (2) _ i = xy — bz Obviously, x and y converge to xr and it can be easily proved that z converges to b~1xr 7. Notice that the two nontrivial unstable fixed points of the uncontrolled Chen system E± are of the form (xr,xr,b~1xr'). Numerical simulations, illustrated in Figure 1, show that the system trajectories are globally stabilized to the equilibrium point E+.
246
f
(a) (b) Figure 1. (a) Convergence to the point for different initial conditions, (b) The controller starts to work at t = 2.
3. Linear feedback control The above designed controller is not easy to implement, since it is nonlinear and it requires the exact measurements of all the state variables x, y and z. Hence we design a linear controller, which preserves the properties of stability and robustness of the linear one, but requires the measurement of the state y only. By using the strategy proposed in 1 , we propose the following linear feedback controller:
u = -f2,e-o.(y-xr), where f2,
e
(3)
is an estimate of f2 — {c — a)x — xz + cy such that:
ke = r~xe = T-\h - / 2 , e ) = T-\y - « - / 2 , e ) , (4) where r is a positive parameter. Since we have supposed to know only the measurement of y, the above form (4) of the estimate / 2 , e can not be implemented. We introduce the variable w = T/2 ) C — y so that the equation (4) can be written as: w = -uwhere:
f2,e,
h,e =r
w(0)=wo, 1
{w + y)
(5) (6)
and the feedback controller is defined by the formula (3) associated with the equation (6). This new controller is linear and its implementation requires only the measurement of the state variable y. Let us prove the following: Theorem 3.1. The controller (3), together with (5) and (6), globally regulates the Chen system to any given point of the form (xr,xr, b~1x^), VT > 0.
247
The governing equations of the controlled Chen system are: ' x = a(y — x) y = (c-a)x-xz + cy~ ^(w + y)- a(y - xr) z = xy — bz ^ ' xr) Kw = a(yWe want to prove that this system stabilizes to any given point with the form E(xr,xr,b~lxl,T[(2c - a - r _ 1 ) x r - 6 _1 x^]), i.e. equivalently that the system arising by the coordinate transformation: £i=x-xr; £4=w-
£,2=y~xr; T ((2c
- a-
T~1)XT
z-b~lxl;
£3 = -
b~xxl).
stabilizes to the origin. We fix a = 35, b = 3 and c = 28 and define the function V : R4 -> R: a ar It is positive definite and its time-derivative along the system trajectories: ^ = - ( ( « - c + ^ ) ^ + ( « - c + r-1)e22 + 6 ( 6 - | 6 )
2
)
0)
4
is less or equal to zero for all £ = (£1 ,£2,£3,£4) e R . Moreover, the set: S = {£ € R 4 : V = 0} = {(0,0,0,£ 4 ),£ 4 € R] (10) does not contain any nontrivial system trajectories. Being the hypotheses of the Krasovskii-La Salle invariance principle 10 all satisfied W > 0 and \/xr the Theorem 3.1 is proved. Numerical simulations, illustrated in Figure 2, show the global stabilization of the Chen system to the equilibrium point E+ (7.9373,7.9373,21). 4. Tracking We design a linear feedback controller such that one of the state variables tracks a given signal defined by a function yr(t) by using the same strategy of the previous section. Let us assume that yr{t) is a bounded function, with bounded first and second derivatives. Let us define the following change of variables: m = x - yr m = y - yr - aTxyr (11) and choose the driving signal u of the form: u = ~h,e + Vr + a"1^ - a(y - yr - a~~V), (12) where / 2 , e is defined as in the previous section. Then the controller is composed by the feedback function (12) and the estimator (6). If we differentiate the equation for the estimation error, we obtain: e = h-f2,e.
(13)
248
lw
i
(a) (b) Figure 2. x(0) = 15, j/(0) = - 1 0 , z(0) = 37. The controller starts to work at t = 2. The regularization of the Chen system holds independently on initial conditions of w: in numerical simulations showed in figures (a) w(Q) = T / 2 ( 0 ) — 2/(0), in figure (b) it is completely arbitrary, (a) T = 0.1; (b) r = 100, u;(0) = 100.
Being / 2 , e — Tj e by definition and indicating / 2 = $(77, e), where 77 = (771,772,2), the above given equation (13) writes: rfe = - e + $(r7,e), (14) then the controlled Chen system is governed by the following equations: ' TJI = 7J2 =
a(T]2 e -
-
771)
A772
z = xy — bz Tfi = —e + $(77, e)
(15)
This system can be considered as a singularly perturbed system 8 , where Tf plays the role of the perturbation parameter. The fast subsystem e = —e, obtained rescaling time t' = t/rj and imposing r/ = 0, is globally asymptotically stable about e = 0. Furthermore, from the slow subsystem: 'Vi
=a(?72-77i)
7?2 = -0772
(16)
z = xy — bz obtained by imposing 77 = 0 in the last equation of the system (15), it can be easily seen that 771 and 772 converge to zero, i.e. x converges to yr and 772 converges to yr +a~~lyr. Applying Lemma 2.1 in 7 to the equation for z we obtain that z converges to yr(yr + a _1 2/r)- AH the hypothesis of Theorem 3 in 8 are satisfied, then there exists Tfmax > 0 such that V77 G (0, r/ m o x ) the controlled Chen system (15) is globally asymptotically stable about (0,0, b~1(yr + a _ 1 y r ) , 0 ) . The tracking is, then, achieved Vr/ G (0,r/ moa .).
249 Numerical simulations, illustrated in Figure 4.a, show t h a t t h e state variable x tracks the given signal yr{t) = sm(t). In Figure 4.b it is shown t h a t t h e other state variables remain bounded.
t
t
(a) (b) Figure 3. a = 35,6 = 3,c = 28,-ry = 0.1. x(0) = 15, j/(0) = -10, z(0) = 37 and tu(0) = 84. (a) State variable x. (b) State variables y and z.
References 1. Alvarez-Ramirez J., Cevantes I., Femat R., An equilibrium point stabilization strategy for the Chen system, Phys. Lett. A, 326 (2004), 234-242. 2. Chen G., Controlling Chua's global unfolding circuit family, IEEE Trans. Circuits Syst.-I, 40, (1993), 829-832. 3. Chen G., Lai D., Anticontrol of chaos via feedback, Int. J. Bifur. Chaos, 8, (1998), 1585-1590. 4. Chen G., Ueta T., Yet another chaotic attractor, Int. J. Bifur. Chaos, 9 (7), (1999), 1465-1466 5. Chen G., Ueta T., Bifurcation analysis of Chen's equation, Int. J. Bifur. Chaos, 10 (8), (2000), 1917-1931. 6. Gambino G., Lombardo M. C , Sammartino M., Global linear feedback control for the generalized Lorenz system, Chaos, Solitons and Fractals, in press. 7. Gao F., Liu W. Q., Sreeram V., Teo K. L., Nonlinear feedback control for the Lorenz system, Dynamics and control, 11, (2001), 57-69. 8. Hoppensteadt F., Asymptotic stability in singular perturbation problems. II: Problems having matched asymptotic expansion solutions, J. Differential Equations, 15 (1974), 510-521. 9. Ott E., Grebogi C , Yorke J. A., Controlling chaos, Phys. Rev. Lett., 64, (1990), 1196-1199. 10. Sastry S., Nonlinear systems, Analysis, Stability and Control, Springer, (1999). 11. Yassen M. T., Chaos control of Chen chaotic dynamical system, Chaos, Solitons and Fractals, 15, (2003), 271-283.
NONCLASSICAL S Y M M E T R Y R E D U C T I O N S OF T H E CALOGERO-DEGASPERIS-FOKAS EQUATION IN ( 2 + 1 ) DIMENSIONS.*
M.L. G A N D A R I A S A N D S. S A E Z . Departamento de Matemdticas, Universidad de Cadiz, PO.BOX 40,11510 Puerto Real, Cadiz, Spain E-mail: [email protected]
In this paper we consider a (2 + l)-dimensional integrable Calogero-DegasperisFokas equation. We apply the nonclassical method in order to obtain new symmetry reductions. From these partial differential equations in (1 + 1) dimensions by further reductions, we get second order ordinary differential equations. These ODE's provide several new solutions; all of them are expressible in terms of known functions, some of them are expressible in terms of the third Painleve trascendents. The corresponding solutions of the (2 + l)-dimensional equation, involve up to three arbitrary smooth functions. Consequently the solutions exhibit a rich variety of qualitative behaviour.
1. Introduction In this paper we discuss the generalized (2 + l)-dimensional integrable generalization of the Calogero-Degasperis-Fokas (CDF) equation 1 4
1 2
w
4
u
2 u2
1 i 1 , l,9«z 1,9 n 1 ( 1 A + -abuz + -a6« x + -f-2 + -b2uxd~l ^-^ = 0, where d~lu = J udx. This equation has been derived by Toda and Yu in Ref. 17 by using a method proposed by Calogero. "This work is partially supported by proyectos BFM2003-04174 of the DGES and FQM 201 from Junta de Andalucia.
250
251 Although this equation arises in a non-local form it can be written as Q^tx
Q^t^XX
5
8
ux .UXUXZ
+4
r
. n^XXXZ
rt^XX^XXZ
ux
ux
hL
ux ^
-Uxllxz
Z
„fztiMi
nUxx'U-xz
I
l
UUT
UZUX
r
UUr
4 u n
z^xx
hz U
2 , n 2
2
A^XXZ
u u u
,
e
-—h b u
xz
r— U
n
z xx
2
u u
z xx
(2)
+2
r 6—5—h ouuza + 2a 2a uz uA u u* x 2 2 + 4 o b ^ - Aab^^ - 6b2^ 6 = 0. + 2 b z ^ - 2b ^^ z z ux ux u ux uux u It is well known that similarity reductions of the best known soliton equations lead to second order Painleve equations 1 . In Refs. 6 ' 7 classes of solutions of (1) has been derived, by using Lie classical method, all of them expressible in terms of known functions such as the second and third Painleve trascendents. In this paper we apply the nonclassical method to study the symmetry reductions of (2). The basic idea of the method is to require the CDF equation in (2 + 1) dimensions (2) and the invariance surface condition £,UX + TjUz +TUt — (j) = 0,
(3)
which is associated with the vector field v = £(x, z, t, u)dx + rj(x, z, t, u)dx + +T(X, Z, t, u)dt +
(4)
to be both invariant under the transformation with infinitesimal generator (4). 2.
Nonclassical symmetries of the (2 + l)-dimensional C D F equation.
By applying the nonclassical method to the (2 + l)-dimensional PDE (2), we get an overdetermined, nonlinear system of equations for the infinitesimals £(x,z,t,u), T](x,z,t,u), r(x,z,t,u) and (x,z,t,u). By solving this system we get: f = a(t)x + 0(t), rj = T)(Z, t),T = l,
(5)
252
We can distinguish two cases depending on whether a ^ 0 or Q = 0. When a / 0, then 77 = - ( £ + 2a)z + 7(t), <> / = -au. In this case the reduction can be obtained by Lie classical symmetries 6 ' 7 . On the contrary, when a = 0, by solving the corresponding surface condition, we obtain the nonclassical symmetry reductions u = f(w,ri), w = x - f 0(t)dt, rj = rj(z,t)
(6)
and the partial differential equation (1 + l)-dimensional E ifdf
d2fr> drj dw2
3
d2f df drjdw dw
df (dfV d2f 0J77 \ dw y dw2 , 2,5#^/ drj dw2
|
2
n
2
3
h2^_^J_
drj dw
2
2df
d2f df Cpf drjdw ' dw dw;2
f3#f*V
2
drj dw
(d2f " dr\ \ dw2
d 3 / d2f drjdw2 dw2
-df_ f^C4 dr\ \ dw ,
drjdw2 \dw J
drydu; \dwJ l3b2df
df d 3 / drj dw dw3
2df
fdf\ d?7 \dw ) 2 d/ L,3 d / d?7dw dw
4
d / d/ drjdw3 dw 2 d/ l2, d / drydw dw 3
drj \dwJ 2
Q2/5
d / df drjdw dw
„
where 77 = rj(z,t) is a function that satisfies the Riemann equation rjt ~ Wz = 0.
(7)
3. Symmetry reductions to ODE's The reduced PDE E in (1 + 1) variables admits further reductions to ODE's. Case 1. For r) = w + V,
/ =
ff(i?),
(8)
we obtain the autonomous ODE -g'g'g"" + g3g"g'" + 3g2(g')2g'" +3(g'f-3a2g\g')3 + 3b2(g')3 = 0 2
2
W?a" (9)
By dividing by g (g') , integrating once with respect to w and then multiplying by g3(g')~2 equation (9) can be reduced to the following second order autonomous ODE
253 3(g')2
„
a2
36 2 1
3
,
By multiplying by g~3g', integrating once with respect to w we get (g'f = -a2g* - 2fc2
(11)
The integration may be completed in terms of elliptic functions. Case 2. For b = 0 we get 0 = ^¥jw,
f = y/rjg.
(12)
and the ODE g3g'g""V - g3g"g'"$ - 3g2(g')29'"0 + 5
2
+ Zg g'g>" - 2g (g")2 - hg\g'fg"
-3G?') tf + Za g\g'fw 2 e
-a g g"
W?g"w
3
4
4
3
2 6
2
3
+ 4g g" + 3 fl ( ff ') + 5a g (g') 2
2
+ 8g (g')
(13)
= 0.
2
By dividing (13) by g (g') , integrating once with respect to i9, setting the integrating constant ki — 0, multiplying by "dg~~2g' and integrating again with respect to w we get the following second order ODE 3(
„
{2k2
\
The change of variables g = (4(i))V(£($)))
nv?
2
leads to 8
,_
„,
ii
2 r = - Y + ^-zv---,
(is)
the solutions can be written in terms of the second Painleve equation (PII) where $,2(4 + a2 = 0 and ( must satisfy the linear equation
C"+fi-^C =o 20V whose solutions are expressed in term of Bessel functions. Case 3. Equation E, for b = 0 , admits the following similarity variable and similarity solutions ,9 = wei,
f = 0(0)e",
(16)
and the ODE -g3g'g""$
+ 93g"g'"# + 3g2(g')2g"^
+3(g')^
- ±kg\g" 2
3
W)39"$
3
- 3g g'g'" + 2g (g")2 4
2 5
2
+ 5 f f V ) V + 8
(17)
254
By dividing (17) by g2(g')2, integrating once with respect to •#, and then multiplying by g~3(g') and integrating again with respect to fl we arrive at Painleve III (PHI) (9')2
,, 9
=
g'
.,
, h
k2g2
q
(18)
a v In the following we present some explicit solutions of the second order ODE's as well as the corresponding travelling solution of the (2 + 1)dimensional CDF equation. Equation (11) can be integrated in terms of elliptic functions. Setting in (11) a = 0, b — 0 an exact solution is given in terms of the Weirstrass V function. Clearly any of the rational, hyperbolic or trigonometric degenerations of the V functions also give solutions. In particular, solitary waves result. We get 9
= ^ rcosh (UJ + 77) By considering the corresponding symmetry reductions (6) and (8) we obtain that a "curve" soliton solution for the (CDF) equation in (2 + 1) dimensions can be written as /*(*) cosh 2 (z + TJ(Z, t) + J P(t)dt)
(19)
In Fig. 1 we can see solution (19) with p 2,7j(z,t) = - and/?(t) 0 for t = 1 and t = 4 respectively. We observe that this soliton evolves on x = r](z,t) with 77 satisfying (7).
Figure 1.
The solution at t = 1 and t = 4.
255 4.
Conclusions
In this work we have considered the (2 + l)-dimensional integrable generalization of the C D F equation. T h e nonclassical invariance study of these P D E and further reductions lead t o second order integrable O D E ' s whose solutions are all expressible in t e r m s of known functions, some of t h e m expressible in terms of the second and third Painleve trascendents. For the C D F equation in (2 + 1 ) dimensions we obtained families of solutions which have a rich variety of qualitative behaviors. This is due t o the freedom in the choice of t h e arbitrary functions /?(£) p(z) and r](z,t).
References 1. Ablowitz M. J., Clarkson P. A. LMS Led. Notes, 149, Cambridge, University Press (1991). 2. Ablowitz M. J., Ramani A. and Segur H.,Lettere al Nuovo Cimento, 23 (1978). 3. Calogero F. and Degasperis A., J. Math. Phys., 22 23 (1981). 4. Fokas A.S. J. Math. Phys., 21, 1318 (1980). 5. Gambier B., Acta. Math., 33, 1 (1909). 6. Gandarias M.L., Saez S., Physica A, in press (2005). 7. Gandarias M.L., Saez S., Theor. Math Phys (2005). 8. Ince E. L. , Ordinary Differential Equations, Dover, New York (1956). 9. Lou S. J. Math Phys., 39, 2112 (1998). 10. Olver P. J. Applications of Lie Groups to Differential Equations, Springer, Berlin, (1986). 11. Ovsiannikov L. V., Group Analysis of Differential Equations, Academic Press, New York (1982). 12. Painleve P., Bull. Soc. Math. France, 28, 201 (1900). 13. Painleve P., Acta. Math. 25, 1 (1902). 14. Pavlov M.V., Phys. Lett. A, 243, 245 (1998). 15. Ramirez J., Bruzon M.S., Muriel C., and Gandarias M.L. J. Phys. A: Math. Gen., 36 1467 (2002).. 16. Rosenau P., Phys. Lett. A, 211, 265 (1986). 17. Toda K. and Yu S., Reports on Math. Phys., 48, 255 (2001). 18. Wang J. P., J. Nonlin. Math Phys. 9 213 (2002). 19. Weiss J., J. Math Phys., 26, 258 (1983). 20. Weiss J, Tabor J. M. and Carnevale G., J. Math Phys. (1983).
A LINEAR INSTABILITY ANALYSIS OF THE B E N A R D PROBLEM FOR D E E P CONVECTION*
A. GEORGESCU Faculty of Mathematics and Computer Sciences University of Pitesti s. Tirgu din Vale 1 Pitesti, 11004.0, Romania E-mail: adelinageorgescu&yahoo. com A. LABIANCA and L. PALESE Department of Mathematics University of Bari v. Orabona, 4 Bari, 1-70125, Italy E-mail: [email protected], palese @dm. uniba. it
A linear stability analysis of the Benard problem for deep convection is performed. An estimate of the critical Rayleigh number that reduces to the classical one for vanishing depth parameter is obtained.
Consider a homogeneous viscous and thermoconducting fluid of viscosity v and thermometric conductivity k confined in a horizontal layer of depth d subject t o an adverse temperature gradient 8 and with stress free and perfectly thermoconducting bounding planes. Let us choose a reference frame {O, i,j, k} with k pointing upwards in which the planes bounding the layer have dimensionless equations z = 0 , 1 . T h e non dimensional equations governing the Benard problem for deep
*This work is performed under the auspices of the italian I.N.dA.M./G.N.F.M. and supported by the italian M.I.U.R.
256
257
convection [1] [2] are: [3] dv at
-VrV-K + VrAv +
RVrTk, r
jrn
— = Rv • k + n{z)AT + 2-^{z)D(v) dt R
: D(v),
(1)
V - t > = 0,
where v, T, -K are the velocity, temperature, pressure fields, D(v) is the strain rate tensor, i.e. D(v) = - (Vt; + V v T ) , Vr v 2
''
'
k
is the Prandtl num-
gctfi IH and V, = ~—dA is the Rayleigh number, q is the acceleration kv of gravity and a is the coefficient of volume expansion. R ••
her,
The function /j,(z) of the deep convection is ii(z) =
-, 0 < ; 1 + d(l - z) 5 < 1. As 5 —> 0 we regain the classical Boussinesq equations for the Benard shallow convection. Let us consider the mechanical equilibrium v = 0, T = TQ—(3Z, n = ir(z) as basic flow and the perturbation fields, u, 1?, p of the velocity, temperature and pressure respectively. The linearized perturbation equations are: 1
_,
9M
.
„.,
— — = - V p + A u + J&?fc, Vr at [l +
5{l-z)
dt
Ru-k)
(2)
= Atf,
[ V - u = 0, d2(u-k) = 0 = 0, z = 0 , 1 . dz2 We assume that the perturbations are normal modes, i.e.
with the boundary conditions: u • k =
{w, C, $} = {W(z), Z{z), e ( z ) } exp[i{axx + avy) + at], 2
22
(3)
2
periodic in x and y of assigned wave number a = a, -+ a , and % u • k, £ = k • V x u. We limit ourselves to study the problem into the periodicity cell fi 0, :
2TT
"1
0,
2TT
OL2
x [0,11 and, by introducing D — —, we have: dz ^(D2-a2)-(D2-a2)2 ^T~(D2-a2)
k
W
-Ra2&,
z = o,
[1 + 5(1 - z)] ( a 9 - RW) = (D2 - a2) 9 ,
(4)
258
with W = D2W = DZ = Q = 0,z = 0,l. From (4)2, taking into account the boundary conditions, we have Z = 0. If we suppose the validity of the principle of exchange of stabilities, so that a = 0, the instability sets in as a stationary convection. In this hypothesis the system (4) becomes:
{{D2-a2fw
= Ra2@,
{ [1 + 5{1 - z)\ RW = - (D2 - a 2 ) 9 , whence we obtain the boundary value problem: f (D2-a2)3W
= - [ 1 + 5(1 -z)}TZa2W,
2
[W = D W = 0,
ze[0,l],
2 = 0,1.
remembering that 71 = R2. Let us write now W^asa sum of its Fourier series by using the total set {•^n}nepjj Fn(z) = sm(mrz) in the subspace of £ 2 ([0,1]) of odd functions [7], e.g. OO
W=
£
WnFn(z),
zG[0,l]
(7)
71=1
n odd
where the sum is on the odd indices because W is supposed to be symmetric with respect to z = 1/2. Furthermore we apply a method of BudianskyDiPrima type [5]. By using the backward integration technique, the derivatives are: 00
D2k+lW{z)
00
D2kW(z)
= J2 W^+^Eniz),
= ] T W^Fn(z),
n=l n odd
where
n=l n odd
En = cos (nirz) and, for n odd Vfc e N : wi2k+2) Thus the (6)1 becomes:
=
-n2v2w!?k).
00
£ (nV+a 2 )V n F n (z) = 71=1
n odd 00
00
= (1 + 5)Ka2 J2 WnFn(z) - 6Ua2z ^ n=l n odd
WnFn(z).
(8)
n=l n odd 00
We consider the expansion zFn(z)
= ^
ankFk{z),
jt=i fc odd
z e [0,1] where
259 the coefficients are: r 1 4 Otnk =
8n27T2'
fc = n,
<
2 A;n 2 2 I, 7T (n - fc2)2' By using the orthogonality of Fn in (8) we obtain, for n odd: oo
( n V + a 2 ) 3 W„ = (1 + <5)fta2Wn - 5Ka2 ] T a „ m W m .
(9)
m=l m odd
Denoting 6„ = nzWn then (9) becomes the relation _ "
(1 + flfta2 (n 2 7r 2 +a 2 )3°"
n 3 (n 2 7r 2 + a 2 ) 3 ^—rf rr 3
that can be written in the form (5nm + anm)bm 1
+
4
+
m
>
(10)
m=l m odd
= 0, n, m odd, where
8 n 2 7 r 2 ; (n27r2+a2)3'
^a2 2J 2 2 2 — m 2 ) 2 (n27r2 + a 2 ) 3 ' k 7r m (n
"
_
m>
n^m.
2TZa2 „ „ and 7 m = — j — - , obviously we have that
Introducing (3n = Vn,m : |a„ m | < /?„7 m . Let be
A(N)
1+an
oi2
a2i
1 + CS22
ajvi
0-2N
«JV2 • • • 1 + ajviv
oo
Because the series V^ /?fc7fc is convergent, it follows [6] limAr-^oo A(iV) fe=i k odd
is convergent too. Denote by \^
the summation over n indexes ii,i2, •••,*n each one
(n,oo)
ranging from 1 to oo. Then the previous limit is the sum of the absolutely
260 convergent series (as n —* oo)
&i-ii-\
1 + E Oiiii + ^i
2
>
* %2l\
(2,oo)
(1,°°)
&i "'12*2
+•••4E
+ ••
(n,oo) a
i„ii
a
in*2
Table 1. Critical Rayleigh numbers at a 1 s t order approximation. •R
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
4.9611681 4.9611402 4.9611160 4.9610948 4.9610761 4.9610595 4.9610446 4.9610312 4.9610190 4.9610080 4.9609979
654.7700637 608.3743483 568.1185857 532.8595890 501.7213846 474.0214482 449.2200972 426.8849808 406.6656609 388.2750905 371.4759005
This series, truncated at the 2 n d term, gives an equation of the 1 s t order in U:
*+y 4 / ^E ' nn odd =l
(n27r2+a2)3 v
'
8 ^ nn=l odd
n2ir2(n2n2
+ a2)3
x
n=o
'
(ii) that, for 5 = 0, becomes 1 — Y J fto = 0. -^ (n27r2 + a 2 ) ; n odd oo
o
Taking into account the asymptotic expansion of } and ^—' (n2-ir2+a2)J
£*-—' n 7r (n 7r 2
2
2
n=l n odd
2
+ a2Y
x
'
, from the last equations we can explicitly obtain
n odd
the values of TZQ and 11(5) (Table 1) (remember that — ~ 4.9348022 and 2 27 —7T4 ~ 657.5113644).
261
Prom the first 2 x 2 minor of the infinite determinant: 1 + a n +Q33
a n ai3
1 + an 013 031 1 + 033
we have another approximation (of the 2 n d order
&31 O33
in ft):
(A1A2-B)Tl2-(A1+A2)n where A, = ( l + |<5 + ^ )
^^f^,
+l
A 2 = ( l + \8 + ^ )
J ^ J S ,B =
9 s* 217^1-^2^2)3 (9;r2C!fa2)3 t h a t satisfies the relations AiA2 > B and Ai > A 2 . T h e minimum eigenvalue, i.e. the critical value, is then ft = Ax+A2y / ( A i - A2)2 + 4 B , t h a t for 5 = 0 reduces to [4] ft0 = (-7T2 + 2 ( A i A 2 - B) a2)3/a2, and from these we can explicitly obtain the values of fto and 1Z(S) (Table 2). Table 2. Critical Rayleigh numbers at a 2 n d order approximation. 5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
a2 4.9348022 4.9348022 4.9348022 4.9348023 4.9348024 4.9348024 4.9348025 4.9348026 4.9348027 4.9348028 4.9348029
n
657.5113644 610.9187207 570.4924202 535.0843052 503.8146060 475.9978488 451.0920251 428.6629246 408.3586055 389.8907914 373.0210980
References 1. 2. 3. 4.
R. KH. Zeytounian, Int. J. Engng. Sci., 27, 1361 (1989). M. Errafiy, R. KH. Zeytounian, Int. J. Engng. Sci., 29, 625 (1991). B. Straughan, Int. J. Engng. Sci., 30, 739 (1992). S. Chandrasekhax, Hydrodynamic and hydromagnetic stability, Clarendon Press, Oxford (1968). 5. R. C. DiPrima, Quart. Appl. Math., 18, 4, 375 (1960). 6. R. E. Eaves, SIAM J. Appl. Math., 18, 3, 652 (1970). 7. S. Goldstein, Proc. Phil. Soc, (1935).
M U L T I C O M P O N E N T R E A C T I V E FLOWS
VINCENT GIOVANGIGLI Centre de Mathematiques Appliquees and CNRS Ecole Poly technique, 91128 Palaiseau Cedex, France E-mail: vincent.giovangigli@polytechnique. edu
We discuss multicomponent reactive flow models derived from the kinetic theory of gases. We review the evaluation of multicomponent transport coefficients as well as the mathematical hyperbolic-parabolic structure of the resulting systems of partial differential equations. We address the anchored wave problem, various numerical algorithms specifically devoted to complex chemistry flows, and recent extensions to chemical equilibrium flows and ionized/magnetized mixtures.
1. Introduction Multicomponent reactive flows with complex chemistry and detailed transport phenomena arise in various engineering applications such as combustion 24 , crystal growth 23 ' 8 , or atmospheric reentry 1 . This is a strong motivation for investigating the corresponding governing equations and analyzing their mathematical structure and properties 12 . We first present the governing equations for multicomponent reactive flows as obtained from the kinetic theory of gases 9 ' 12 . We address the mathematical structure of the transport linear systems and fast, accurate, evaluation of the transport coefficients4'5'6. We next investigate the Cauchy problem and present global existence theorems around constant equilibrium states as well as asymptotic stability and decay estimates 15 . The method of proof relies on the normal form of the governing equations, on entropic estimates, and on the local dissipativity properties of the linearized equations 18 ' 15 . We then address the anchored wave problem 11 as well as various numerical simulations of complex chemistry flows2'3,7'20'21'22'23 and finally discuss various extensions that have recently been obtained for partial chemical equilibrium flows17 and ionized/magnetized multicomponent flows13,14.
262
263
2. Governing equations The equations governing multicomponent reactive flows are derived form the kinetic theory of polyatomic gas mixtures 9,4 ' 12 . These equations can be split between conservation equations, thermochemistry properties, transport fluxes. 2.1. Conservation
equations
The equations for conservation of spacies mass, momentum and energy can be written in the form12 dtpk + dx-(pkv)
+ dx-Fk = mkoJk,
dt (jm) + dx • (pv®v +pI)+dx-U dt{£ + \pv-v)
+ dx-({£
+ \pv-v +p)v) +dx-(Q
k <= S,
(1)
= pg,
(2)
+ U-v) = pvg,
(3)
oil 6\ denotes the time derivative, dx the space derivative operator, pk the mass density of the kth species, v the mass average flow velocity, Fk the diffusion flux of the fcth species, mk the molar mass of the fcth species, u)k the molar production rate of the fcth species, S = { 1 , . . . , n) the set of species indices, n the number of species, p = J^keS Pk t n e total mass density, p the pressure, II the viscous tensor, g the gravity, £ the internal energy per unit volume and Q the heat flux. 2.2.
Thermodynamics
Thermodynamics obtained in the framework of the kinetic theory of gases is valid out of equilibrium and has, therefore, a wider range of validity than classical thermodynamics introduced for stationary homogeneous equilibrium states. The internal energy per unit volume £ and the pressure p can be written in terms of the state variables T, pi,..., pn as
£(T, pi,...,Pn) = J2 Pkek(T), kes
p(T, Pl,..., pn) = J2 kes
R T
^ 7T' k
where T is the absolute temperature and Rg the gas constant. The internal energy ek of the kth species is given by efc(T) = ef + fTBfivk{T) dr, k e 5, where ef is the standard formation energy of the &th species at the standard temperature T s t and Cvk the constant volume specific heat of the fcth species. The (physical) entropy per unit volume S and the specific entropy of the kth species Sfc can be written in the form S(T, p±,..., pn) = 5Zfces Pksk{T, Pk)
264
with sk(T, pk) = sf + / T ^ £ P dT' - £ • log ( - £ st- ) , JTst T' mk \7 mkJ
kES,
where sf is the formation entropy of the fcth species at the standard temperature T s t and standard pressure p s t — p a t m , and 7 s t = p s t /iZ g T s t is the standard concentration. Similarly, one can introduce the mixture enthalpy Ji and Gibbs function Q which can be written TC = ^2ke$ pkhk(T), and Q = EkesPk9k(T,pk), where hk(T) = ek{T) + RsT/mk and gk(T,pk) = hk(T)-Tsk(T,Pk),keS. 2.3.
Chemistry
We consider a system of n r elementary reactions for n species, which can be formally written as fees fees
where 2% is the chemical symbol of the kth species, vki and uTki the forward and backward stoichiometric coefficients of the kth species in the i t h reaction, R = { 1 , . . . , n r } the set of reaction indices, and vki = vTki — vki the overall stoichiometric coefficients. The molar production rates that we consider are the Maxwellian production rates obtained from the kinetic theory 4 ' 12 . These rates ujk, k € 5, are compatible with the law of mass action and are in the form Wfc =
:«.-^.)(^n(i)""-^n(^)";'). ***
^
where fCf et KL\ are the forward and backward rate constants of the ith reaction, respectively. The reaction constants ICf and K,\ are functions of temperature and are Maxwellian averaged values of molecular chemical transition probabilities 4 . However, forward and backward chemical transition probabilities are always proportional—as are nonreactive cross sections in any Boltzmann equation—and this implies the reciprocity relations 4 ' 12 mT)
= g|||,
logKJCT) = - £
^gk(T,mk).
i e R,
where /C|(T) is the equilibrium constant of the ith reaction. In practice the reaction constant K.f is also evaluated from Arrhenius law K-f = AiT0i exp(—Ei/RsT) where At is the preexponential factor, /% preexponential exponent and Ei the activation energy of the ith reaction.
265
2.4. Transport
fluxes
The transport fluxes Tl, Fk, k G S, and Q due to macroscopic variable gradients can be written in the form 9 ' 4,12 F
k = - Eies °ki{di + Xaidx
logT),
n = - ( K - %rj)(dx-v)I - v(dxv Q = £ f c e S hFk
-XdxT
I e 5,
(4)
+ dxv*),
(5)
+ RgTEk&s(Xk/mk)Fk.
(6)
where Cki, kj G S, are the multicomponent flux diffusion coefficients, dk = (9xPk)/p, k £ S, the diffusion driving forces, pk = pkRgT/mk, k e S, the partial pressures, Xk = Pk/p, k G S, the mole fractions, Xk, k € S, the reduced thermal diffusion ratios, K the volume viscosity, r\ the shear viscosity, A the thermal conductivity and * the transposition operator. When mass fractions do not vanish, it is also possible to define the species diffusion velocities Vk, k G S, with
Vk = Fk/Pk = -J2
D
ki{di + Xaidx logT),
k G S,
where we have defined the diffusion coefficients Dki = Ckij'pYk, k,l G S. 2.5. Assumptions
on the system
coefficients
The assumptions on the coefficients are deduced form the kinetic theory of gases and are typically in the following form12. (Hi)
The molar masses rrik, k G S, and the perfect gas constant Rg are positive. The formation energies ef, k G S, and entropies sf, k G S, are real constants. The specific heats cvk, k G S, are C°° functions of T G [0, oo). There exist positive constants cv and cv with 0 < cv < Cyk(T) < cv, forT>0 and keS.
(H2)
The stoichiometric coefficients v^ and vTki, k G S, i G R, are nonnegative integers. The reaction vectors Vi, i G R, defined by Vi = (fu,..., v-nif, satisfy the mass conservation constraints (z/j,m) = 0, i G R, where m = ( m i , . . . ,m„)*.
(H3)
The reaction constants K-f, and K.\, i G R, are C°° positive functions ofT > 0 and satisfy K,f(T) = K\{T)K\{T), i G R.
(H4)
The flux diffusion matrix C = (Cki)k,ii=s, the reduced thermal diffusion ratios x — (Xi>--->Xn)S ^e volume viscosity K, the shear
266
viscosity rj, and the thermal conductivity X are C°° functions of T > 0 and p% > 0, 1 < i < n, and satisfy the mass conservation constraints N(C) = MY, R(C) = UL, x G XL where Y = (Yi,.. .,Yny, Yk = Pk/p, U = {l,...,lY and X = (Xu ..., Xny. (H5)
The thermal conductivity A and the shear viscosity 77 are positive. The volume viscosity K is nonnegative. For Y > 0, the matrix D = (l/p)y~1C is symmetric positive semidefinite and its nullspace is N(D) = MY, where y = diag(Y1, ...,Yn).
2.6. Entropy
production
Prom the Gibbs relation T US = B £ — ^Zk£s9k^Pk: where D denotes the total derivative, one can derive a governing equation for ps — S dt(ps) + dx-(pvs) f (dx-v)2 +±dxT-dxT
+ £(dxv
+ Ox-($-
£ f e e 5 fFk)
+ dxv* - \[dX'v)I):(dxv
+ $ £ M € S Dkl(dk + XkXkdx
+ EieRRzR({^M"i)
~
faMis!))
+ dxvl
= -
logT). (dj + Xtxidx
\{dx-v)I) logT)
(exp{fi,Mv?)-exp(ix,Mvl)),
where vf = ( 4 , . . . , z A ) ' , "i = M " ^ * G R, M = d i a g ( m i , . . . , m „ ) , pk(T,pk) = gk/RgT, k e S, p = (pi,... ,//„)*, and where /Cf is defined from log/Cf = log/Cf - (Mi/f,/i u ) = log/CJ - {Mv\,pa), with /uJ!(T) = pk(T,mk), k £ S. Entropy production therefore appears as a sum of nonnegative terms.
3. Transport coefficients The multicomponent transport coefficients K, rj, C = (Cki)k,i€S, D = {Dki)k,ies, \ and x = (Xk)kes, are not explicitly given by the kinetic theory of gases.. Evaluating these coefficients—which are functions of state variables—requires solving "transport linear systems." The mathematical structure of the transport linear systems can be obtained form the kinetic theory and allows their solution either by direct methods or by efficient iterative techniques 4,5,6 . Similarly, the mathematical assumptions on transport coefficients can be derived from the properties of the transport linear systems 16,11,12,14 .
267
4. The Cauchy problem The equations governing multicomponent reactive flows as derived from the kinetic theory of gases have local regular solutions 16 and global solutions around constant equilibrium states 15 . 4.1. Vector
notation
Denoting by U the conservative variable U = ( p i , . . . , pn, pvi,...,
pvd, £ + \pv-v)
,
(7)
the governing equation can easily ce recast in the compact form
8tU + Y,MU)diU=
J2
diiBijWdtf+tyU),
where di the space derivative operator in the i t h direction, C = { 1 , . . . ,d} the set of direction indices, d > 1 the space dimension, Ai(U) = dvFf the jacobian matrix of the convective fluxes Ff in the ith direction, Bij ,i,j € C, the dissipation matrices such that Ti = —J2jecBij(U)djU, i & C, is the th dissipative flux in the i direction, and Q the source term. All the system coefficients Ai(U), i G C, Bij(U), i,j G C, are smooth functions of U on an open convex set Ov. 4.2. Entropy
and
symmetrization
Symmetrization properties of second order dissipative systems 19,15 generalize the classical results about hyperbolic systems and can be applied to the system of equation governing multicomponent flows15. Proposition 4.1. Let us consider the mathematical entropy —S and the corresponding entropic variable V = — (9u<5) given by V = (1/T)(gx-
\vv,...,
gn-\v-v,
u i , . . . , vd, - l ) .
(8)
Then U —> V is a C°° diffeomorphism from the open set Ov onto Ov = R"+ d x (—oo,0). The corresponding governing equations can be written AodtV + £ AidiV =Y.di tec i,jec
(SVdiV)
+ Q,
(9)
where AQ = dvU, Ai = AIAQ, B^ = B^AQ, 0. = fl, and A0(V) is symmetric positive definite, Ai(V), i G C, are symmetric, we have the reciprocity relations Bij(VY = Bji(V), i,j e C, and B(V,w) = Ylijec Bij(V)wiWj is symmetric positive semidefinite for w in the sphere E d _ 1 and V € Ov-
268
4.3. Normal
forms
The symmetrized system can now be rewritten into a normal form, where hyperbolic and parabolic variables are split 19 ' 15 . Theorem 4.1. Consider the diffeomorphism V i—> W from Oy onto the open set Ow = (0, oo) x R™-1 x M.d x (0, oo) given by
w = (P, i o g ( # / # ) , . . . , log^"/^1). *i, • • •. ««*. r)*, where rj. = Rs/mk- The equations in the W variable obtained by the change of variable V = V(W) and upon multiplying on the left hand side by the jacobian matrix <9WV* can be written
AQ(W)dtW+J2MW)diW
= Y,
di(Bij(W)djW)+T(W,dxW)+n(W)J
with A0 = 5 w F * A 0 a w y , Btj = d^BijdwV, Ai = dwV'AidwV, T = —^3j j
the matrix B ' (W, w) — X^ -eC B^ haveT(W,dxW)
=
(W)wiWj is positive definite and we
(0,Tn(W,dxWIt)y.
This normal variable W generalizes the one used by Kawashima and Shizuta 19 . It is also possible to characterized all possible normal forms15 thanks to the invariantce of the nullspace naturally associated with the dissipation matrix B(V,w). 4.4. Local
dissipativity
We present in this section the dissipativity properties around equilibrium states that are needed in order to establish global existence and asymptotic stability. The existence of equilibrium points is a consequence from the structural properties of thermochemistry 12 . Proposition 4.2. For Te > 0 and (pf,..., p£)* £ (0, oo)™, there exists a unique equilibrium point Ue with v\ = 0, i € C, and (pf — p\,..., p£ — p£)* £ Span{Mvi, i G R} where M = diag{m\,...,m„).
269 Proposition 4.3. The matrix Ao(We) is symmetric positive definite, the matrices Ai(We), i G C, are symmetric, we have Bij(We)t = Bji(We), i,j G C, and the linearized source terms L(We) = — dwQ(We) is symmetric positive semidefinite. Moreover, the linearized normal form is strictly dissipative in the sense that the eigenvalues A(£, w) of the problem
\A0(We)
- C2 ^
Bij{We)wiwj
+ L(We)}cf> = 0,
for C G iM\{0} and w G £ d _ 1 , have a negative real part. Proposition 4.4. The smallest linear subspace containing the source term Q,iV) = Vl(U(V)), for all V G Ov, is included in the range of L(Ve) = — (dvCl)(Ve). In addition, there exists a neighborhood of Ve in Oy such that a | fi(V)| < — (V — Ve, fi(V)) where a is a positive constant. 4.5. Existence
of
solutions
The local dissipativity properties now imply global existence and asymptotic stability of equilibrium states 18 ' 15 . Theorem 4.2. Letd>l,l> [d/2] + 2 and W°(x) such that \\W°-We\\Hi is small enough. The Cauchy problem with initial conditions W(0,x) = W°(x) has a global solution such that Wj - W; G C°([0, oo); Hl) n C 1 ([0, oo); Hl~l),
(11)
Wa-WZeC°{[0,oo);Hl)
(12)
nC^fO.ooJjff'- 2 ),
More over we have the estimates Jo and supmii |W(t) — We\ goes to zero as t —> oo. Theorem 4.3. Letd >1, I > [d/2]+3 and assume that the initial condition W°(x) is such that \\W° — We\\Hi is small enough. Further assume that W° - We G Hl{WLd) n LP(Rd), where p = 1, if d = 1, and p G [1,2), if d>2. Then if \\W° - We\\Hi + \\W° - We\\jj, is small enough, the global solution satisfy the dacay estimate \\W(t) - We\\m-2 < 0(1 +t)-^(\\W°
- We\\Hl-2 + \\W° -
We\\u,),
for t G [0, oo), where J3 is a positive constant and •y = d x (l/2p — 1/4).
270
5. Anchored -waves Traveling waves in inert or reactive flows can be classified into deflagration and detonation waves 24 . In the context of combustion—which does not decrease the problem generality but makes things more explicit—weak deflagrations correspond to plane laminar flames. The anchored flame problem has been investigated with complex chemistry and detailed transport by using entropic estimates and the Leray-Schauder topological degree theory 11 . A key point is that entropy production estimates associated with multicomponent diffusion yields estimates in the form
VJ „
pD\\dxXkf T
• dx,
Xk
where D is a typical diffusion coefficient. An important tool is also the exponential decay of entropy production residuals close to equilibrium 11 . 6. Numerical simulation Numerical simulation of compressible flows is a very difficult task that has been the subject of numerous textbooks and requires a solid background in fluid mechanics and numerical analysis 10,20 ' 21 . The nature of compressible flows may be very complex, with features such as shock fronts, boundary layers, turbulence, acoustic waves, or instabilities. Taking into account chemical reactions dramatically increases the difficulties, especially when detailed chemical and transport models are considered. Interactions between chemistry and fluid mechanics are especially complex in reentry problems 1 , combustion phenomena 2 ' 3,7 , or chemical vapor deposition reactors 23 ' 8 . An important aspect of complex chemistry flows is the presence of multiple time scales. For compressible flows, we already know that the presence of acoustic waves introduces small characteristic times for small Mach number flows. However, chemical characteristic times can range typically from 10~ 8 seconds up to several seconds. In the presence of multiple time scales, implicit methods are advantageous, since otherwise explicit schemes would be limited by the smallest time scale 21 ' 12 . A second potential difficulty associated with the multicomponent aspect is the presence of multiple space scales. In combustion applications, for instance, the flame fronts are very thin and typically require space steps of 10~ 3 cm whereas typical flow scales may be of 10 cm. The multiple scales can only be solved by using adaptive grids obtained by successive
271
refinements or by moving grids for unsteady problems 2 ' 3 ' 10,21 ' 22 . Nonlinear discrete equations can be solved by using Newton's method or any generalization. The resulting large sparse linear systems must then be solved by using a Krylov-type method, such as GMRES. More sophisticated methods involve coupled Newton-Krylov techniques. Evaluating aerothermochemistry quantities is computationally expensive since they involve multiple sums and products. Optimal evaluation requires a low-level parallelization, e.g., by using vector capabilities of computers, depending on the problem granularity. Finally, it is preferable, when writing numerical software, to clearly separate the numerical tools from the special type of equations that are under concern. In the context of multicomponent flows, it is therefore a good idea to write codes for general mixtures and use libraries that automatically evaluate thermochemistry properties and transport properties.
7. Conclusion and extensions The models developped in the previous sections can also be used to describe gas mixtures in full vibrational desequilibrium when each vibrational quantum level is treated as a separate "chemical species" allowing detailed state-to-state relaxation models 16 . When the vibrational quantum levels are partially at equilibrium between them but not at equilibrium with the translational/rotational states—allowing the definition of a vibrational temperature—a different structure is obtained The case of infinitely fast chemistry, that is, the case of chemical equilibrium flows can also be embedded in the same framework12. In this situation, one has to solve the momentum and energy equations together with equations expressing the conservation of atomic elements These results have recently be extended to the situation of partial chemical equilibrium 17 . In this situation, only a subset of the chemical reactions are considered to be infinitely fast. Note, however, that the mathematical structure of numerous simplified chemistry methods is still obscure at variance with partial equilibrium. The system of partial differential equations modeling reactive ambipolar plasmas can also be embeded in the same framework13. The ambipolar— or zero current—model is obtained from general plasmas equations in the limit of vanishing debye length. In this model, the electric field is expressed as a linear combination of macroscopic variable gradients and the resulting system can be recast into a symmetric hyperbolic-parabolic composite form.
272 Asymptotic stability of equilibrium states, decay estimates, and continuous dependence of global solutions with respect t o vanishing electron mass can then be established 1 3 . T h e system of partial differential equations modeling partially ionized/magnetized reactive gas mixtures has also been investigated. In this model, dissipative fluxes are anisotropic linear combinations of fluid variable gradients and also include zeroth order contributions modeling the direct effect of electromagnetic forces. There are also gradient dependent source terms like the conduction current in the Maxwell-Ampere equation. Upon introducing a notion of partial symmetrizability and of entropy, the system has been recast into a partially normal form, t h a t is, in the form of a quasilinear partially symmetric hyperbolic-parabolic system. Using a result of Vol'Pert and Hudjaev, local existence and uniqueness of a bounded smooth solution can then be established 1 4 . Global existence and asymptotic stability is an open problem for such nonisotropic systems. Various extensions could consider initial-boundary value problems with the possibility of surface reactions. Various numerical analysis theoretical results could also be extended to the case of mixtures like convergence results of Petrov-Galerkin 'Streamline-Diffusion' finite element techniques.
References 1. J. D. Anderson, Jr., Hypersonics and High Temperature Gas Dynamics, McGraw-Hill Book Company, New-York, (1989). 2. B. A. Beth and M. D. Smooke, Local Rectangular Refinment with Application to Combustion Problems, Comb. Theor. mod., 2, (2098), pp. 221-258. 3. E. Burman, A. E m and V. Giovangigli, Busen Flames Simulation by Finite Elements on Adaptively Refined Unstructured Triangulations, Comb. Theor. mod., 8, (2004), pp. 65-84. 4. A. Ern and V. Giovangigli, Multicomponent Transport Algorithms, Lecture Notes in Physics, New Series "Monographs", m 24, (1994). 5. A. Ern and V. Giovangigli, Fast and Accurate Multicomponent Property Evaluations, J. Comp. Physics, 120, (1995), pp. 105-116. 6. A. Ern and V. Giovangigli, The Structure of Transport Linear Systems in Dilute Isotropic Gas Mixtures, Phys. Rev. E, 53, (1996), pp. 485-492. 7. A. Ern and V. Giovangigli, Thermal Diffusion Effects in Hydrogen/Air and Methane/Air Flames, Comb. Theor. Mod., 2, (1998), pp. 349-372. 8. A. Ern, V. Giovangigli and M. Smooke, Numerical Study of a Three-Dimensional Chemical Vapor Deposition Reactor with Detailed Chemistry, J. Comp. Phys., 126, (1996), pp. 21-39. 9. J. H. Ferziger and H. G. Kaper, Mathematical Theory of Transport Processes in Gases, North Holland Pub. Co., Amsterdam, (1972).
273 10. J. H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics, Springer-Verlag, Berlin, (1996). 11. V. Giovangigli, Plane Flames with Multicomponent Transport and Complex Chemistry, Math. Mod. Meth. Appl. Sci., 9, (1999), pp. 337-378. 12. V. Giovangigli, Multicomponent Flow Modeling, Birkhauser, Boston, (1999). 13. V. Giovangigli and B. Graille, Asymptotic Stability of Equilibrium States for Ambipolar Plasmas, Math. Mod. Meth. Appl. Sci., 14, (2004), pp. 1361-1399. 14. V. Giovangigli and B. Graille, The Local Cauchy Problem for Ionized Magnetized Reactive Gas Mixtures, Math. Meth. Appl. Sci., 28, (2005), pp. 16471672. 15. V. Giovangigli and M. Massot, Asymptotic Stability of Equilibrium States for Multicomponent Reactive Flows, Math. Mod. Meth. Appl. Sci., 8, (1998), pp. 251-297. 16. V. Giovangigli and M. Massot, The Local Cauchy Problem for Multicomponent Reactive Flows in Full Vibrational Nonequilibrium, Math. Meth. Appl. Sci, 2 1 , (1998), pp. 1415-1439. 17. V. Giovangigli and M. Massot, Entropic Structure of Multicomponent Reactive Flows with Partial Equilibrium Reduced Chemistry, Math. Meth. Appl. Sci, 27, (2004), pp. 739-768. 18. S. Kawashima, Systems of hyperbolic-parabolic composite type, with application to the equations of magnetohydrodynamics, Doctoral Thesis, Kyoto University, (1984). 19. S. Kawashima and Y. Shizuta, On the Normal Form of the Symmetric Hyperbolic-Parabolic Systems associated with the Conservation Laws, T6hoku Math. J , 40, (1988), pp. 449-464. 20. B. Lucquin and O. Pironneau, Introduction to Scientific Computing, Wiley, Chichester, (1998). 21. E. Oran and J. P. Boris, Numerical Simulation of Reactive Flows, Elsevier, New York, (1987). 22. T. Poinsot, S. Candel, and A. Trouve, Application of Direct Numerical Simulation to Premixed Turbulent Combustion, Prog. Ener. Comb. Sci, 2 1 , (1996), pp. 531-576. 23. K. F. Roenigk and K. F. Jensen, Low Pressure CVD of Silicon Nitride, J. Electrochem. Soc, 134, (1987), pp. 1777^1785. 24. F. A. Williams, Combustion Theory, Second e d . The Benjamin/Cummings Pub. Co. Inc., Melo park, (1985).
N O N - L I N E A R WAVES IN FLUIDS N E A R T H E CRITICAL P O I N T
H. G O U I N E.A.
Laboratoire de Modelisation en Mecanique et Thermodynamique, 2596, Universite d'Aix-Marseille, 13397 Marseille Cedex 20, France E-mail: [email protected]
A non linear model associated with a Landau-Ginzburg-like behavior in mean field approximation forecasts phase transition waves and solitary kinks near the critical point. The behavior of isothermal waves is different of the one of isentropic waves as well in conservative cases as in dissipative cases.
1. Introduction The study of plane waves propagating in fluid second gradient theory is a main subject in continuum mechanics 1 _ 7 . Prom our paper (8) where strong variation of density for the matter constituent occurs in isothermal conservative motions, we propose an extension to other thermodynamical conditions and for dissipative motions. Such cases appear in phase transitions through interfacial layers 9 ' 1 0 . Near the critical point the thickness of the interface separating the two phases of a fluid gets non molecular dimension. The theory of second gradient corresponding to an extended van der Waals model with thermodynamical potentials modeling the mean field approximation allows to study the interface in equilibrium 11>12. With a convenient rescalling, the governing equations of material waves can be expressed for conservative and viscous motions with a Landau-Ginzburg-like behaviour 13 ; they are compatible with the second law of thermodynamics and we can rewrite the one-dimensional equations of non homogeneous fluids in terms of density 14 ' 15 . To obtain wave motions, we study the thermodynamical potentials of a fluid near the critical point and extend the expression given in 9 ' 10 to the local internal energy. Thermodynamical potentials are introduced in equation of motions in two cases: when the temperature is fixed (isothermal motions) and when the specific entropy along trajectories is constant (isentropic motions). We find the possibility
274
275
of non-linear waves in form of solitary waves or phase transition waves for conservative motions, but no kinks as traveling waves can appear for non conservative motions. Finally we discuss the fluid velocity near the critical point depending on the boundary conditions. 2. Thermodynamic potentials near the critical point In mean field approximation, the thermodynamical potentials of fluids can be expressed in an analytical form near the critical point 10 . The chemical potential of a homogeneous fluid can be written in the following form 9 , i e
p = nc + pc01 (T - Tc) + p^p - pc) (T - Tc) + 1 /4 0 (p - pcf
(1)
where p—pc and T—Tc denote the difference between the density p and the Kelvin temperature T from their values pc, Tc at the critical point. This simple expansion depends on coefficients corresponding to the values of the partial derivatives of p at the critical point; that is to say, di+> , ^ ' ~ dPidTJ[pc' c
c)
The chemical potential p has been expanded using a cubic polynomial in terms of density and temperature; they are not the natural variables but the most convenient for the calculations. Denoting the coefficients by A = yufj > 0, B = - p^0 > 0 (the sign conditions are at the critical point) and p0 = pc + /xgi (T — Tc) depending only on T, Eq. (1) reads H = fi0-A(Tc-T)(p-pc)+B(p-pc)3
(2)
Expression (2) allows us to solve the problems of waves in an universal way where T—Tc plays the role of an order parameter. Such a chemical potential expansion agrees with the van der Waals equation of state n where only the values A and B are representative of a particular fluid. Consequently, the free energy per unit volume ip can be written in the form tf = PoP - j(Tc
- T)(p - pcf + | ( p - pcf - f{T)
(3)
where / is an additive function of T only. The pressure p verifies p = pp.—ip and we get near pc the expansion p = Pc + D(T - Tc) - (v - vc)^(T
- Tc) + Bi{v-ve))
(4)
276
where v = — denotes the specific volume, D = —— (vc,Tc),
f(Tc) = pc
and Ai = A pi, B\ — B p\. Due to the general properties of two-phase regions near the critical point the two-density version of specific internal energy of the fluid is expressible in the form 10 a(v,s) = (x2 -y)2+y2 with
x = ar + ba, y = cr + da
+ j3x + -yy + 6
(5)
where r = v — vc, a = s — sc,
(3,7,5, a, b, c, d are constants and s denotes the specific entropy of the fluid. 3OL da Taking into account that p = — —— (v, s), T = —- (v, s) and denoting ov os a\(v,s) = a(v,s) + pc(v — vc) — Tc(s — sc) — r 4 , we obtain the partial differential equation —
+
{D-A1r)^r=0
which yields the general solution for the specific internal energy Ai r4 2 a = g(a - Dr + -^-r ) + Bi— -pcr + Tca Due to the fact that a must verify the general form (5), g(u) is necessarily a Ai polynomial with respect to u = a—Dr+ — r2 in the form g(u) = Eu+Gu2. The term Gu2 yields a term of four degree in r and r must be a function of x only. No term in the form xy appears in (5); consequently a — Dr does not contain the variable y and the general form of a is a = E(k2 y + ^ k2 x2) + G(k2 V + ^ k2 x2)2 + ^ kf x4 -
Pc
r + Tc a
where E and G are two constants. By straightforward calculations, an identification with the expression (5) yields the general expression for the specific internal energy a{v, s) = x4 - 2x2y + 2y2 - pc(v - vc) + Tc{s - sc) with x = {/—- (v - Vc) and y=—J-± V
^
(6)
[D(v - vc) - (s - sc)}
Ji\ V Z
As a consequence the pressure gets the form (y-x2)x
+ —^2B~1(x2-2y)
(7)
277
and in the case where s is constant, the specific enthalpy reduces to the form h
=~X^v3c
[Vc(P
- Pcf
+ ^ ( P ~
Pc)j
+ ho
(8)
where h0 is a constant depending on s. Near the critical point, a fluid behaves like a gas but with a high density 17 . As for interfaces separating two bulks, the fluid is not homogeneous. The view that a non-homogeneous fluid near its critical point may be treated as matter in bulk with a local energy density that is that of a hypothetically uniform fluid of composition equal to the local composition with an additional term arising from the non-uniformity, and that the latter may be approximated by a gradient expansion typically truncated in second order is most likely to be successful and perhaps even qualitatively accurate 10>18. The first study has been done on the theory of the near-critical interface within the framework of the van der Waals theory of capillarity 9 ; the simplest model able to take into account the density and its gradient uses a unique supplementary quantity represented by the constant C of internal capillarity. In S.I. units, the value of C for water at 20° Celsius is of the order of 1 0 - 1 6 (see 1 5 ). In the mean field approximation the expression of the internal energy anh of a non homogeneous fluid near the critical point is in the form 10 ' 12 - 14 Q-nh = a + — (Vp) 2 The supplementary term due to the non-homogeneity of the medium in the expression of the internal energy is effective only in interfaces and for fluids near the critical point. 3. Motions of a fluid near the critical point Equations of motions are classically given in the literature with the additive second gradient term 6 , s . Fluid motions are associated in the literature with the capillary fluid equation of motion u: pT + Vp + pV{n-CV2p)-divav=0
(9)
where p is the previous thermodynamic pressure associated with the medium considered as homogeneous, (i.e. the pressure entering in the thermodynamic potentials of the homogeneous fluid near the critical point), tt is the body force potential, av — A t r ( A ) / d + 2/xA is the viscous stress tensor in the classical form where A is the velocity deformation tensor,
278
r the acceleration vector, V the gradient operator and V 2 the Laplacian operator. The mass balance yields ^+d±vpV
=0
(10)
where V denotes the velocity of the fluid. Let us notice that we can obtain the equation of energy such that the system is compatible with the second law of thermodynamics 19 ' 20 . Capillary fluids belong to the class of dispersive systems because the internal energy depends not only on density but also on its derivatives with respect to space variables; we have previously seen that the constant solutions are nevertheless stable 21 . Let us study the one-dimensional problem when the velocity V and the density p are only functions of the variable £ = z — c t, where z is the space variable, t the time and c the wave celerity with respect to a Galilean frame: V = V(z-ct),
p=
p(z-ct)
The mass balance equation yields _cdp
d{pV)__
and by integrating, we obtain p(V-c)
=q
(11)
where q is constant in the motion. In the cases of waves we obtain
We consider that body forces are negligible as in space with weightlessness. In the uni-dimensional case, diva*, = (\ + 2p)q
d?_ J A 22
dC
\p,
In the following, we assume that v = (A + 2p)/p is constant in the fluid as assumed in interfaces 17 . Then Eq. (9) yields d (I q2
d2p
d / 1 \
1 dp\
n
We consider two cases a) isothermal motions corresponding to fluid motions near equilibrium conditions and b) isentropic motions corresponding to fast ,
••
T
„
1 dp
dH
velocities. In the two cases, we can write - — = —— where H is the pdQ
dQ
279 chemical potential of the fluid in case a) and the specific enthalpy in case b). In all the cases, near the critical point p « pc; consequently v q/p2 « v q/p2 and the equation of motion becomes ^ - ^ ^ 2= HH - H 'd? p d(-
0
^ + Mo+
2p
(12)
where H0 is constant. 4. Example of waves of a fluid near the critical point Two main cases are generally considered for traveling waves : isothermal processes and isentropic processes. Motions can be conservative as a mathematical limit of the dissipative case. Nevertheless, in thin interfaces the viscosity may be neglected and conservative cases may be considered as realistic physically 2 2 . The cases of viscous fluid involve inequality due to Liapounov functions. 4.1. Existence
of solitary
waves
Let us notice that for viscous capillary fluid, solitary waves as kinks cannot appear. Multiplying Eq. (12) by — we obtain by integration :
£(8,-™+i?-0(3)'« ™ where Co is a constant and K'(p) = H(p) — H0. Eq. (13) allows to obtain a first integral only when v = 0 . Due to the fact that 2
for v > 0 and C > Co ,
/ ~ [ ^ ) dC > 0
it is not possible to obtain p(—oo) = p(+oo) and no solitary wave can appear in dissipative motions. 4.2. Isothermal
waves
In the isothermal case the wave motion is obtained by using the chemical potential // given by Eq. (2) 1 d2p
vq dp
2 d?"Tk i
A .„,
= ~™
( T
_. . -
r ) (
^
B , )+
,,
277 <™>
3+
q2 4^ + * °
. ,. {14)
280
where k0 is a constant. Near the critical point \(p — pc)/Pc\ < 1; consequently, we get the following expansion to the second order in (p — pc)/pc ~ 1 (i _ 2(P-Pc)
1 2
P ~ Pc\
KP-PC)2
+
Pc
2
P c
and to the fourth order Eq. (14) yields 1 d2p 2 dC
q2 + 2Bp*
vq dp B ( Z 2 p\C -J7 d( =7TF,\P-Pc 2C V
3
wherefciis constant. We define the following change of variables:
' = (* - 2 ^ )
(1+£y)
' C = ^ and Q =
on with
L2
A(TC
( 2 g )
,/2Bl/^c/ («)
A(TC-T)
^
=
V
t -T) + ^
a n d £2 =
Pc
+ ^ Pc
B / ' - I* ~ V rc 2£p;
Then, the equation of isothermal waves writes 1 d2Y
vQ
HY
\w- %'Y*-Y+kl Conservative
(16)
case, v = 0: we obtain the first integral ^
= {l-Y2)2-a,Y-b1
(17)
where oi and b\ are two constants. The intersections of the straight line a{Y + b\ and of the quartic (1 — Y2)2 yield the density range Interfacial propagation - We recall the main results of ( 8 ): in the bulks phases, the densities are constant and thus the first and second derivatives of Y are zero. The straight line is tangent to the quartic at the associated points. If the bulk phases are different on both sides of the interface, the straight line has to be bitangential to the quartic which implies a± and b\ are null (case of two phases). A liquid-vapor interface wave is similar to the one obtained in the equilibrium case (q = 0) but mass flows through the
281 interface. Vaporization or condensation phenomena depend on the sign of q. This case corresponds to a shock wave in the sense of Slemrod 2 and
P=\Pc-
2Bpi
l+etanh(
-
(18)
(see left side of fig. 1). Solitary waves - It is also possible to obtain a solitary wave moving in one of the bulk phases (liquid or vapor). The straight line is tangent to the quartic at the point associated with the bulk Y = Y0 and intersect the quartic at the point associated with the middle of the wave Y = Y'Q. When Y belongs to the interval between Y0 and Y£, the right hand side of Eq. (17) must be positive and this authorizes only two possibilities: a) — 1 < Y0 < —1/y/S and Y0 < Y„ < 1, the density increases in the wave (see right side of fig. 1), b) 1/y/Z < Y0 < 1 and — 1 < Y0' < Y0, the density decreases in the wave.
Po
Figure 1. On the left side of the figure a phase transition wave is represented (densities Pl and pg are associated with the liquid and vapor bulks); on the right side a traveling wave is a kink associated with case a) (density p0 corresponds to the bulk of the fluid).
Consequences - The change of variables (15) assumes that Tc — T +
M
is positive; Then, for T > Tc the flow through the interface must be more important than the limit qm = y/Ap%(T - Tc) and the wave velocity £ must be greater than the limiting value £m — y/Apc(T - Tc) which is the celerity of isothermal acoustic waves. Dissipative
case, v > 0 : In this case no solitary wave appears (see
282
dY section 4.1). Let us define Z(Y) = — ; then Eq. (16) yields l
-ZZ'
= Y3 -Y
-vQZ
+ kx
We look for the phase transition waves with a solution in the form Z(Y) = axY2 + pxY + where the polynomial a\Y2 differential equation
7l
+ f3\Y + 71 has two real roots. Then, the
r\Y — = a 1 y 2 + /3 1 y + 7 i at,
has a solution in the same form than (18); (see left side of fig. 1). Straightforward calculations prove that the polynomial has two real roots when v2q2 (A (TC - T) + ^ ^ < 24 C3Bp4c which corresponds to a velocity q/pc of the wave small enough. This condition does not appear in the conservative case where v = 0. 4.3. Isentropic
waves
For an isentropic conservative motion, s = Cte and the equation of motion is in a form deduced from Eq. (12),
c—
=h-h0
-72
+
where h is now the specific enthalpy. We consider the new change of variables p = pc(l+Y),
£ = L(
and
q = bQ
with L2 =
2CpjA1 DBl
b2 =
'
D]h pcAx
and the equation of wave motions yields 1 /' AY\2
It)
Y3
=Y{1
Y2 + 3Q2)
+
T{T-2Q2) + a°Y
+ b
°
(19)
283
with T = 2
2D B1
2pcD Ax
and a0 and b0 are two constants. We denote by £n
a limit celerity of the waves with respect to the fluid. Due to
the form of the second member of Eq. (19), we notice immediately that it is not possible to obtain phase transition waves. Consequently traveling waves cannot appear in dissipative motions when the entropy is constant in the fluid. The two cases are represented on fig 2.
f(P
f(p)
Figure 2. On the left side of the figure, the case where the line f(Y) = a0Y + b0 is above the cubic representing the left side of Eq. (19) is presented; on the right side the opposite situation: the line f(Y) = a0Y + b0 is below the cubic.
Case 1: Q2 < T/2 or £ < £m. In this case solitary waves cannot appear (see left side of fig 2). Case 2: Q2 > T/2 or £ > £m. In this case solitary waves are possible depending on initial conditions (see right side of fig 2). In the van der Waals model of pressure we obtain by straightforward calculations D =
4p c
and consequently, tm = \ I
6pcPc
A1 4pc
=
Bi
ZVcPl
.
3 pc 4.4. Fluid velocity
near the critical
point
For a given density, the fluid velocity is deduced at time t = 0 from Eq. (11). Consequently, V(z) = c +
P{z)
284
where V(z) is the fluid velocity at z and c is an arbitrary constant. At any time t,
V(z-ct)
= c+
q
p(z — ct) where initial conditions yield the arbitrary velocity c. Interface propagation - In a phase transition wave, the fluid changes from liquid to vapor as its volume increases. Consequently such a phenomenon cannot occur in a closed tube but only in a tube with only one closed end. For example, the tube is closed at the other end with a piston whose displacement is imposed. The fluid velocity at the fixed end is zero (for example liquid bulk) and c = —q/pi- If we impose a value U for the piston velocity in the vapor bulk, we deduce the value of q
,(1-1)-* \Pv Pi J where pv and pi are the values of the density in the vapor and liquid bulks. From the values of pv and pi deduced from expression (18), we determine the flow q and the velocities I ~ q/pc and c. Solitary wave - The volume of the interface that moves in only one bulk phase is constant. Such a wave may be moving in a closed tube such as Natterer tube 23 . At the ends (assumed far from the region of the wave) the velocity in the bulk phase is zero and c ~ —q/pc. When the temperature is close to T c , the model of fluid endowed with internal capillarity allows to obtain traveling waves in a tube which depends on two arbitrary parameters. 5. Conclusion The mean field approximation for fluid near the critical point is able to predict solitons and transition of phase waves. Two kinds of waves are investigated: a) liquid-vapor waves in the case of isothermal medium whose the celerity depends on the distance between the temperature and its critical value. Above the critical temperature, the fluid behaves like an elastic medium and the waves are supersonic with respect to the isothermal sound velocity 8 . Below the critical temperature, the fluid behaves like a nonrigid medium and the wave velocities can take any value in conservative motion but are bounded in dissipative motion proportionally to the viscosity of the fluid, b) Traveling kinks appear in the conservative case as well as for isothermal than for isentropic motions.
285 Thermodynamics functions and parameters vanish or diverge at t h e critical point proportionally to some power of the distance from t h a t point currently measured as T — Tc. T h e critical exponents are central t o the discussion of critical phenomena and can be generalized by nonclassical value of critical point exponents for the potentials given in section 2. T h e problem can be extended for multi-component fluid mixtures but critical points are not unique; for a mixture of fluids there is a curve of critical points. Acknowledgments T h e paper has been partially supported by P R I N 2000 (Coordinator Prof. T. Ruggeri) and by G.d.R. C N E S / C N R S 2258. References 1. 2. 3. 4.
P. Germain, J. Mecanique, 12, 235 (1973). M. Slemrod, Arch. Rat. Mech. Anal., 81, 301 (1983). M. Grinfeld, Proc. Roy. Soc. of Edinburgh, 107 A, 153 (1987). J. Poujet, Continuum Mechanics and Discrete Systems, Ed: G.A. Maugin, Longman Publ., vol. 1, 296 (1990). 5. H. Gouin, IMA vol. Math, and its Appl., 52, Springer, 111 (1994). 6. L. Truskinovsky, IMA vol. in Math, and its Appl., 52, Springer, 185 (1994). 7. H. Gouin and T. Ruggeri, Eur. J. Mech. B'/fluids, 24, 596 (2005). 8. H. Gouin, C.R. Acad. Sci. Paris, 317, 1263 (1993). 9. B. Widom, Phase transitions and critical phenomena, Ed: C. Domb and MS. Green, Academic Press, vol. 2, 79 (1972). 10. J.S. Rowlinson and B. Widom, Molecular theory of capillarity, Clarendon Press (1984). 11. J.D. van der Waals, Archives Neerlandaises, 28, 121 (1894). 12. J. Cahn and J. Hilliard, J. Chem. Phy., 31, 688 (1959). 13. L. Landau and E. Lifshitz, Statistical Physics, Mir (1967). 14. P. Casal and H. Gouin, C.R. Acad. Sci. Paris, 300, II, 231 (1985). 15. P. Casal and H. Gouin, C.R. Acad. Sci. Paris, 306, II, 99 (1988). 16. J.S. Rowlinson and F.L. Swinton, Liquid and liquid mixture, Butterworth Scientific (1982). 17. Y. Rocard, Thermodynamique, Ch. V, Theorie Cinetique, Masson (1956). 18. B.A. Malomed and E.I. Rumanov, Dokl. Akad. Nauk. SSSR, 284, 6 (1984) 19. P. Casal and H. Gouin, C.R. Acad. Sci. Paris, 300, II, 301 (1985). 20. J. Dunn, New perspectives in thermodyanmics, Ed: J. Serrin, Springer, 187 (1986). 21. S. Gavrilyuk and H. Gouin, Trends in applications of mathematics to mechanics, Ed: G. Iooss et al, Chapman & Hall, 106, 306 (2000) 22. D. Langevin, Light scattering by liquid surface, Marcel Dekker, 161 (1992). 23. G. Bruhat, Thermodynamique, Masson, 188 (1968).
ON EULER CLOSURES FOR R E A C T I V E BOLTZMANN EQUATIONS *
M. G R O P P I A N D G. S P I G A Dipartimento di Matematica , Universitd di Parma, Parco Area delle Scienze 53/A, 43100 Parma, Italy
The problem of an hydrodynamic closure at the Euler level of the complicated set of integro-differential non-linear Boltzmann-like equations describing the evolution of a chemically reactive gas mixture at the kinetic level is addressed. In comparison to other physical situations, the case in which the process is driven by mechanical collisions between molecules of the same species is worked out and discussed.
1. Introduction A kinetic description of chemically reacting gases represents quite a difficult mathematical task 1 , also because of its intrinsic physical complexity 2 . In recent years, a bimolecular chemical reaction has been dealt with starting from the kinetic model proposed in 3 , and extended in 4 to the case in which the four species are allowed a structure of N > 1 discrete energy levels, to mimic non-translational degrees of freedom. The gas may be considered as a mixture of 4iV mono-atomic gases, for which collision equilibria and i?-theorem have been determined. An encounter may result in either a chemical reaction or in a mechanical scattering, with global conservation of mass, momentum and total (kinetic plus internal) energy in each collision. Crucial point for practical applications is the availability of a closed set of balance equations for the main macroscopic observables (moments of the distribution functions), usually obtained as hydrodynamic limit when the proper Knudsen number tends to zero, according to the dominant microscopic process. In particular, one is interested in the equations of chemical kinetics, and in the relevant rate coefficients 2 . *Work performed in the frame of the activities sponsored by MIUR (Project "Mathematical problems of kinetic theories"), by INDAM, by GNFM, and by the University of Parma (Italy), and by the European Network HYKE "Hyperbolic and Kinetic Equations: Asymptotics, Numerics, Analysis" (e-mail: m a r i a . g r o p p i 8 u n i p r . i t )
286
287 The present paper is aimed at moving some steps towards the derivation of a closed set of moment equations for the macroscopic evolution following the fast initial kinetic transient, in which distribution functions relax rapidly to the local equilibrium relevant to the dominant collision operator. The simplest closure can be achieved by a zero-order approximation with respect to the small parameter, to provide reactive Euler equations which take into account not only streaming, but also the slow part of the collision integral. The same procedure has been applied in the past, as a starting point for more accurate hydrodynamic descriptions, in different collision dominated regimes, in which the dominant role was played either by elastic scattering only 5 , or by all mechanical encounters 6 . Here we shall focus on the physical situation when the typical mechanical collision times (or mean free paths) for encounters between molecules of the same species are much shorter than the corresponding quantities relevant to all other processes (inter-species scattering and chemical reaction), taken in turn to be comparable to the macroscopic scales. Most frequent collision partners for any molecule are then molecules of the same species in whatever energy level (affine interactions). Collision invariants and collision equilibria for the fast process are derived, and their use into the exact macroscopic balance equations, in order to achieve the desired Euler closure, is discussed. The AN different components are labelled according to a single index and ordered in such a way that the s-th chemical species, s = 1, 2, 3, 4, may be regarded as the equivalence class of the indices i which are congruent to s modulo 4 (i = s mod. 4, or simply i s s if no confusion arises). If Ai, 1 < i < 4/V, denotes the general component, and Ei the energy level corresponding to its state, the general binary interaction is written as Ai + Aj ^± Ah + Ak, and is described, under the same validity hypotheses as for the Boltzmann equation 2 , in terms of a cross section a^f, relevant to the process in which particle i gets transformed into particle h, and particle j into particle k. Energies are monotonically increasing with their index in the frame of each species, and normalized in such a way that Ei > 0, with equal sign only for one (or more) of the fundamental levels 1,2,3, 4. Given s, 1 < s < 4, all molecules Ai with i = s share thus the same mass m s , with m i + m 2 = 7713 + 7714, and we may always take AE = E3 + E4 — E1—E2 > 0. Symbols x , v , i are used to denote position, molecular velocity, and time, fi stands for the general unknown distribution function, and / represents the 4JV-dimensional vector made up by the /,.
288
The kinetic equations for the evolution of the distribution / reads 4 dfi dt
,
dfi = Ml}= E [[K?hk[l}(v,w,n')dwdn', dx (j, h, k) e DP (j, h, k) € Di
Kiihk[f](y,
l
w, n') = G(g2 - 6%)ga%(g, n • n')
(1)
fra->)3A(v?/)A(w?/)-/i(v)/J-(w) XH'hk /
where g = |v - w|, n = (v - w)/g, and Vy = r y v + r ^ w + w;•w= =
rkhgifri
1/2
2
^(g - 5$) Here r y r y v + r i ( w - rhfc ^ n ' , with g™ = denotes the mass ratio mi/(m,i + rrij), fiij = r^mj the reduced mass, AEff = Eh + Ek — Ei — Ej the energy variation (positive or negative), while <5yfc stands for 2A£y fc //zy; when the latter is positive, the unit step function 0 actually introduces a threshold for the collision. Moreover, the set Di describing the allowed collisions can be viewed as Df U Df, where the superscript refers to either fast or slow processes, and the slow part can be separated in turn in a mechanical part DfM and in a chemical one Dfc. Such disjoint sets are defined as Df = {(j, h,k),l< M
Df
c
Df
j , h,k<
= {(j, h, k), l<j,h,k<
AN, j , h,k = i} 4N, j^i,h
= {(j, h,k),l<j,h,k<4N,i£j£h£k,i
= i,k = j}
(2)
+ j = 3,h + k = 3}.
2. The collision-dominated regime The main properties of the collision operator Jj (collision invariants, detailed balance, equilibria, H-theorem) have been investigated in 4 , where also all major moments, defined in the usual way, are listed. They include number density n* of each component and Ns of each species, the corresponding drift velocities Uj and u*, total number density n, mass density p, and mass velocity u, pressure tensor P , thermal energy density U = \tr¥ (defining temperature T as in a perfect gas), thermal heat flux q, excitation energy density U*, and excitation heat flux q*. After introducing typical values for density, n, and for cross sections relevant to fast and slow processes, Of and <7S, respectively, we measure distances in units of the slow mean free path Xs = (nas)~l and times in units of a suitable time T. Assuming as usual that the Strouhal number is of order unity, all quantities in the kinetic equations (1) may be rescaled in the spontaneous way, which leads merely to the appearance of the dimensionless parameter - = As/A^
289 (e small in our hypothesis) in front of the fast mechanical collision integrals. If we keep the same name to all adimensionalized quantities, the Boltzmann-type equations (1) take the singular perturbation form: f ^ + v - g
= ^f[/]+Jf[/]
l<^
(3)
with obvious meaning of symbols and with the same formal expression for all collision integrals. For the fast collision operator defined by jf, after introducing a set of 47V smooth test functions ipi, we can define the functionals C
M =£
/ ^ W f [/]
s = 1,2,3,4,
(4)
and such a weak form can be cast, after some careful algebra, as
• [h(^)fk«k)-fiMfjM]
(5)
• [ph (v?/) +
Es) ,
i = s, 1 < s < 4, (6)
characterized by 20 parameters as, hs, cs, with as,cs e R, and b s e R 3 . A basis in this space can be chosen as:
{
(l, m s v, ^msv2) K
'
for i = s
s = 1,2,3,4
(7)
(0, 0, 0) T otherwise and represents particle number, momentum, and kinetic energy of each species. At this point fast collision equilibria, namely distribution functions making J vanish identically, can be found explicitly by a detailed balance principle, upon resorting to the functionals Ws[f] = C s [log(/)], where log(/) is the vector made up by log / j . This leads, through the same
290 steps exploited in 4 , to the conclusion that the class of the mechanical collision equilibria / is exhausted by the Maxwellian distribution functions N ,M, x s ( /* ( v ) = V 7 ^ T e x P
^(Ts) = ^ e x
P
i=s
r -
E
Ei-Ea\ 7F '
E T s
^
^
( n.s • U^H i = s,
3/2 ex
"Wis ,
\j
P ~2i;(v-U5)
s = l,2,3,4,
(8)
/
where Z s is the partition function. They depend on 20 (dimensionless) parameters NS,TS > 0 and u s e l 3 , representing number density, temperature, and drift velocity of each species. Collision equilibria for the whole Boltzmann collision operator are of the same form, but with a common value u for the u s and T for the Ts, and with densities Ns related by the chemical law of mass action 4 . 3. Moment equations Macroscopic transport equations are now nothing but the weak forms of the kinetic equations (3) corresponding to suitably chosen test functions. With reference to the fast collision invariants (7), contributions from the collision integral J obviously disappear, and we are left with
dl E J ^( v )/*( v )rf3V+ ^ • Y^ J vVi(v)/i(v) d3v = i=s
i=s
E E ///^(v)^"[/](v,w,n')rf3vd3wd2n' i=s (j,h,k)€Df JJJ
(9)
for s — 1,2,3,4. The explicit form of such exact balance equations is not shown in this short note, and only their main features will be commented on. Integrals on the left hand side can be handled in standard way and expressed in terms of the 13 canonical moments relevant to each of the single species. On the right hand side instead, the triple integrals may be rearranged in the equivalent form HI
[Pi ( v f t ) 0( f f 2 - 6jik)gaiik(g, n • n')/ h (v)/ f e (w)
-
- 5%)ga%(g, n • n ' J / i W / ^ w ) ] d3vd3wd2n'
(10)
summation can be split in a mechanical part, in which one has to all species r — 1,2,3,4 the contributions coming from &lli,h = s = r, and in a chemical one, in which we identify a single species as of a given s. In particular, concerning mass balance for species s
291 (first collision invariant in (7)), the mechanical part vanishes (mass of any species is conserved in mechanical encounters) and we get
^ T +l ^ X ) =Qa=E
E
f[[KiJh fc[/](v,W,irV3Vd3W^'.
(11) Manipulations similar to those of 6 lead to the physically expected result Qi = Q2 = ,—Q3 = —Qi, with, for instance,
Q1=
E
JIJ[KiJhk[l}+Krkh[f]\d^d3^d2n'.
(12)
t = l , j = 2, h = 3,k = 4
Momentum and energy balances are instead affected by the slow interspecies mechanical scattering, since these quantities are actually exchanged in microscopic non-affine collisions, and an overall conservation is obtained only by summing on all species. The desired Euler closure would be finally achieved by substituting the Maxwellians / M (8) for / into the transport equations (9), where then all terms become in principle amenable to the 20 moments Ns, u s , Ts. In particular, we would have u* = u s and all viscous stresses p 5 and heat fluxes q s would vanish. This closure differs from those of 5 and 6 , where the closed set of partial differential equations was involving AN + 4 and 8 macroscopic fields, respectively, in agreement with the different physical conditions they were coming from. As usual, these Euler equations do retain collision terms on the right hand sides, as a left over of the slow microscopic processes. It is easy to see that such terms, of algebraic nature now, vanish for all equal us and Ts, and Ns satisfying the mass action law. References 1. V. Giovangigli Multicomponent Flow Modeling, Birkhauser, Boston, 1999. 2. C. Cercignani, Rarefied Gas Dynamics: from Basic Concepts to Actual Calculations, University Press, Cambridge, 2000. 3. A. Rossani and G. Spiga, "A note on the kinetic theory of chemically reacting gases", Physica A, 272, 563-573 (1999). 4. M. Groppi and G. Spiga, "Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas", J. Math. Chern., 26, 197-219 (1999). 5. M. Groppi, G. Spiga, "Kinetic theory of a chemically reacting gas with inelastic transitions", Trans. Th. Stat. Phys., 30, 305-324 (2001). 6. M. Groppi, G. Spiga, "Stability in extended kinetic theory", in: Proceedings of WASCOM 1999, V.Ciancio, A.Donato, F.Oliveri, S.Rionero Eds., World, Singapore, 2001, p. 245-256.
O N T H E G E O M A G N E T I C D I R E C T I O N PROBLEM: A N O N E X I S T E N C E RESULT
RALF KAISER Department of Mathematics, Universitat Bayreuth, D-95440 Bayreuth, Germany E-mail: [email protected]
Consider the following nonlinear boundary value problem in the exterior space V = {x 6 R 3 : |x| > 1} of the unit sphere S: Given a vector field D : S -> R 3 we ask for all harmonic vector fields B : V —• R 3 which decay at least as fast as a dipole field at infinity and are parallel to D on S, i.e. there is / : S -> R such that B = / D . This problem is related to the problem of reconstructing the geomagnetic field outside the earth from directional data measured on the earth's surface. It is shown in this note that in any neighbourhood of a single multipole field (restricted on S) D * , n £ N, \k\ < n, there are direction fields d*, close to D * in any reasonable norm, for which the above problem has no solution.
1. Introduction The problem to determine the magnetic field outside the earth if only the direction of the field is known on the earth's surface is well-known in geomagnetism. It is especially relevant for the interpretation of historical or palaeomagnetic data sets which provide directional information only. Neglecting deviations from the spherical shape S of the earth's surface and assuming the exterior region V to be insulating and to be free of sources of magnetic field this problem can be formalized as follows: Given a direction field D € C° (S, K3) we ask for all nontrivial vector fields B G C 1 (V", R 3 ) for which a scalar function / : S - > E exists such that the conditions V x B = 0, V - B = 0 3
inV,
|B(x)| = 0 ( | x | - )
for |x| -> 00,
B = /D
on S
(1)
are satisfied. It is the nonlinear boundary condition (1)3 which makes the problem different from the standard boundary value problems of potential
292
293 theory, which specify either the normal component or the tangential components on the boundary. The type of function / which is appropriate here is specified in the next section. Let us just remark that / need not obey a sign condition ("unsigned direction problem"), which makes the set of solutions B for fixed direction field D a linear space Z-D- Having in mind the reconstruction problem the dimension of this solution space, in particular, the question of uniqueness or nonuniqueness, is of major interest. Uniqueness is, of course, always understood up to a multiplicative constant which remains free. An early attempt to prove uniqueness for arbitrary direction fields went astray [1]. This became obvious as Proctor & Gubbins gave the example of an axisymmetric direction field, for which they found numerically three different solutions [2]. On the other side, Hulot et al. derived an upper bound on the dimension of the solution space, dim L D < ' D — 1 with l-o being the number of "poles" of the direction field D [3]. Kaiser & Neudert investigated the case of single multipole fields as direction fields: Neglecting the boundary condition the general solution of (l)i, (1)2 in spherical coordinates (r,6,ip) allows the representation 00
B = V*,
-.
n
*=E £
c
«* ^
Yn(0,
(2)
n=lk=—n
with Y£ being the usual spherical harmonics and cnk arbitrary constants. Defining the basic direction fields or (simply) multipole fields
Dkn(6,V):=v(^TYj;(6,V))\r=i,
n e N, k € {-n,.. .,n},
(3)
there is obviously dimL^* > 1- In the case of axisymmetry it turned out that dimL D o = n, n £ N [4], which is precisely the upper bound in ref. [3]. For all other single (nonaxisymmetric) multipole fields there is uniqueness, i.e. dim LDib = 1, n € N, 0 < |fc| < n [5]. In the present note it is shown that in any neighbourhood of a single multipole field there are (nonaxisymmetric) direction fields d*, n E N, |fc| < n, which allow no solution at all: dimZ/d* = 0. This is quite in contrast to the axisymmetric situation, where dimLo depends on a single property of D, viz. a suitably defined rotation number [6]. As a consequence, small axisymmetric perturbations of D, which do not change the rotation number, cannot change dimLD either. The result of this note is based on a Hilbert space formulation of the problem [4], which is summarized in the following section; section 3 contains the precise statement of the result, its proof, and some concluding remarks.
294
2. Hilbert space criterion Let us introduce the Hilbert space L2(S) = \h : S ->• c l f \h\2dil < 00} with scalar product (/ii,/i2) := Jghih^dCl and norm \\h\\ := {h,h)ll2. Boundary values are explained in the sense of traces. Introducing a potential \t for the harmonic field B and using spherical coordinates (r, 8,
in V,
|*(x)| = 0 ( | x | - 2 ) for |x| -> 00, V* = / D
(4)
onS.
In order to simplify the proof in the next section the formulation of the Problem is slightly more general than that in ref. [4]. In fact, only those solutions make physical sense, where at least one of the quantities / and D is real. Taking the real or imaginary part of the other quantity yields then a reasonable solution. We have now the following [4,5] Criterion: Let D G C°(S, C 3 ), as in the Problem, and f : S ->• C such that / D G L2(S)3. Moreover, let fornE^ke {-n, ...,n} T-i := D;, TO
:=D*esm6Y°-^D*rY?,
rpk — n * ninfl Yk
eO *->0
I r>*(n-1
. , / " * r * 2 Vk
-
/("+1)27fc2"vfc
"\
~D;sm9Y0°, := (-ikD;
+ (n + 1) D*v sin0) Y„fc,
where Yk = Yk (9,
and(f,T_1)
= (f,Tk)
= (f,Skn) = 0, n € N 0 , |A| < n } .
Then, f is a solution of the Problem if and only if f G L D •
295
3. Nonexistence in the vicinity of single multipole fields Let us define the 1-parameter family of direction fields Dff(A) := -(N + l ) l f f e r + d9Y* ee +
%{K + A )
. . F # e„ , A € K, (5) sin u which differ from Djy as given by (3) only in the ^-component. We have then the following Theorem: Let D = D^(A), A € M, be the 1-parameter family (5) associated with the single multipole field D $ , N £ N, \K\ < N. Then, A ^ 0, A 7^ — K implies L
r>%(\) = {°}>
i.e. the Problem has no nontrivial solution. PROOF: The proof is based on the Criterion and follows the lines of ref. [5]. Let / e LDKW {—n, ...,n}>
for fixed A G E, A ^ 0, A ^ -K.
Since IYA n £ N 0 , A; €
is a complete orthonormal system in L2(S),
the condition
2
/ Dr € L {S) from the Criterion implies the representation ..
n
oo C
Y
f = yKEY, "* n-W
(6)
n=0k=-n
Inserting (6) in the condition (/, 5*) = 0 yields (-k(N
+ l) + (n + l)(K + \))cnk=0,
n € N o |fc| < n.
(7)
So, cnfc ^ 0 only if A; = ^ (if + A) € Z, n £ N 0 , |fc| < n. Let (nj.fc/) with / € { 1 , . . . , L} or € N denote all pairs such that cn,k, ^ 0. Obviously, there is m = n\ o- h = k[ <£> I = V and (ni,ki) ^ (N,K) for any Z. The representation (6) has thus the form
and from the condition fDg € L2(S) follows the representation: sin0 deY$ £
cn|)fc, Y„*< = V # E
E
> « ^
<9>
for some coefficients 7„ K . We make now use of the representation (cf. [7]) y n *(0,
(10)
296
where Pn (•) are associated Legendre polynomials and dnk are some constants ^ 0. Inserting (10) in (9) and comparing coefficients of exp(i(K + ki)(p) leads to
cnikl(l-x2)P'NW(x)P^(x)=pW(x)
5 3 Mk,iPlk,iW
with x = cos9 and suitable coefficients %\ki\, x2r\K\/2p\K\{x)
=
. p\K\{x)
and
v
(11)
> \ki\- Note that (1 —
{l_x2)-\K\/2+lp,N\K\{x)
=
.
p^\K\(x)
are polynomials in a; of order N — \K\ and N — l-Pfl + 1, respectively. The zeroes ofPJf' and -FAT ' are all real, simple, and different from each other. Therefore, (11) would imply that all zeroes of PN are zeroes of some Pn, as well, i.e. PN is a factor of P„, , which is not the case. Thus, all cnik, must vanish. This argument does not work in the case \K\ = N, where P$ — const, and \K\ = N — 1, where Pj!} has only the trivial zero x = 0. In the case 3 K = N — 1 we consider instead the condition (f,f£) = 0 with T ' - ^ + A i + ^ T * . ! , Akn:=^£^, n £ N , |*| < n. Applying the identities [8] sinflcosfldflY^-^ (Ncos26-
1)Y^_1,
cosfly n *=^* + 1 y f * + 1 +,4*y n *_ 1 ,
JVeN, n€N,
|*|
and a similar one for cos2 8Yk, the condition (/,T*) = 0 amounts to (JV-S=2(JV + l ) ) A * _ 1 A * c „ _ 2 J b
+[(2JV + 1 ) ( ^ ) 2 + (iV - ^(N
+ l))(Akn+1)2
- l]cnk
(12)
+ ( 2 J V + l ) ^ * + 1 ^ * + 2 c „ + 2 f c = 0. Let now (ni,ki) ^ (N,K) such that cn,k, ^ 0. Setting n := n\ + 2, fc := fc; in (12) and observing that c„l+2k, = c„,+4fc, = 0 we have
which is a contradiction. Thus, cn,k, = 0. The case K = -N + 1 leads again to (12) and the (simpler) case \K\ = N can be excluded as in ref. [5].
• a
This case has been missed in the proof of the Theorem in ref. [5]. The subsequent argument fills this gap.
297 We conclude with some remarks: 1. T h e case A = —K, excluded in the Theorem, refers in fact to an axisymmetric situation: D ^ is equivalent to the axisymmetric direction field _ . j-)| According t o ref. [6], d i m i n depends on the rotation e-iKj)K number of D ; therefore, in general, Lo ^ {0}. 2. Theorem 4.5 in ref. [4] asserted the unique solvability of the problem for all axisymmetric direction fields which are close in a certain sense (roughly: in the norm of the Sobolev space W3'2) to the axisymmetric dipole field D i . T h e Theorem from above demonstrates t h a t this assertion cannot be extended to nonaxisymmetric perturbations of D i . 3. In the 2-dimensional as well as the axisymmetric 3-dimensional case a suitably defined rotation number g of the direction field D turns out to be decisive for d i m i n [6]. This implies t h a t continuous deformations of D , which do not change g, cannot change d i m L n either. According t o the Theorem above there are in the general 3-dimensional case arbitrarily small continuous deformations D of single multipole fields D * with dim L& ^ dim Lj^k. Thus, although there are n a t u r a l generalizations of the concept of rotation number to higher dimensions, viz. the degree of mapping, these concepts do not seem to be useful in the general 3-dimensional case.
References 1. Kono, M.: Uniqueness problems in the spherical harmonic analysis of the geomagnetic field direction data. J. Geomag. Geoelectr. 28, 11-29 (1976). 2. Proctor, M.R.E., Gubbins, D.: Analysis of geomagnetic directional data. Geophys. J. Int. 100, 69-77 (1990). 3. Hulot, G., Khokhlov, A., Le Mouel, J.L.: Uniqueness of mainly dipolar magnetic fields recovered from directional data. Geophys. J. Int. 129, 347-354 (1997). 4. Kaiser, R., Neudert, M.: A non-standard boundary value problem related to geomagnetism. Quarterly of Applied Mathematics 62, 423-457 (2004). 5. Kaiser, R.: On the geomagnetic directional problem: A uniqueness result. Nonlinear Analysis 63, e2129-e2133 (2005). 6. Kaiser, R.: The geomagnetic direction problem: The case of axisymmetry. In preparation. 7. Abramowitz, M., Stegun, LA. (eds): Handbook of mathematical functions. Dover Publications, New York 1972. 8. Varshalovich, D.A., Moskalev, A.N., Khersonskii, V.K.: Quantum Theory of Angular Momentum. World Scientific, Singapore, 1988.
ON T H E MODELS OF NONLOCAL N O N L I N E A R OPTICS *
BORIS G. KONOPELCHENKO AND ANTONIO MORO Dipartimento di Fisica dell'Universita di Lecce and Istituto Nazionale di Fisica Nucleare, Sezione di Lecce via Arnesano, 1-73100, Lecce E-mail: [email protected], [email protected]
We show that under certain assumptions a general model of nonlocal nonlinear response in 1 + 1—dimension is equivalent to the model considered by Krolikowski and Bang for a Kerr-type medium. We derive the limit of weak nonlocality in high frequency regime and discuss the integrable cases.
1. Introduction A paraxial laser beam propagating in a medium with response of the form D = e 0 E +
298
299 it can be solved analytically for weak nonlocality and it admits soliton-like solutions. In the section 3 we analyze a class of weak nonlocal responses in the high frequency regime. The study of high frequency limit can help to detect interesting properties of the beam such as singular phases and consequent vortex type behaviours. Moreover, this study could be useful to construct new nontrivial ansatz for applying a variational method. Looking for solutions which depend "slowly" on the coordinate along the propagating direction, we show that the phase of the electric field is given by an overdetermined system of partial differential equations for the phase. Their compatibility and some integrable cases are also discussed. 2. The nonlocal nonlinear Schrodinger equation. Equation (1) in 1 + 1—dimensional case looks like as follows «92E
„. <9E 2lw
+
aJ ft?
w
-
„
+waD(3)
(2)
= °-
We set £o = 1 without lost of generality. Let us assume a constitutive relation of the following general form +oo
/
R(x - x'; a) N (I{x')) E(x')dx'.
(3)
-oo
Distribution R(x — x'\ a) characterizes the nonlocal response around the point x and a is the "width" parameter (in the following it will be assumed to be depending on the frequency u>). N(I) is an arbitrary nonlinear response depending on the intensity of the electric field I = |E| . In what follows we will propose a way to simplify the general model (3) which will lead us to model considered in the paper [9]. Let us consider a nonlocal distribution R(x — x';a) of width 6R defined as the minimum such that R(x-
x'; a) ~ 0,
\/x' <£[x-SR,x
+ 5R].
(4)
Analogously we introduce the widths 5E and 5N of the electric field and the nonlinear response respectively. Suppose they verify the following conditions 5R ~ SN,
5R «
SE.
(5)
Let us assume a nonlinear response of the form N (I(x)) = N (X),
(6)
300
where X — 72; and 7 = l/SN. Expanding E(x') and N(I(x')) in Taylor series around x, one gets the following approximation of the formula (3) D(3)~/Y
°° R(x-x';a)N(I(x'))dx'^E(x),
(7)
where we kept into account that due to the equation (6) higher orders of the expansion of N (I(x)) are not negligible. Note that the formula (7), in the case of nonlocal Kerr-type medium, leads to the nonlocal nonlinear Schrodinger equation discussed in the paper [9]. For instance, given a bell-shape electric field E = E 0 exp [—x2/2
a
2 exp —(72)\ /o~2 /2a
where 7 = V2a, satisfies the condition (6). Nevertheless, it's easy to see that validity of relations (5) is sufficient to obtain the model (7). For instance, choosing E = Eo/cosh (x) and N (I) = Ia, condition (5) is verified for a large enough. 3. Weak nonlocality and high frequency limit. Let us consider a limit of small nonlocality (5R « 1) such that it is reasonable to expand both E and N in Taylor series. This expansion gives D =N (I(x)) E(x) + R, (N^ „ /lAr92E R ^ 2 N ^
+
(-N— \ 6
^ E ld2N\
dNdE ^^+2-dx^E)+
I^^!5
dx3
+
2 dx dx2
W
If^/V^E 2 9a;2 dx
3
ld N
6 9a:3
\ /
where the distribution R (x — x'; a) is assumed to be normalized c+00
R{x-x';a)dx'
/
= 1
(9)
and Rn are the n-moments f+00 +00
/
R(x-x';a)
{x - x')n dx'.
(10)
-00
In the present section, we are interested to perform the high frequency limit. In particular, we assume that for W—> 00 the effective nonlocality decreases because of rapid oscillation of the electric field. So, one has lim R(x-
x'-a(uj)) = 5(x - x'),
(11)
301 where the parameter a is a function of the frequency such that there exists its limit for to —> oo and 8 (x — x') is the Dirac 5-function. As illustrative example, we focus on the distribution defined as follows - ( - a 3 ( x - x')2 + a)
x' £
X —
x' i
X —
R(x — x';a) = < 0
1 a 1 -,x+ a
1" a 1" a
-,x+ -
(12)
It is straightforward to verify that distribution (12) tends to the 5—function as a —> oo, that is
f
R(x — x';a) dx' = 1
R(x-
x'; a) fix') dx' =
/•OO
/
(13)
J —c
+oo
fix).
-oo
A direct calculus shows that Tin
R2n+i = 0 , n G N U {0} . (14) 2n' a We would like to stress that the present discussion still holds for any nonlocal response whose moments are of the form (14). Let us consider a general dispersion law of the following form Rln —
N = N0 +
N
(15)
2v '
1 + (ZT0W)x
where TQ is usually called relaxation time. In the high frequency limit w —> oo, expanding I = IQ+ u~2l/Ii + ..., one gets N = N0(I0) + ^
^ ,1v l
+O
1
(16)
where Nx is complex-valued and 2u > 0. For 2v < 1, relation (15) is known as Cole-Cole dispersion law10 which substitutes the "classical" Debye law (2i/ = 1) for a wide class of liquid and solid polar media. We perform the high frequency limit in usual way looking for solutions of the form E = E 0 eiujS.
(17)
Moreover, we assume that the electric field depends slowly on the z—variable according to the rule d_ dz
.-21/
d_ ~d~z'
(18)
302
It is straightforward to see that under the assumption given above and under the choice a = 1 + v, the high frequency limit of equation (2) leads, at u)2 and w2~2v orders, to the following system
S^-^NoSl+'-Nu
(19)
where fx = df/dx. System (19) is over-determined and its compatibility condition is equivalent to the following nonlinear equation ±N~ * N0z = -r2N0 N0x + Nlx.
(20)
For Ni ^ 0 solutions of equation (20) are complex-valued, and they implies complex-valued phases. Such solutions are connected in principle with absorption or amplification of the electric field. We note that, once R is assigned, the model (7), at the order w2~2v, provides us with the equation Sz = \NX,
(21)
and the compatibility condition can be written down in the form of conservation law M0z = Nix where M 0 = ±v^VoIf we assume iVi = 0, setting iVo = y 3 equation (20) is reduced to the well known Burgers-Hopf equation (pz = ±ip
(22)
It is solvable with the hodograph method and the solutions are given in terms of the following implicit relation x ±
(23)
where H is an arbitrary function of its argument. Now, let us assume N\ — N\ {No). Even in this case, equation (20) is reduced to the following quasilinear equation N0z = ±7V| (N[ (N0) - r2N0) N0x,
(24)
where iV{ = dNi/dNo. Analogously to the Burgers-Hopf equation it is solved by the hodograph relation x ± i v j (N{ (No) - r2N0) z + L (N0) = 0,
(25)
303 where L is an arbitrary function of JVnOther integrable equations can be obtained considering nonlocal response of the following form L
" V
n
(-an+1(x-x')n+a),
x'&D
0, x' i D
1
where n € N and D = [x — £, a; + ^ ] . In this case, t h e n—moments are (27) ^ - — 2 ^ For any even n one gets the system (19). If n is odd, setting a = 1 + 2v we obtain t h e system
i n
1
(28)
Compatibility condition is equivalent t o the equation NQz = i-riN0N0x
± v^Vo^i*•
(29)
Acknowledgments A.M. is pleased to t h a n k Prof. W. Krolikowski for useful discussions. References 1. L.D. Landau, E.M. Lifshitz and L.P. Pitaevski, Electrodynamics of continuous media vol.8, Pergamon Press (1984). 2. C. Sulem and P. Sulem, The nonlinear Schrodinger equation, Springer-VerlagBerlin (1999). 3. A.W. Snyder and D.J. Mitchell, Science 276, 1538 (1997). 4. C. Conti, M. Peccianti and G. Assanto, Phys. Rev. Lett. 92, 113902 (2004). 5. O. Bang, W. Krolikowski, J. Wyller and J.J. Rasmussen, Phys. Rev. E 66, 046619 (2002). 6. D. Briedis, D.E. Petersen, D. Edmundson, W. Krolikowski, O. Bang, Optics Express 13(2), 435 (2005). 7. B. Konopelchenko and A. Moro, Paraxial light in a Cole-Cole nonlocal medium: integrable regimes and singularities, Proc. of "SPIE Int. Conf. on Optics and Optoelectronics" (August 28 - September 03, 2005, Warsaw) vol. 5949. Preprint arXiv:nlin.SI/0506012 (2005). 8. B. Konopelchenko and A. Moro, Stud. Appl. Math. 113, 325 (2004). 9. W. Krolikowski and O. Bang, Phys. Rev. E 63, 016610 (2000). 10. K.S. Cole and R.H. Cole, J. Chem. Phys. 9, 341 (1941).
CLOSURE OF T H E B A L A N C E LAWS FOR GASEOUS M I X T U R E S N E A R CHEMICAL EQUILIBRIUM
GILBERTO M. KREMER* Departamento
de Fisica, E-mail:
Universidade Federal do Parana, [email protected]
Brazil
MIRIAM PANDOLFI BIANCHI* Dipartimento
di Matematica, Politecnico di Torino, E-mail: [email protected]
Italy
ANA JACINTA SOARES* Departamento
de Matematica, Universidade do Minho, E-mail: ajsoares©math.uminho.pt
Portugal
The relaxation process towards mechanical and chemical equilibrium of a reacting gas mixture is modeled starting from the generalized Boltzmann equation which is linearized following a BGK-type approach. A first-order perturbation scheme of Chapman-Enskog type is performed in a flow regime close to chemical equilibrium and the reactive Navier-Stokes equations of the model are presented in the hydrodynamic form. Moreover, the transport coefficients of diffusion, shear viscosity and thermal conductivity are characterized in an explicit form which shows the dependence on elastic and chemical contributions.
1. Introduction In this paper, the BGK kinetic model for an inert gas mixture proposed by Garzo et al.1 is extended to a four-constituent reacting gas with a bimolecular reversible reaction. In the considered flow regime the reaction is close to its final stage where the affinity, which indicates the tendency of a reaction to proceed, is considered to be a small quantity and the elastic and reactive frequencies are of the same order of magnitude. The chemical time scale is then the one proper of a "fast" chemical reaction. The model equation, * Brazilian Research Council (CNPq). + INDAM-GNFM and the National Research Project COFIN 2003 "Non linear mathematical problems of wave propagation and stability in models of continuous media" (Prof. T. Ruggeri). *Minho University Mathematics Centre and Portuguese Foundation for Science and Technology (CMAT-FCT) through the research programme POCTI.
304
305
according to the content of paper 2 , is deduced replacing each collision operator of the exact generalized Boltzmann equation 3 with relaxation-time terms containing, as reference distributions, perturbed Maxwellians. Their parameters are all determined by requiring that the approximate and exact models share the same balance laws. Such distributions are different from each other when referred to either elastic or reactive interactions, but preserve the same functional form. The relaxation mechanism then differs from the BGK-type approaches of papers 4,5 . Moreover in the cited paper 2 relevant applications are developed connected with transport phenomena and harmonic plane wave propagation. Aim of this paper is to treat the closure of the balance laws at the Navier-Stokes level, in order to derive the hydrodynamic form of the reactive Navier-Stokes equations of the model and the transport coefficients of diffusion, shear viscosity and thermal conductivity. 2. Chemical a n d kinetic framework In this section, some preliminaries on chemistry and kinetic features of a reacting mixture with reversible reaction of type A1+A2 ^ A3+A4 are briefly recalled 3 ' 6 . Chemical potentials. The chemical potential of a-constituent for mixtures of ideal gases where internal degrees of freedom of molecules are neglected is 3, 3. /2-7rm (2nmaafc\ k' Ma — E<x kl - In T - In na + - In (1) 2 2 where ea and na denote the energy formation of a molecule and the particle number density of the a-constituent, respectively, k is the Boltzmann constant, h the Planck constant, and T the temperature of the mixture which is the same also for each constituent. The chemical equilibrium condition S Q = i va^ea — 0 f° r t n e considered reaction becomes
nT + ^ =& +
tf,
(2)
since the stoichiometric coefficients are v\ = z/2 = — ^3 = — ^4 = — 1. Law of mass action. Thanks to (1) and (2), the law of mass action reads
^ . ^ ± £ i - £ i ^ =| l n ^ )
+
l n ( ^ ) ,
(3)
kT 2 \mim 2 / \nznA J where E* is the ratio between the binding energy difference and the thermal energy of the mixture. Above the index 'eq' denotes equilibrium values. Affinity of the reaction. For the forward reaction the affinity7 is v^ mi f nin2nfln^i\ .
VocfJ-a = Ml + M2 - M3 - M4 = kl *—••
In I gq-gq 1 , \Tl3Tl4?l^ 71.2 /
(4)
306
thanks to (1) and (3). Due to (2), A vanishes in chemical equilibrium and tends to a small value during the final stage of the reaction. Balance laws. The linearized Maxwellian distribution function of a-constituent with respect to the differences uf = vf — Vi and Aa = Ta — T is
fa
na
l^fcrJ
ma£a ' 2kT
exp
kT «,-
+
2kT
T
.(5)
where £ a is a peculiar velocity. The underlined term in (5) will be neglected, since only the case where all constituents have the same temperature will be analyzed. The assumption that (5) depends on only one vector quantity, namely uf, implies that the crossed thermal diffusion and diffusion-thermal effects are not expected, so that the Soret and Dufour transport coefficients are absent. The knowledge of fa, given by (5), permits to determine explicitly the linearized elastic and reactive production terms, as shown in detail in paper 2 . Such terms are then equated to the corresponding ones evaluated by means of the model equation (6) leading to the identification of the collision frequencies. So doing, the contributions of mass, momentum and total energy due to both elastic and reactive collisions are the same in the approximate and exact models. This procedure, which assures that the above said models share the same conservation equations for momentum and total energy as well as the same balance laws for the number density of each species, is justified on the basis of the proposed approximation. In fact, each elastic and reactive collision term for each species is separately approximated in the BGK philosophy, as it will be clear in the next section. 3. Model equation According to the discussion previously carried on, the following BGK-type model of the Boltzmann equation is proposed, a = 1 , . . . ,4, dfa C dt + i Q
—
X./ ^a0 v' 0
*a$> ~ ^ a 7 V Q
_
•>a~/) '
(6)
0=1
the first term at the r.h.s. being referred to scattering contribution to constituent a due to elastic collisions with particles of any /3-species, whereas the second one is the inelastic contribution to species a due to chemical reaction. More in detail, the elastic (E) and reactive (R) reference distributions are characterized by position and time dependent fields through (T = E,R) /,cx.8 = Ua
\2^f)
6XP
1
' 2kT
+ AaS + BaS • £a + Ca5 £ 0
(7)
307
where 5 = /3 if T = E, £ = 7 if T = i? with 7 defined in Eq. (10). Coefficients Aca5, B^ 5 , C^g are determined as outlined at the end of Sec.2. The detailed computation carried on in the extended paper 2 leads to
kT
ma + mp
1
'^(^'•-^HHHX -^) m
°® Kf + v^l^M^i
~ f (4 + 2)M 7 « - «?)] J . (86)
fcT
Elastic and reactive collision frequencies. Differential elastic cross sections of rigid spheres are assumed, so that the elastic frequencies ^3 are n ? C & = n^Cfa - § ^2-KkT/ma0
< % X / 3 , a, /? = 1 , . . . , 4,
(9)
^a/3 = (^a + d/3)/2 and d a , d/j being diameters of colliding spheres. Reactive cross sections of the energy model of Present 8 are adopted and the reactive collision frequencies ^ become n ? C £ = n?C& = n > ? 4 ° > ,
( a , 7 ) = (1,2), (2,1), (3,4), (4,3),
(10)
ka being the first approximation to the rate constant. For the forward and backward reactions the first approximation to the rate constant reads fc(°> = ^kT/m12e-^(s1d12)2,
k®l = k^in^n^/inlV^),
(11)
respectively. Moreover, e* = ea/kT is the activation energy such that when a = 1 (reactants a = 1,2) and a = — 1 (products a = 3,4), e\ denotes the forward activation energy, whereas e*_-^ = c\ — E* the backward one. The collision diameters are connected by the steric factor sCT, namely da = sa dap, with s_i = si y/mi2/m34 di2/d34 and 0 < si < 1. 4. T r a n s p o r t p r o p e r t i e s The model equation (6) together with (8a) and (86) is solved through the first-order Chapman-Enskog procedure 9 , for which
where the first approximation fa ' is the known Maxwellian. The insertion of (12) into the model equation (6) yields dJ
^+^d-£r
= -Ec5»(/i 0) +/i 1) -/5 J X-c5 r (/i 0) +/i 1) -/^). (13)
308
On the t.h.s. of (13) only fa ' is considered, since the partial derivatives of fa are connected with the third approximation of fa. When the chemical regime is such that elastic and reactive frequencies are of the same order of magnitude 6 ' 10 , the affinity is small, A ~ 0, and the mixture is near chemical equilibrium. The further step of the procedure consists in eliminating from equation (13) the time derivatives of the partial densities ga, velocity Vi and temperature T, resorting to the balance equations referred to an Eulerian mixture, where diffusion, shear flow and heat flux are absent. Transport fluxes. Insert distributions (12) into the definitions of diffusion velocity uf, pressure tensor pij, heat flux % and production term of particle number density Pa. Performing the integration, the transport fluxes are obtained in dependence on the macroscopic variable gradients and transport coefficients. They are ,a
qi =
A
xe x
af
(
a
~XJx~i +T<(lkT
p\
fdvi,
x
+ 6a
) n«Q^'
Pa =
dvj
2dvr
\
^nan7k^ A , (14)
where x^q = n^/neq denotes molar fractions and d° are generalized diffusion forces defined by df = {dpa/dxi)/p - (dp/dxi) ga/g, JZa=1 df = 0. Equations (14) reproduce the well known laws of Fick, Navier-Stokes and Fourier, respectively. They will be used to determine the transport coefficients of diffusion, shear viscosity and thermal conductivity. The law of Navier-Stokes gives a linear relationship between the pressure tensor and the traceless part of the gradient of velocity. The second term on the right hand-side of the heat flux refers to the transport of enthalpy and of formation energy due to the diffusion of the molecules. Transport coefficients. For the assigned mixture, the transport coefficients of diffusion Dap , viscosity /x and of thermal conductivity A are explicitly displayed in dependence on both elastic and reactive collision frequencies. For sake of brevity, only two diffusion coefficients are presented here 2 mX2 neqkT CR 4. V 4
Cl2 + 3C12 ( e l + 2
C
E
\ I'
S>a7 T 2^/3=1 S>Q/g /
(15)
5 A ( ~ 2 2s I a=l
SR \ ^7
+
neqk2T/ma , \p4 rE
2^/3=1 S>a, /3
309 Navier-Stokes equations. Constitutive equations (14) for the t r a n s p o r t fluxes form 22 constraints, which permit to reduce t h e 30 unknown macroscopic variables Pa, na, uf, Vi, Pij, qi, T to 8 independent unknowns. Consequently, t h e balance laws associated to model equations (6) constit u t e the closed set of reactive Navier-Stokes equations, whose hydrodynamic form reads -g-
+ -Q-. (naUi
dpvi
d_ | -nkT dt
d
+ naVi
=
'
,
"
(16o)
.
(166)
+ Y, nQea + ^pv2
qi + PijVj +
1.....4,
Pc
\
-TlkT + Y
n
a£a + 7;PV2 J Vi
0.
(16c)
a=l
T h e above system of equations was solved in the paper 2 for the problem of plane harmonic wave propagation.
References 1. V. Garzo, A. Santos and J. J. Brey, Phys. Fluids 1, 380 (1989). 2. G. M. Kremer, M. Pandolfi Bianchi and A. J. Soares, "A relaxation kinetic model for transport phenomena in a reactive flow" (submitted). 3. A. Rossani and G. Spiga, Physica A 272, 563 (1999). 4. M. Groppi and G. Spiga, Phys. Fluids 16 4273 (2004). 5. R. Monaco, M. Pandolfi Bianchi and A. J. Soares, "BGK-type models in strong reaction and kinetic chemical equilibrium regimes" (submitted). 6. A. W. Silva, G. M. Alves and G. M. Kremer, "Transport phenomena in a reactive quaternary gas mixture" (submitted). 7. I. Prigogine, R. Defay and D. H. Everett, Chemical thermodynamics, (Longman, London, 1973). 8. R. D. Present, Proc. Natl. Acad. Sci. U. S. 4 1 , 415 (1955). 9. S. Chapman and T. G. Cowling, The mathematical theory of non-uniform gases, 3rd Ed. (Cambridge UP, Cambridge, 1970). 10. B. V. Alexeev, A. Chikhaoui and I. T. Grushin, Phys. Rev. E 49, 2809 (1994).
CHAOTIC PATTERNS IN LOTKA-VOLTERRA SYSTEMS W I T H BEHAVIORAL ADAPTATION
D. LACITIGNOLA Department of Mathematics, University of Lecce Via Provinciale Lecce-Arnesano 1-73100 Lecce, Italy E-mail: [email protected] C. T E B A L D I Department
of Mathematics, Politecnico of Corso Duca degli Abruzzi, 24 I- 10129 Torino, Italy E-mail: claudio. tebaldiQpolito. it
Torino
We study the properties of a n 2 -dimensional Lotka-Volterra system describing competition among species with behaviorally adaptive abilities, in which one species is made ecologically differentiated with respect to the others by carrying capacity and intrinsic growth rate. The case in which one species has a carrying capacity higher than the others is considered here. Stability of equilibria and time-dependent regimes have been investigated in the case of four species: an interesting example of chaotic window and period-adding sequences is presented and discussed.
1. I n t r o d u c t i o n a n d t h e M o d e l Systems of Lotka-Volterra type 1 , 2 are among the fundamental qualitative models in community ecology and are still the most commonly used theoretical framework for describing multispecies interactions 3 . We consider Lotka-Volterra systems describing interacting species with the "ability to adapt", a mechanism based on a kind of learning, discussed in 4 , which is intended to occur over short time scales and therefore not in an evolutionary sense. On the same line of 5 , adaptive abilities are considered by introducing a distributed lag in the interaction coefficients, which assume an integral form. More precisely, we consider the general Lotka-Volterra system describing competition among n species
310
311 dNN-£-=ri[l--g}Ni-'£jaijNiNj with ai:j{t) = / ^
N^NJ^KT.
1 < »,j < n, j + i, {t - u)du
l
(1)
j ^ i,
Ni(t) denotes the density of the i-species at time t, the positive parameters Vi and ki stand respectively for the intrinsic growth rate and the carrying ft e't/T capacity of i-species. The delay kernel KT is chosen as in , KT = —^— since it provides a reasonable effect of short term memory. Having the aim to discuss the role of the interactions, we consider species with the same adaptation rate T and the same carrying capacity k, except for one 7 . In this case, the set of integro-differential equations (1) is equivalent to the following set of ordinary differential equations 4 , ( r/ATi -j± = ritfi(l - citfi) - E j aijNiNj -^
= rNi(l~cNi)-J2jaijNiNj
daij _ NiNj dt T
—
1 < i,j
j jt 1
1 < i, j < n, j / i, i±\ 1
(2)
j j=i
where c\ = — and c = —. k\ k The analysis of the equilibria in (2) has been completely described according to the size of ecological advantage or disadvantage of the first species: the case C\
312 R(v,N%,...,NZ) with v « JV£, S ( S l , A ^ , . . . , A ^ ) with 5 l » A ^ and S*(s 2 , N^2,..., Nj?) with s 2 > AT*2 and s2 < si; the (n - 1) equilibria £ i ; ( ^ S & i , JV£> • • •, Nfr), ( N t , ^ ^ . . .
,Nfr),...,
(Nb\Nb\..
.,Nbh\h)
with h > Nbl; the (n - 1) £ * with b2 > Nbh2 and b2 < h, 2 < i < n (Nb\b2,
Nbh2,..., Nbh% (Nt,Nb2,b2,...,
TVt), ...,(Np,Nfr,...,
Nbh\b2)
To characterize the equilibria, we report only the n-ple of JVj's since a^ = NiNj at the equilibrium. We remind to 4 for a complete discussion about existence conditions and stability properties of equilibria. 3. A Period-Adding Example In this section we analyse adaptive competition among four species. Choosing appropriately the time scale, without losing generality, we can assume r = 1 and consider the following intervals for the relevant parameters: 0.1 < n < 3.5, 0.01 < ci < 0.2, 0.2 < c < 0.8, 0 < T < 180. We consider c\ = 0.01, well representative of the case c\ « c; the equilibria for system (2) are represented in Figurel, where the plane cr\ remains divided into four regions. We focus our attention on Region 1, in particular
Figure 1. Equilibria for the case c\ «
c, i.e. c\ — 0.01.
on the analysis of the time dependent behaviors after the equilibria Bj's have become unstable. We recall that the Bj's are coexistence equilibria
313 with a strong dominance of the i-species on the others and, in this region, they lose stability at T = Tg(c, n ) via Hopf bifurcation, either supercritical or subcritical. The case c = 0.35 provides a variety of interesting dynamical patterns: for T >«s Tg and initial conditions near Bi, according to the value of r\ it is possible to have exclusion by the first species, i.e. the equilibrium S, or species coexistence in the form of periodic or complicated patterns. A
Figure 2. Projection in the {Ni,N2,Nz) T = 26.5: a strange attractor
phase space, c = 0.35, r\ = 0.2 and
further analysis in the time dependent regimes for the case 0.1 < r-\_ < 0.42, turns out to be of great interest when T is varied. In the following, we present results for r\ = 0.2, representative of the above range, and to distinguish different periodic solutions we denote a periodic solution having TO spikes around £?2 by order-m periodic solution. As shown in Figure2, at T = 26.5 and for a set of initial conditions near B2, the system exhibits complicated behaviors showing a chaotic attractor surrounding this unstable fixed point. Purtherly increasing the value of the parameter T and for the same set of initial conditions, a stable order-7 cycle is found, Figure3(a)- Figure4(a), which persists up to T = 29.3 when it is replaced by a chaotic attractor, Figure3(b). At T = 29.5 such an attractor disappears and trajectories are attracted by a stable order-6 solution, Figure3(c)- Figure4(b) which persists up to T = 30.5, changing again into a chaotic attractor, Figure3(d). This phenomenology clearly reveals an interesting alternance of chaotic and peri-
314
(a)
(b)
(c)
(d)
Figure 3. Projection in the (Ni,N2,Ns) phase space (a) The order-7 cycle at T = 28.9 (6) The chaotic attractor at T = 29.3 (c) The order-6 cycle at T = 29.5 (d) The chaotic attractor at T = 30.5
(a)
(b)
Figure 4. Time dependent behavior (a) The order-7 cycle at T = 28.9 (b) The order-6 cycle at T = 29.5
odic windows which are strictly linked to each other. Progressively increasing the value of T, the shape of the cycle is changed by removing, step by step, a loop around B^ and equivalently the period decreases because of the disappearence of a spike. Figure3(b)-Figure3(d) also suggest that chaotic solutions are related to the period-adding sequence. Moreover, Figure3(a)Figure3(c) provide an interesting example of period-adding phenomenon as
315 reported in 8 when read for T decreasing. This interesting scenario deserves further investigations in order t o understand the essential mechanisms underlying t h e involved period-adding sequence: these phenomena are in fact very interesting in t e r m s of bifurcation theory because they can be interpreted as successive local bifurcations. In t h e present case, we think t h a t period-adding is likely t o be related to sequences of saddle-node bifurcations for periodic solutions which deserve further investigations for a complete description. Poincare m a p s 8 have to be defined for this purpose and bifurcation of the fixed point corresponding t o the periodic solution studied. T h e computational effort is therefore much heavier but these investigations t u r n out to be relevant since the phenomenon has been reported in m a n y different fields such as biological neurons 9 , electric circuits 1 0 , chemical reacting systems 1 1 . Furthermore, with a detailed description available, existence of scaling-law properties can be investigated as for the case of a cubic Van der Pol oscillator 1 0 , very much in t h e line of t h e period-doubling phenomenon 1 2 .
References 1. A.J. Lotka, Elements of Physical Biology, William and Wilkins, Baltimore, 1925. 2. V. Volterra, Variazioni e fluttuazioni del numero di individui in specie animali conviventi, Mem. Ace. Lincei 2 (1926) 31-113. 3. J.D. Murray, Mathematical Biology, Springer, 2002. 4. D. Lacitignola, C. Tebaldi, Effects of ecological differentiation on LotkaVolterra systems for species with behavioral adaptation and variable growth rates, Math. Biosci. 194 (2005) 95-123 5. D. Lacitignola, C. Tebaldi, Symmetry breaking effects and time dependent regimes on adaptive Lotka-Volterra systems, Int. J. Bifurcation and Chaos 13 (2003) 375-392. 6. V.W. Noomburg, Competing species model with behavioral adaptation, J. Math. Biol. 24 (1986) 543-555. 7. R.M. May, Will a large complex system be stable, Nature 238 (1972) 413-414. 8. E. Ott, Chaos in Dynamical Systems, Cambridge University Press, 1993 9. W. Ren et al., Period-adding bifurcation with chaos in the interspike intervals generated by an experimental neural pacemaker, Int. J. Bifurcation and Chaos 7, (1997) 1867-1872. 10. M.A.F. Sanjuan, Symmetry-restoring crises, period-adding and chaotic transitions in the cubic Van der Pol oscillator, J. Sound Vibr. 193 (1996) 863-875. 11. M.J.B. Hauser et al., Routes to chaos in the perioxidase-oxidase reaction: period-doubling and period-adding, J. Phys. Chem. B101 (1997) 5075-5083. 12. M.J.Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Stat. Phys. 19 (1978) 25-52.
AN EULER-POISSON MODEL BASED ON M E P FOR HOLES IN SEMICONDUCTORS
S. LA ROSA AND V. ROMANO Dipartimento di Matematica e Informatica Universita di Catania Viale A. Doria, 6, 95125 Catania, Italy E-mail: [email protected], [email protected]
1. T h e kinetic m o d e l A consistent hydrodynamical models for electron transport in semiconductors, free of any fitting parameter, is formulated on the basis of the maximum entropy principle (MEP) for the transport of holes in Si semiconductors. We follow the same approach used in 1 (to which the interested reader is referred for the notation, value of the physical constants and basic assumptions in the kinetic framework), but with a more accurate description of the energy bands, here assumed warped parabolic. The scatterings of holes with both non-polar optical phonons and acoustic phonons are taken into account. The hole energy spectrum in Si is represented by three bands 2 . The first two are the heavy and light valence bands which are degenerate where they reach their maximum. The third band is usually neglected because of its low density of states. The dispersion relations for the two degenerate energy bands of light and heavy holes have a quite difficult analytical expression. In the present paper we assume (at variance of 1 where a single parabolic band has been considered) the warped parabolic approximation e(k)
h2 Ak2 T y/BW 2m*H
+ C2 {k2xk2y + k2yk2z + k2k2x)
(1)
where m*H is the heavy hole effective mass and hk is the crystal momentum which varies on all R 3 , with H the reduced Planck constant. =p refers to 316
317 light and heavy holes respectively and kx,ky, kz are the component of k with respect to the principal crystallographic axes. The parameters A, B and C depends on the specific material. The constant energy surface have a warped form. In the semiclassical approximation the behavior of holes inside the crystal is described by a distribution function fH(x,k,t) satisfying the Boltzmann equation ~
+ Vh • Vx/tf + Y
• V*-^ =
C
^ -
(2)
VH = ^Vkf(k) is the hole velocity2. E represents the electric field that is related to the electric potential by E = — V. This latter satisfies the Poisson equation - V (e(x)V0) = e (iVD(x) - NA(x) - n(x) +p(x)) ,
(3)
where e is the dielectric constant of the material and n, p ND, NA the electron, hole, acceptor and donor densities. C [ / H ] represents the collision term which for each type of interaction can be schematically written as C[fH] = /
[ P ( k ' , k ) / H ( k ' ) - P ( k , k ' ) / n ( k ) ] dk'.
P(k, k') is the transition probability per unit time from a state k to a state k'. The expression of P(k', k) is obtained by means of the detailed balance principle according to which P(k', k) = exp [— (e — e')/ks TL] P(k, k'). For the non-polar optical phonon scattering the transition probability reads 3 P o p (k,k') = #
N0. T N„ op
5[sCk') - e(k) =F hwop]
(4)
> J-
with Kov = i-r—~, where DtK is the optical deformation potential, uop is the optical phonon frequency, p is the silicon density and Nop is the optical phonon distribution at equilibrium op
1 exp(frwop/kBTL)
- 1'
ks is the Boltzmann constant and Tj, is the lattice temperature. The 5 is the Dirac delta function. The acoustic phonon transition probability is Pac(k,k')=#ac9
Nq + 1
- ( 1 + 3 cos2 6>)<5(e(k') - e(k) =F hqv3
(6)
318 where q, the acoustic phonon wave vector, is given by q = \f2k\/\ — cos 6, with cos# = n • n', where n = k/fc and n' = k'/fc'. The coefficient Kac -.2
is given by Kac = 8 '2'' • z-d is the acoustic deformation potential, vs the longitudinal component of the sound velocity and Nq is the acoustic phonon distribution at equilibrium having the same form as (5) with the acoustic phonon energy fkoq — hqvs instead of fajjop. 2. The moment equations and the maximum entropy principle Starting from the Boltzmann equation (2), it is possible to obtain the macroscopic equations for the holes (for details see 4 and reference therein) by multiplying equation (2) by a weight function %l> = ip(k) and integrate with respect to k on R 3 . If one sets M^= /
^(k)/tf(x,M)dk,
JR3
which is the moment of ffj relative to the weight function ip(k) we obtain the following equation dMj,
.,
<Jt
, / < K k ) v - V x / H d k + ^ - / V(k)V k / H dk= / ^(k) C\jH]dk. Jm.3
n,
J^3
yK3
(7) We set the weight function ip(\a) equal to l,fik,e, getting the continuity equation a and the balance equations of momentum and energy b
For this reason and recalling that one has also to include the equation for the electric potential, the resulting model is called of Euler-Poisson type. P, Vfl, WH are the density, macroscopic velocity and macroscopic energy of holes and are assumed as fundamental variables. Therefore there is the problem of closing the system (8)-(10) by expressing the average crystal a
T h e coupling with the electrons is not considered here Einstein summation over repeated indexes is understood
319 momentum PH, the fluxes U1^ and the production terms Cj, and CwH as functions of p, VH, WJJHere we employ MEP to get the required closure relations for the system (8)-(10). According to this principle, if we have a finite number of known moments MA, where MA = / R 3 Vu/dk, A = l,...,n„ then the distribution function /ME, which can be used for an evaluation of the unknown moments, corresponds to the extremum of the entropy functional, under the restrictions MA=
i>AfMEdk:, A = l,...,n. (11) m3 It is important to remark that the maximum entropy distribution represents, in a statistical sense 5 , the least biased estimator of the exact distribution / on the base of the knowledge of a finite number of moments of /• In the case of a sufficiently dilute hole gas the entropy functional is given by the classical limit of the expression arising in the Fermi statistics s = -kB
{fH log f„ [ UH
- fH)dk.
(12)
JR3
By introducing the Lagrangian multipliers A^, we get the maximum entropy distribution function 4 fHIE = exp f — A^A ) . With our choice of the weight functions ipA — (1, v )£)i the maximum entropy distribution function reads /jT=exp
(13)
kB
In order to complete the program, it is necessary to express the Lagrangian multipliers in terms of the fundamental variables by evaluating the constraints (11). By proceeding as in 4 , we expand fjfB with respect to a parameter of anisotropy 5 and solve the resulting equations up to first order in 5. From the constraints (11) we get for the lagrangian multiplier relative to the energy the same expression as the simple parabolic case iW A"
3
2WH
while for the lagrangian multipliers relative to the velocity and density respectively one has Xv = b(Wn)VH, A = -kB log 2 ^ 1 / 2 ( ^ / ^ / 3 ) 3 / 2
320
where b(WH) = - ^ - ^ ,
Jx = j
D-V*dSl,
J2 =
^ - ^ c o s
2
^ a
dfl = sin tidfldip is the elementary volume of solid angle, \JB2 + C2 sin2tf(sin21? sin2
D = AT
2B 2 + C 2 sin2 0
2A=p
-1/-B2 + C 2 sin2 $(sin 2 •d sin2 y> cos2 tp + cos21?) 3. Closure relations Once the explicit expression of fjfE is known, all the closure relations can be obtained from the kinetic definition of S%H, U^, CJp and CwH (see l). The fluxes are evaluated up to first order as ?H = -\WH
b{WH)YH,
SH = | WH VH,
XJ% = \WH
Sij. (14)
The production terms are the sum of a term due to the acoustic phonon scattering and one due to the non-polar optical phonon scattering. The contribution of the interaction of holes with non-polar optical phonons is given by C™ = Kop W~3/2 J, [Nop Bx - (Nop + 1)B2],
C<£> = c™ (WH)VH, where c^\W„)
=- - ^ -
J 3 b{WH) W^'2
[Nop B{ + (Nop + 1)B'2].
J
op
The prime refers to the derivative with respect to Xw while
BX{XW) = exp (huop^ B2{\w)
= exp (-twopXw)
^
KX ( W ^ ) ,
B^XW),
J 3 = 2 f - ^ cos2 tfdfi. Jo. D
with Ki the modified Bessel function of second kind of index 1 and Kop = 3fuuop(m:tJ)3'
2
\ / 3 / 7T
r
,
„.
..
r
l
•
-
r
Ine contribution of the interaction of holes with acoustic phonons is more involved to evaluate because the anisotropy of the scattering. We expand Nq in Laurent's series with respect to hoq/kBTi, as
321 well as the transition rate, getting, after a lengthy calculation, up to first order in 5 and fajq/ksTL C£>
= cM(WH)VH,C%> = - 8 m ^ /
2
i f > ^ < (\VH - \kBTL
where c^
= K'acW]j\WH)
^WHkBTLIn
- (l2h3 + h13)
m*Hv2skBTL+
(I6/23 - 4/13) WH m*H v2 + | (I 13 - I23) m*Hv2s W2H/kBTL\ AT * 3 / 2
.
^<*
c =
MitA •7/1 '
I2=
J D~z'2{d',
hr = \f
T ^ac'
n
• n ' = sin ?9 sin # ' cos(y> -
lb (J Q U \
D~3^',
2
^ ,
/i3 = ^ | ^ - 3 / 2 ( ^ < p ' ) ( l + 3 ( n - n ' ) 2 ) ( l - n - n ' ) | ^ y c o s 2 M f i d Q ' /»» = ! / T ^ , vO (1 + 3 (n • n') 2 ) (1 - n • n') n • n'
^
dQdfl'.
^
Acknowledgments T h e authors acknowledge t h e financial support from P.R.A (ex 60 %) University of Catania, from by M.I.U.R. (PRIN 2004 Problemi Matematici delle teorie cinetiche) and from t h e R T N Marie Curie project COMSON, grant n. MRTN-CT-2005-019417. References 1. G. Mascali, V. Romano, J. M. Sellier, II Nuovo Cimento B 120, 197 (2005). 2. N. W. Ashcroft and N. D. Mermin, Solid State Physics, Philadelphia, Sounders College Publishing International Edition, 1976. 3. Jacoboni C., Lugli P. The Monte Carlo Method for Semiconductor Device Simulation, Springer, Wien, New York, 1989 4. A.M. Anile, G. Mascali, V.Romano. Recent developments in hydro dynamical modeling of semiconductors (2003) 1:54 in Mathematical Problems in Semiconductor Physics, Lecture Notes in Mathematics 1832, Springer (2003). 5. E. T. Jaynes, Phys. Rev. 106, 620 (1957).
A H Y D R O D Y N A M I C A L MODEL FOR SILICON BIPOLAR DEVICES
S. F. LIOTTA ST Microelectronics, Stradale Primosole 50, Catania (CT), 95121, Italy E-mail: salvatore-fabio. UottaQst. com G. MASCALI Dipartimento di Matematica and INFN-Gruppo Collegato di Cosenza, Ponte Bucci cubo 30 B, Arcavacata di Rende (CS), 87036, Italy E-mail: [email protected] In bipolar electronic devices both charge carriers contribute to the total current. Here we present a complete hydrodynamical model of hole and electron coupled transport in silicon semiconductors based on the maximum entropy principle, following an approach already used for unipolar devices1. Generation-recombination effects are taken into account. We employ this model for studying a p-n junction.
1. Introduction The simulation of hole transport by the drift-diffusion model is adequate under the assumption of isothermal charge flow. This assumption is no more valid for devices in which the contribution of holes to current is of the same order or even greater than that of electrons, such as p-n junctions and bipolar junction transistors (BJT's). BJT's are basic elements in manufacturing modern electronic devices such as tiny rectifiers, luminance photocells and many others. For such devices it is necessary to use models for holes of the same type as those already utilized for electrons, that is which include at least the average energy as a fundamental variable. These models are usually known as hydrodynamical models. 2. Silicon Energy Bands The shape of the valence and conduction bands plays a key role in the determination of the charge transport properties of a material. In silicon 322
323
there are six equivalent lowest minima in the conduction band and three valence bands 2 . The energy bands are schematically represented in fig. (1) as functions of the wavevector k along some crystallographic directions. The heavy and light valence bands are degenerate at k = 0, where they reach their maximum, while the third band is the so-called split-off band, which is separated from the first two by the spin-orbit energy A = 0.0443 eV at k = 0. Because of its low density of states and its energy separation the split-off valence band is usually neglected. The energy dependance on the wave vector in the neighborhood of each conduction band minimum or valence band maximum, that is in a valley, is usually approximated by means of analytical expressions. Among these approximations, the most r = (0,0,0)
AdirecHon
X a (1,0,0)
Conductor) bands
enwrjy gap
Forbidden regton
Figure 1. A schematic representation of the silicon band structure along some crystallographic directions.
common ones are the parabolic band approximation
£ being the energy and m* the carrier effective mass, and the Kane dispersion relation £(k)[l + a£(k)] =
n2k2 2m*
, ke
involving a non-parabolicity factor a. 3. Kinetic transport model At a kinetic level charge transport is described by two Boltzmann equations (BE's), one for electrons the other for holes, coupled to the Poisson
324
equation for the electric potential
^ + v a . V x / 0 + ^-Vk/0=C0[/0]+!„[/„,/a], at n eA
a = e,h,
(1)
All quantities referring to electrons and holes are respectively labelled by e and h, if o = e then a = ft, and vice versa. / 0 ( x , k , t ) , a — e,h, are the one-particle distribution functions, v 0 = ^Vk£ a (k), a = e,h, the group velocities, g a , a = e,h, the particle charges, E the electric field coming for the potential (f), K the reduced Planck constant, e the dielectric constant, and AT+(x) and iV_(x) the donor and acceptor densities respectively.
i a (x) =
fadk,
a = e,h,
JB
are the electron and hole densities. Ca[fa] are the collision terms, which in the non-degenerate case have the general form a
C[f] = f [W(k',k)/(k') -w(k,k')/(k)]dk', JB
where tu(k, k') is the transition probability per unit time from a state k to a state k' and it is the sum of the transition probabilities corresponding to the various scattering mechanisms that charges may undergo in silicon. These mechanisms are the scattering with acoustic phonons, with non-polar optical phonons and with impurities 2 . 2a[fa, /a], a = e i h, take into account the electron-hole generationrecombination whose effects are not negligible in the devices under consideration. We consider only the most relevant among these mechanisms, that is the Auger and the Schockley-Red-Hall ones in their relaxation time approximations 3 2o[/o, fa] = - r „ [na na fa - na n*Ma] - Td [na rig. fa - n a nJMa] + na fa -nfMa , +—, ; r—1-, ; T> a = e,h, Th(ne + m) + Te(nh + Hi) where Ta, are constants 3 , Ma the Maxwellians normalized to unit density, ra the carrier life times and rii the intrinsic concentration. a
T h e index a is dropped for simplicity
325
4. Macroscopic transport model Starting from the BE's (1), it is possible to obtain macroscopic equations describing electron and hole transport. To this end, it is sufficient to multiply equations (1) by suitable weight functions ip = ip(k) and integrate with respect to k on B b in order to have
dt
k
/ / ^ ^
+
^ /
e
^
k
= / /
C
^
k
+
/
/
^
*
which are equations for the moments of the distribution functions M^ = fgipfdk. We have used the Kane approximation for the six equivalent valleys in the conduction band, while for the valence band we have utilized a simplified energy band model consisting of a single spherical parabolic band, that of the heavy holes, with an effective mass related to some plausible average in the k-space. Considering the following weight functions both for electrons and holes: ip(k) = 1 , Kk, £{k), hv£{k), we obtain two systems of equations of the form dn
dinV1)
dinPt) ^W):3 dt 9(n5 < ) dt +
d(nU^) ')+nqVtEk= dxi djnFv) + nqEjGij dxi
{ n S
n C w t
=nCl;.
The unknowns n = I fdk, Vi = - [ v*fdk, W=f £(k)fdk, JB n JB n JB
Sl = n- /
v^ityfdk
JB
respectively are the number densities, the average velocities, the average energies and the energy fluxes for electrons and holes, while the extravariables Pi = - f htffdk n JB F» = - / vVSfdk, n
b
JB
= m*Vl + 2aS\ Gij = - f n n •JBh
T h e index a is dropped for simplicity
Uij = - f v'hkt n JB \^-{v*£{k))fdk akj
fdk,
326
and Cn = - / I[fJ}dk, n JB
(%, = n- [
Cw = - f £(C + X)dk, n JB
k\C[f}+l)dk,
JB
Ci = - [ v'SiC + l)dk, n JB
respectively are the fluxes and the production terms. The moment systems are not closed since the unknown variables are more than the equations. The maximum entropy principle (MEP) gives a systematic way to get constitutive relations for the extra-variables on the basis of information theory6'1. The MEP allows one to find distribution functions which depend on the fundamental variables. Substituting these distribution functions in the integral expressions of the extra-fluxes and productions, it is possible to close the moment systems 1 ' 5 .
5. Application t o a p-n j u n c t i o n We have applied the above-described bipolar hydrodynamical model to a ID p-n test diode of length 2/xm. The p and n regions are both l^im long, the lattice temperature is 300-ftT, while the doping profile is the following
{
1 x 10 1 6 cm - 3
p region
j 1 x 10 18 cm~ 3
p region
1 x 10 1 6 cm - 3 n region I 1 x 10 1 0 cm - 3 n region The numerical method which has been used in order to solve the ID spatial system is based on a splitting strategy consisting of combinations of convective steps and relaxation steps 4 . The convective steps use a NessyahuTadmor scheme for homogeneous hyperbolic systems, while the relaxation steps use an implicit Euler numerical integration. We have compared the results of our model (HD) with those obtained by means of a commercial drift-diffusion (DD) simulator (DESSIS). In Fig.( 2) we compare the electron and hole densities obtained with the two models, while in Fig. (3a) the electrostatic potentials are presented. In Fig.( 3b) we show a comparison between the DD I-V characteristic curve (continuous line) and the HD one (asterisks). The two models show a good agreement on this simple device, taken into account that the DD results depend also on the chosen macroscopic mobility model (doping and electric field dependence), while the HD
model depends on microscopic scattering parameters. For this reason we are currently working on a comparison with Monte Carlo results.
327
Figure 2. Electron (n) and hole (p) densities vs x, V=2.829 V.
Figure 3. a Potential vs x, V=2.829 V. b Characteristic Curve
References 1. A. M. Anile, G. Mascali, V. Romano, Recent developments in hydrodynamical modeling of semiconductors, in Lecture Notes in Mathematics, vol. 1823, Springer-Verlag 2003. 2. Jacoboni C., Lugli P. The Monte Carlo Method for Semiconductor Device Simulation, Springer: Wien, New York, 1989 3. P. Gonzales, J. A. Carrillo, F. Gamiz Deterministic Numerical Simulation of Id kinetic descriptions of Bipolar Electron Devices, to appear on SCEE Proceedings 2004. 4. S. F. Liotta, V. Romano and G. Russo, Central Schemes for Systems of Balance Laws, Internat. Ser. Numer. Math. 130, pp. 651-660, Birkhauser, Basel, 1999. 5. G. Mascali, V. Romano, J. M. Sellier, MEP Parabolic Hydrodynamical Model for Holes in Silicon Semiconductors, preprint, 2004 (www.dmi.unict.it/~ romano). 6. I. Miiller and T. Ruggeri, Rational Extended Thermodynamics, Berlin, Springer-Verlag 1998.
ANALYSIS OF A MATHEMATICAL MODEL FOR T H E I N T E R A C T I O N B E T W E E N ALGAE A N D LIGHT
M E R I LISI A N D SILVIA T O T A R O Dipartimento
di Scienze
Matematiche ed Informatiche "Roberto Pian dei Mantellini 44i 53100 Siena, Italy E-mail: [email protected], [email protected]
Magari",
In this paper we study a mathematical model for the interaction of algae with light consisting in two integro-differential equations. In particular, we prove existence, uniqueness and positivity of the mild solution, in a suitable Banach space. Estimates for the number of photons and the algal biomass concentration are also given.
1. The mathematical model Many physical phoenomena can affect the growth dynamics of algae in water: diffusion, settling, absorption of light, self and mutual shading, temperature variation, limitation of nutrient. On the other hand, also the light intensity can be affected by the presence of the algae, because both algae and water can absorb or scatter light. Several different mathematical models outlining the above phoenomena have been studied in literature. We can divide these papers into two groups according to the approach used toward the evolution of the light intensity. In fact, the first group studies only the evolution of the algal biomass (phytoplankton or macro-algae), disregarding the evolution of the light intensity (see, for example, [6] and [8]), whereas the second group gives also an evolution equation for the light intensity (see, for istance, [10]). Obviously, the list of papers on the evolution of algae in dependence of light intensity is very long: we quote, for example, [4], [5], [7], [11] and [12]. There are also several numerical models and probabilistic ones which are often used to estimate the parameters that appear in the other models. In this paper, we study a mathematical model for the growth dynamics of algae, in a bounded region, under light interaction.
328
329 The model we propose here consists of two coupled nonlinear integrodifferential equations. The first one is a Boltzmann-like equation for the radiation intensity / = I(z,n,t), where z represents the sea depth (z = 0 is the water surface, z = b is the bottom, b < +00), y is the cosine of the angle between the direction of the light and the vertical z axis ( — 1 < / Z < 1 ) and t > 0 is the time variable: 1 f+1 1 f~l dtI = -cfidzI - c2I - cpal + - c E s / I dp.' + -casp f Idp!, (1) 2 2 J-i J+i In (1), dt — d/dt, dz = d/dz, E and E s are the total and the scattering cross sections of photons with water respectively, whereas a and as are the total and the scattering cross sections of photons with algae respectively. We recall that E = S a + E s and a — aa + as, where E a and aa are the absorption cross section water-light and algae-light respectively. The unknown function p = p(z,t) is the algal biomass concentration in a water column of depth z, at time t. Finally, c is the light speed. In order to simplify the model, we assume that the quantities I, E, E s , a, as do not depend on the photon frequency. In any case, this dependence does not change the results proved. We assume that I — I(z,fi,t) satisfies the following conditions: I(z,fj,,0)=I0(z,p,),
0
I(0,H,t) = $(ji,t),
+ c o , - l
(2) (3)
I(b, y,t)=0, - l < (j, < 0, t > 0, (4) where Io(z, (i) is a positive quantity and $(//, t) is a given positive function. Note that conditions (3), (4) are consistent from a phisycal point of view. In particular, condition (3) describes the fact that light "comes into" water from the surface z — 0, i.e., the water surface is illuminated, whereas condition (4) means that light cannot come out from the water-bed z = b. The second equation we consider for the model is a diffusion equation for the algal evolution process. In particular, the algal dynamics is assumed to be influenced by diffusion, settling, growth and death processes and remotion by external agents. Along a vertical water column, we consider the following equation: dtp = Kd2zp-wdzp-(3p
1 + PF{-j
f+1
Idfj,'),
(5)
where dt = d/dt, d2z = d2/dz2,dz = d/dz. In Equation (5), the function p = p(z,t) is the algal biomass concentration
330
in a water column, at depth z (0 < z < b < +00) and time t, K is the vertical diffusivity, w is the settling velocity and /3 is a positive constant such that the term (3p represents the loss of algae due to predation, death or removal processes (made, for example, by external agents). Finally, the growth term (due to light absorption) reads as follows, [8], [12]: p(z,t)F(g)
= {acp{z,t)g)(l+Xv(z,t)g)-\
where g = - J_x I(z,fx',t)d/j,'
and a, x
are
(6)
suitable positive constants.
We assume that p = p(z, t) satisfies the following conditions: p(z, 0)=p0{z),
0
+00,
(7)
Kdzp(0, t) - hp(0, t) = 0,
t > 0,
(8)
dzp(b,t)=0,
t>0,
(9)
where po is a given (positive) function, and h represents the speed of the removal process. We assume, h >w, where w is the settling velocity. In this way, there is no accumulation of algae on the water-bed z — b. Moreover, there is no algal flux through the bottom of the water (see (9)). Existence and uniqueness results on the solution of the model are obtained by using linear and nonlinear semigroup theory techniques, [l]-[3]. Estimates of the number of photons and the algal biomass are also given. Note that we are perfectly aware that the time scale for the light intensity evolution and the time scale for the algae lifetime are different, (moreover, the effect of the factor c in (1) is by no means elementary). As a matter of fact, a more realistic model could obtained by putting the left hand-side of (1) equal to zero and making an asymptotic analysis. However, we are also convinced that, only after an accurate analysis of our model, we can achieve a more appropriate asymptotic analysis and face the problem of initial, boundary and corner layer corrections . The asymptotic analysis of the model will be made in a further paper. 2. Abstract formulation and existence of the solution Let us consider the Banach spaces: • Xx = L\V{), V1 = ((0,6) x (-1,1)) and \\h\U = J*dz j \ |A(z,»)\d», • X2 = LX{V2), V2 = (0,6) and norm ||/ 2 || 2 = £ \f2{z)\dz, • X = Xi x l 2 , with norm ||/||x = II/1II1+6H/2II2, where 6 is a dimensional constant. Their relative positive cones are X^,X2 and X+.
331 Let us define the following operators: *) Jfi = 3 / ^ / 1 ^ , with D(J) = Xi; ii) T / j = -codsh, where D(T) = {fx G Xi : 3 z /i € XfJ^fi) = 0,/i G (0,l);/ 1 (6, M ) = 0 , / x G ( - l , 0 ) } ; m ) A/i = - c a c V i , with D(A) = {A G Xi : d ^ G X i ; / i ( 0 , M ) = $(/i,t),/1 G (0,1); A(6,/i) = 0,/x G (-1,0)}; iu) B / 2 = A-^/2 - wdj2, where £>(B) = {/2 € X 2 : tfdz2/2 - wdzf2 € ^ 2 ; # d z / 2 ( 0 ) - h/ 2 (0) = 0; 5 z / 2 (6) = 0}. Note that the operators T,A are different because of their domains. The following assumptions on the function $(/i,t), appearing in the definition of D(A), are also made: ii) ${n,t) > 0, for any t > 0; $(/z,i) = 0, for any t < 0; i 2 ) dt$ G X i , for any t > 0;
i3) 3$ > 0 : ll^e-^/^Hioo <<£,V«; > 0. Note that ||/i||ioo is the norm in X^oo = L°°(Vi), i = 1,2. Since J is a bounded operator (with ||J|| = 1) and T is the generator of a Co-semigroup of contractions, by using perturbation theorems, it is easy to prove that the operator T — cSI + cEs J generates a Co-semigroup in X\, such that | | e x p ( r - c S 7 + cS s J)3i||i < exp(-cE a *)llfli||i> Vgi G Xut > 0. Moreover, the operator B generates a strongly continuous semigroup of contractions in X2 and it can be proved that the type of the semigroup UJ is such that LJ < -w2/4K ([1], [3]). By considering the light intensity I = I(z,fi,t) splitted into two parts, I = Mj = vi + w\, with vi = V\(t) the instensity of radiation of the socalled first flight photons (i.e., those photons which have not yet had any interaction with water or algae) and wi = w\ (t) the other kinds of photons, and putting p = u2, it is possible to rewrite system (l)-(5): ' dfVi = (A — cS)vi — cau2v\, dtWi = (T — cE)wx — cau2wi + cEsJui + ca3u2Jui, du2 = Bu2 - /3u2 + u2F{Ju{), , Wi(0) = vw; wi(0) = ww; vw + ww = u10; u 2 (0) = w2o-
. .
By using affine semigroup techniques, it can be shown that the mild solution of (10) verifies the following system of integral equation: ' Ul(t) = f*[Z(t - s ) e - c ( s + ^ ) ( ' - s W ( z y - « 2 (s))«i(s)+ +cJu1(s){Es+asu2{s))}ds + u10Z(t)e-c^+^t + q1(t), u2{t) = u20e^etB
+ f* e"' 3 ('- s )e B ('- s ) U 2 ( S )F(J( W l (s)))rf S ,
332
with v a suitable positive constant that will be chosen later on, (*fr,t-
— )exp[-(E + zA7)f]
/xe(0,l),
lO,
/J€(-1,0),
and wi(t) = vi(t) + u>i(t),Z(t) = exp(*T),ui(0) = MIO,«2(0) = M20However, system (11) can be written in the following form in the space X: /•'
u(t) = Z(t)u0 + Q(t) +
„
Z(t-
s)G{u(s))ds,
(12)
Jo w i t h u =
(«0'
U O =
(^o)'
( ? =
/Z(t)exp[-c(?:
(o
1
+ v)t}
0
Z(t) = \
0
fca(^~f2)
+
V
f2F(Jfi)
exp(tB) exp(-/ft)
cT,sJf1+casf2Jfi\
G(f)=\
,D(G)=
/Gl,G(/)Gl|
J
It is possible to prove that the family {Z{t),t > 0} is a strongly continuous semigroup such that ||Z(£)|| < e ~ 7 \ with 7 = min{cS a , —w}. The following theorem holds: Theorem 2.1. Let M\,M.2 he two fixed positive constants. If v = M2 and uo e S'"'" (777,1,7712), with mi + $ < Mi,7712 < M2, i/ian system (12) has a unique positive solution u(t) € S+(M\,M2), where »S+(/ii,/12) =
{Si(fci) n x+} x {52(^2) n x+}, with Si(hi) = {fi e xt •. \\fi\\ioo < hi}, hi = m,i,Mi,i — 1,2. It is also possible to give an estimate for the L°°-norm of V\,W\ and U2Prom the second equation of (11), by using the fact that M2 it) belongs to the positive cone of X2, we get: IM*)||2oo < e - ^ m a + / * ^e'^^ds Jo X Hence,
= e~^m2
+ —(I XP
- e -<").
ac ||w2(*)||2oo < ™2 + —5 = "2-
XP
If v = u~2, from the first equation of (11) we get the positivity of ui(t) by a successive appoximation procedure.
333
By taking into account the positivity of U2(t), from the mild version of the first equation of (10), we have that ||vi(i)||ioo < $ + e" cEt ||vio|| 10 o = « ! • In an analogous way, from the mild version of the second equation of (10): t K ( i ) | | i o o < e - c E t | | W l 0 lioo + / e - c E ( ' - s ) c ( E s + tr a u 2 )(vi + \\wi(s)\\loo)ds. Jo /o Multiplying each side of the preceding inequality by exp(—cEt) and applying the generalized Gronwall's lemma, we get: IHWIIloo < l l ^ i o o l l l o o e - ^ - - - ) * +
S
^ + ^ l
i->a -
(
l
_
ec(<,.«-a-=.)t)t
asu2
which, in particular, if S a > asu^, becomes: II
/ . M I ^ i |
\\wi{t)\\loo
||
. ^sVl
< ||wio||ioo H
+ ^sU^Vl
^ ==— = wi^a - crsu2
On the other hand, with v = 1I2 in (10), we get also the positivity of vi(t),wi(t), if vio and ww are in the positive cone of X\. References 1. W. Arendt, Proc. London Math. Soc, 3, 321-349 (1987). 2. L. Barletti, Math. Models Methods Appl. Scl, 10, 877-893 (2000). 3. A. Belleni Morante, A. C. McBride, Applied Nonlinear Semigroups, John Wiley & Sons, Chichester, (1998). 4. E. Beretta, A. Fasano, Math. Methods Appl. Set, 17, 551-575 (1994). 5. G. Frosali, S. Totaro, Transp. Theor. Stat. Phys., 26, 27-48 (1997). 6. H. Ishii, I. Takagi, J. Math. Biol., 16, 1-24 (1982). 7. R. Morton, Math. Biosci., 40, 195-204 (1978). 8. N. Shigesada, A. Okubo, J. Math. Biol., 12, 311-326 (1981). 9. U. Timm, S. Totaro, A. Okubo, Nonlinear Anal, 17, 559-576 (1991). 10. S. Totaro, J. Math. Biol., 20, 185-201 (1984). 11. S. Totaro, Nonlinear Anal., 13, 969-986 (1989). 12. S. Totaro, Transp. Theor. Stat. Phys., 27, 223-240 (1998).
Acknowledgments This work was partially supported by GNFM of Italian CNR, and Par 2004 - Research Project "Nuovi approcci matematici per lo studio di problemi in biologia, medicina, trasporto di fotoni e finanza", Universita degli Studi di Siena, Italy.
SINGULARITIES FOR P R A N D T L ' S EQUATIONS *
G. LO BOSCO, M. SAMMARTINO AND V. SCIACCA Dipartimento di Matematica ed Applicazioni, Universita di Palermo, via Archirafi, 34, 90123, ITALY •{lobosco}{marco}{sciacca}@math.unipa.it
We use a mixed spectral/finite-difference numerical method to investigate the possibility of a finite time blow-up of the solutions of Prandtl's equations for the case of the impulsively started cylinder. Our tool is the complex singularity tracking method. We show that a cubic root singularity seems to develop, in a time that can be made arbitrarily short, from a class of data uniformly bounded in H1.
1. Introduction Prandtl's equations describe the behavior of an incompressible fluid close to a physical boundary in the limit of small viscosity. Short time existence was proved if the initial data are monotonous 11 , and global well-posedness was proved when the outer pressure gradient is favourable16. Short time existence and convergence to the solution of the Navier-Stokes equations was proved 10,12 imposing that the initial data and the Euler matching data are analytic with respect to the tangential variable. A major problem left unsolved for Prandtl's equations is to know whether they are well posed 6,2 in Sobolev (or other non-analytic) spaces without assuming monotonicity, or if they develop singularity in an arbitrarily short time. In this paper we present a numerical study of the process of singularity formation for Prandtl's equations. We will present a class of initial data, contained in Hl, for which the singularity time seems to be arbitrarily short. We consider an impulsively started cylinder, for which Prandtl's equations are: dtu + udxu + vdyu — UdxU = dyyu, dxu + dyv = 0,
(1) (2)
*Work is supported by the INDAM and by the PRIN grant "Nonlinear Mathematical Problems of Wave Propagation and Stability in Models of Continuous Media".
334
335
with the following boundary and initial conditions u{x;Y = 0;t) =v(x;Y
= 0;t) = 0
u{0;Y;t)
=u(2n;Y;t),
u(x; oo; t) = u(x; Y; 0) = U(x) = sin(x).
(3)
Here, u and v are respectively the tangential and normal component of the velocity of the fluid, the domain is 0 < Y < oo and 0 < x < 2TT, in which Y and x are respectively the normal and the tangential variable. Van Dommelen and Shen 15 using lagrangian coordinates found, numerically, that equations (l)-(3) develop a singularity. Here, instead, we solve the above equations using a spectral method for the tangential variable and a finite difference method for the orthogonal variable. Our aim is to use the method of complex singularity tracking 14 ' 7 . The basic idea of the singularity tracking method is to give an initial datum, real analytic, with a complex singularity at distance 5 from the real axis, and to follow the path of the singularity in the complex plane. If the singularity hits the real axis then a blow up in the real solution occurs. To compute the width S of the analyticity strip of a function u, one has to investigate the asymptotic behavior of the spectrum of the solution. In particular if the Fourier coefficients behave like: Uk ~ |fc|(1+a) exp (—5\k\) exp (ix*k),
for large k
then 14 ' 4 u behaves like u(z) ~ (z — z*)a, i.e. has an algebraic singularity at z* = x* + i5 of type a. Evaluating the exponential rate of decay of the spectrum, one knows the distance 5 of the singularity from the real axis. The strategy therefore is: first solve numerically an IVP with a spectral method, second fit the spectrum to locate and classify the singularity in the complex plane. The above strategy, which was first proposed to investigate the possibility of finite time blow up for the 3D Euler equations 14 , has been extensively adopted in the literature, particularly for interfaces problems.
2. The numerical scheme Now we give some detail on the scheme we have adopted to solve the Prandtl boundary layer equations (l)-(3). We use a finite difference method in the normal direction Y and a spectral method in the tangential direction x. The number of grid points in [0,T] and in the normal direction are N and M, respectively. Therefore, we approximate the solution of the Prandtl equations truncating its Fourier series in x-direction at the spatial mode j
336
at time n, as: k=K/2
u(x;jAY;nAt)
=
£
u\/k*'.
(4)
k=-K/2
We write the equation (1) for u as a conservation law, and we treat the diffusion term dyyu with the Crank-Nicolson method. Our scheme is: u 2
k;j-i
, ^„n+l (1 + n)un+l k;j + ^WfciJ + 1
At + -=
Atik[F(u)]ld
[G«
;
• k ' - r ( l - ^ - k
._> -
[G(u)]lJ+i
2
i +
i
+ At sin(x) cos(:c) k;j
2
where F(u) = u , G(u;v) = uv, and /x = At/AY . The expression for the flux in the x direction is standard. For the flux in the Y direction, we use the Richtmyer two steps Lax-Wendroff9'8 scheme: [G(u)}lJ+; u
fc;i+; =
v
*?+u?+i-^+i*(u?+i-u?)
Ui*
2«,+<
J +
(5) (6)
i)-
The normal velocity v is computed using the incompressibility condition, and nonlinearities are treated first inverse transforming back to the physical space, then performing the multiplications and finally transforming to the frequency domain. Aliasing effects are handled using the usual 3/2 — method 3 . All the calculations have been performed using 32-digits arithmetic precision with the multiprecision ARPREC package1; the resolution is K — 1024 and M = 3200 while the time step size is determined by the CFL condition. For more details, and for the stability and convergence properties of the scheme, the reader is referred to 5 , where the authors studied and classified the van Dommelen and Shen's singularity. Some of their results are shown in Fig.l. 3. Early singularity formation We now take the van Dommelen and Shen's solution at t = 1.5 and add a real analytic (in x) perturbation of the form g{x, Y) — Us0 (x) • f(Y) where: ,-50\k\
c«,to
-ia2<\k\
^73
cos(x*k)e
ikx
(7)
and f(Y) is a C 2 function that is 0 at the boundary, reaches its maximum value fmax « 0.54 for Y G [2,3], and decays exponentially fast. The Us0 is
337 i 0.8 0.6
u
i/
°- 4 0.2 i I
of -0.2
i 1
i
/
/
V " > *,\'-
X
\\-.-. *\ -.*\\ -.-. \\ '-\ \ \-X-.V
£
T=0 T=1.5 T=3
-0.4 0
1
x
2
•
it
5(t) 2.4
2.6
2.8
3
2.6
2.8
3
t
2.6
2.8
3
t
0.45 0.3 0.25
oc(t)"2.4
1.96 1.95 1.94
/
K
0.1 0.05 0
x*(t) 2.4
Figure 1. To the left it is shown the tangential velocity u at location V = 4. At the singularity time T = 3 a shock structure is visible at x ss 1.942. To the right the results of the analysis of the spectrum. The strip of analyticity shrinks to zero at t ss 3, as the result of a cubic root singularity (a = 1/3) hitting the real axis. The singularity location x* agrees with the shock position.
analytic in a strip of width J 0 ; moreover for all So > 0, Us0 € Hs Vs < 7/6, and uniformly on So, \\USQWH1 < c - The role of a is to tune the strength of the perturbation. The presence of the cos(a;*fc) in (7) means that we are considering a "dipole singularity" i.e. two singularities of opposite sign placed at x*±e + iS with e —> 0 and of strength <x/(ke). We chose x* « 2.17 which is the location 5 of the singularity of the VDS solution at time t = 1.5. In Fig. 2 we show the result when the perturbation we are imposing to the VDS solution at t = 1.5 has a singularity located at a distance from the real axis So = 0.025, with a = 1.5. In this case a cubic-root singularity forms at time T = 0.297. The perturbation (7) of the VDS solution has led to an early singularity formation. We now consider different perturbations with different SQ. In Fig. 3, to the left, we show the shrinking to zero of the strip of analyticity for different values of So- The closer the singularity is to the real axis, the sooner the singularity develops. Qualitatively, it seems that dividing by four the initial distance of the singularity it leads to the halving of the singularity time. This is evidenced to the right of Fig. 3 where the relationship between the singularity time and the <5o is reported in log — log scale: the available data lay on a straight line of slope 1/2. It is interesting to compare the above behavior of the critical times for Prandtl's equations to Burger's equations with initial data sin (x)+Ug0. The results are shown in Fig.4. This behavior is well understood for Burger's equation, and explains as follows. The data sin (x) + Ug0 are uniformly
338
0.03 0.015
1 0.8 p..
i
0.6
0
'-.
i
"
0.3 •
a(t)
0
/
0.2
0.6
0.2 0
0.1
'-.
• — *-. 1/ V-.
0.4
•
5(t)
0
0
T=0 T=0.15 T=0.297
2.17 2.14
• 0.1
0.2
0.3
x*(t) 0.3
t
Figure 2. Let So = 0.025. To t h e left we show the formation of a shock at Y = 5 after a time t ss 0.297. To the right the shrinking to zero of the strip of analyticity and the classification of the singularity as a cubic-root singularity.
—
log(T) o
50=0.1 8o-0.0S
s
* ».
50=0.025 8<=0.0125
^C;::
* . „ _
' iog(50) Figure 3. To the left one can follow in time the distance of the singularity from the real axis for different initial So- To the right log5o versus logT c . The singularity time seems to go to zero when So goes to zero.
'
' o.i 0 08
•
—
50=0.05
\
0.02
logfTJ •
50=0.025
0.06 0.04
8„=0.1
50=0.0125
^i\ ";-
*-*«..
•
0.097 0.138
0.195
log^
Figure 4. The shrinking of the analyticity strip(left) and the critical times (right) for Burger's equation with initial data sin(i) + Ug0. The behavior of the critical time is the same as for Prandtl's equations.
bounded in H3 for s < 7/6, but are not uniformly bounded in H3/2. In fact sup |9XC/,50| —» oo when So —* 0: in an arbitrarily short time it forms a singularity that drives 13 the solution out of H1.
339 T h e comparison between Fig.3 a n d Fig.4 suggests t h a t t h e situation for P r a n d t l ' s equation is t h e same as for Burger's equation. 4.
Conclusions
We have used a spectral m e t h o d t o solve P r a n d t l ' s equations; we have adopted t h e complex singularity tracking m e t h o d t o classify the van Dommelen and Shen singularity as a 1/3 singularity and to determine t h e singularity time. We have also exhibited a class of perturbations of the VDS solution t h a t lead to early singularity formation. This class is uniformly bounded in H" for s < 7/6. A study of the behavior of the singularity time in this class suggests t h a t t h e singularity time can be made arbitrarily short. This situation for P r a n d t l ' s equations shows a close similarity t o t h e behavior of Burger's equation in H1. Therefore there is some numerical evidence t h a t , starting from a class of d a t a uniformly bounded in H1, the singularity time (after which t h e solution is not in H1) for P r a n d t l ' s equations can be made arbitrarily short .
References 1. Y. Hida, X.S. Li and D.H. Bailey, "Quad-Double Arithmetic:Algorithms, Implementation, and Application", manuscript, Oct. 2000; LBNL-46996. 2. R.E. Caflisch, M. Sammartino, Z. Angew. Math. Mech., 80 733, (2000). 3. C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Springer Verlag, New York. 1991. 4. G.F Carrier, M. Krook, C.E. Pearson, Function of a complex variables: Theory and Technique, McGraw-Hill, New York, 1966. 5. G. Delia Rocca, M.C. Lombardo, M. Sammartino, V. Sciacca, to appear on Appl. Numer. Math., (2005). 6. W. E, B. Engquist, Comm. Pure Appl. Math., 50 1287, (1997). 7. U. Frisch, T. Matsumoto, J. Bee, J. Stat. Phys., 113 761, (2003). 8. L. Hong, J.K. Hunter, Comm. Math. Sci. , 1 293, (2003). 9. J.R. LeVeque, Numerical Methods for Conservation Laws, Birkhauser Verlag, Basel, 1992. 10. M.C. Lombardo, M. Cannone, M. Sammartino, SIAM J. Math. Anal., 35 987, (2003). 11. O.A. Oleinik, V.N. Samokhin, Mathematical models in Boundary Layer Theory, Chapman & Hall/CRC, 1999. 12. M. Sammartino, R.E. Caflisch, Comm. Math. Phys.192 433,(1998). 13. M. Sammartino, V.Sciacca, submitted, 2005. 14. C. Sulem, P.-L. Sulem, H. Frisch, J. Comput. Phys., 50, 138, (1983). 15. L.L. van Dommelen, S.F. Shen, J. Comp. Phys. , 38 125, (1980). 16. Z. Xin, L. Zhang, Advances in Math. , 181 88,(2003).
NONLOCAL B O U N D A R Y CONDITIONS FOR T H E N A V I E R - S T O K E S EQUATIONS *
M. C. L O M B A R D O A N D M. S A M M A R T I N O Dept. of Mathematics, University of Palermo via Archirafi 34, 90123 Palermo, ITALY. E-mail: {lombardo}{marco}@math.unipa. it
In this paper nonlocal boundary conditions for the Navier-Stokes equations are derived, starting from the Boltzmann equation in the hydrodynamic limit. Basing on phenomenological arguments, two scattering kernels which model non-local interactions between the gas molecules and the wall boundary are proposed. They satisfy the global mass conservation and a generalized reciprocity relation. The asymptotic expansion of the boundary value problem for the Boltzmann equation, provides, in the continuum limit, the Navier-Stokes equations associated with a new class of nonlocal boundary conditions.
1.
Introduction
In many practical cases, like geophysical models or turbulence modelling, the no-slip boundary conditions usually imposed for the Navier-Stokes Equations (NSE) fail to correctly describe the interactions between the fluid and a solid boundary and the problem of finding appropriate boundary conditions is a major one. In the framework of turbulence modelling, the approach known as Large Eddy Simulation (LES) seeks to predict local spatial averages of the fluid's velocity above a preassigned length scale. The mathematical problem of finding appropriate boundary conditions in LES has been tackled in 4 ' 5 . The nonlocal boundary condition for the coarse grained Navier-Stokes that have been proposed for the averaged flow u are 5 : Qn = 0
and
(3u • r + 2Re~1n
• D(u) • r = 0,
(1)
where u represents the velocity averaged with a gaussian filter, n and r are "This work is supported by the INDAM under t h e grant PRIN: "Nonlinear Mathematical Problems of Wave Propagation and Stability in Models of Continuous Media 2003-2005"
340
341 the wall normal and tangential unit vectors, respectively, Re is the Reynolds number and D»j(u) = ^(f^- + •%£) is the velocity deformation tensor. Conditions (1) are the Robin boundary conditions for the averaged velocity. Recently1 a mean field approach to the Boltzmann equation, filtering out subgrid scales, led to a subgrid turbulence model. It was shown1 that, as for the Navier-Stokes equations, the Smagorinsky subgrid model enjoys a consistent derivation from the kinetic theory. Motivated by the above considerations, in this paper we want to derive Robin-type boundary conditions for the macroscopic variables taking into account the effect of nonlocal interactions at the wall, starting from a kinetic description. We shall investigate the steady behavior of a fluid on the basis of the Boltzmann equation on a 3-dimensional half space. Two simple models are introduced whose corresponding scattering kernels generalize the Maxwell gas-surface interaction law. The first model describes a situation in which particles can penetrate the wall (thought as a lattice) and can experience a specular reflection from the inner layers of the wall. In the hydrodynamical limit this model leads to BC for the NS equations weakly non local; in the sense that the heat flux at the boundary is driven from (besides the classical term expressing the temperature difference between the fluid and the wall) the divergence of the velocity at the wall. In the second model (from which, in the hydrodynamical limit, we derive (1)) large structures (in the Fourier sense) of the fluid are specularly reflected, while small structures penetrate the wall, get in thermal equilibrium with it and are re-emitted through a Maxwellian. The proposed kernels are shown to satisfy a nonlocal mass conservation and a generalized reciprocity relation. 2. N o t a t i o n s We shall consider a fluid confined to the 3 - D half space D = R+ x R x R , the x = Xi(i = 1,2,3) are the dimensionless Cartesian coordinates of the physical space, xi is the unit vector normal to the boundary wall, y = (X2, £3) is the position of a point on the plane x\ = 0, C = (Ci > C2, C3) is the dimensionless molecular velocity; / ( # , £ ) is the dimensionless distribution function of the fluid molecules; p is the dimensionless density, w^ the dimensionless fluid velocity, T, p are the dimensionless temperature and the pressure of the fluid, Tw, pw, pw, uiw are the dimensionless wall temperature, density, pressure and velocity, respectively and R is the fluid constant per unit mass. We also introduce the Maxwellian /o with Vi = 0,p = po and T = To:
h = 7^fFF\E^
£(C)^exP(-C2), C=(Cf)1/2 = i a
(2)
342
In what follows we shall consider the state of the gas close to the Maxwellian distribution function /o given by Eq. (2). The nondimensional perturbed variables are given by: <j> = f/E-\, Tyj
==
±w
w = p-\, 1,
LU-w
==
Pw
r =t-l, ^j
*w
==
Pw
P = p-1,
(3)
*• •
The steady Boltzmann equation in dimensionless form reads:
C;|^
TO+W<«],
^=^Kn
(4)
where J{4>) and £(
P = W + T + UT.
{ )
Let us consider a particle hitting the wall: let £' = (Ci J C21 C3) a n d C = (Ci 1C2, C3) be the velocity of the impinging and of the outgoing particle. For a simple boundary one usually assumes a fluid particle-surface interaction law of the following form: for x\ — 0, C, • n > 0 |C-n|£(C)(i + # y , 0 ) = / , JQ
\C'-n\R(C,^C,y)E(C')(i
+
(6)
n
where n is the unit vector normal to the boundary and R(C' —> £; y) is the scattering kernel, i.e. the probability that a molecule impinging the wall at point y with velocity £' is scattered with velocity between £ and £ + d£. The scattering kernel has to satisfy the positivity condition, the conservation of mass and the Reciprocity relation 3 . A widely used scattering kernel is the one proposed by Maxwell:
R«' -C,y) = (i-a)tf(Ci+Ci)g(C2-C2)+« ^1
NV2Ci
exp [-te-"™) 2
where ([ < 0, Ci > 0 and uiw is the wall velocity. The above model prescribes that an 1 — a fraction of the molecules is specularly reflected at the surface of the wall, while the remaining a fraction of the molecules is in thermal equilibrium with the wall. All the interactions are local in space. 3. The nonlocal scattering kernel I In this section we want to propose a different model of the interaction between the gas and the wall. We suppose that, other than being reflected
343
at the surface, the molecule can pass some layers of the wall without experiencing any impact and then can be specularly reflected by some inner molecule of the wall lattice. This introduces a nonlocality effect into the scattering kernel: in fact if the molecule hits the wall at point y' on the wall xi = 0, it will travel for some distance inside the wall, will hit the lattice and will come out at a different point y. Since it is specularly reflected, the impact will take place half-way between y ' and y. Let y = (x2,Xs),
where (py,ay) are the polar coordinates on the plane (12,13) centered in y', and (p^,a^) are the polar coordinates on the plane (C2,Cs)- Let C'y be the tangential velocity of the incident particle. The probability of the above process taking place is the product of three different probabilities: the probability of the particle travelling for a distance py/2 = (|y — y'|)/2 without hitting any other molecule; the probability of having an impact between py/2 and py/2+dpy and the probability of travelling again for py/2 without impacts. We shall assume each process to be governed by a Poisson distribution function with ^ as mean value. Moreover the tangential part of the incident velocity C'y has to be parallel (with the same sign) to the vector y — y', which will introduce the term 5(ay — al). Then one has: Probability of having only one ~ ^ | exp (— —) 5(ay - a'^)dpyday (8) £ £ y impact at y' + ^ This term will affect the specular reflection part of the scattering kernel. On the other hand, we suppose that the molecules which experience multiple scattering inside the solid will obey the diffusive reflection law at the boundary. Hence the nonlocal scattering kernel which takes into account both the nonlocal specular reflection and the diffusive reflection is: R(C' - C,y' -» y) = (1 - eP) J(Ci + Ci) S«y - C'y) 4 e " ^ S(ay - a'() 2
+ £P - m — ^ 2 &
exp
«i
"^l^y-y'),
forCi<0,Cl>0
(9)
The above scattering kernel satisfies the positivity condition. Moreover it satisfies a nonlocal mass conservation law and the reciprocity relation in a nonlocal form2. The asymptotic procedure to derive the Navier-Stokes equations and the corresponding boundary conditions for the fluid dynamic variables is standard 6 . To first order in e we get 2 the following BC: ui = 0,
(10)
344
v / ^ 7 i g | i - 8 / 3 ( K i - u i w ) = 0,
(i = 2,3)
Vf,|l-Wr-^-^(g + g).0,
(11)
(.2)
Equations (10), (11) and (12) are the boundary conditions for thefluiddynamic equations: Eq. (10) is the usual no-flux boundary condition. Eq. (11) are the Robin boundary conditions for the tangential component of the velocity. Eq. (12) is the usual Robin BC for the temperature plus an extra term which is proportional to the tangential divergence of the velocity. Introducing the Fourier transform with respect to y, denoting by u; = (ci>2, W3) the dual variable of y, namely: / ( £ , w) = !F{f{C,,y)), and denoting with K the specular reflection part of the scattering kernel (9), one easily verifies that:
Me1 - c,«) - w c - c,y)) = (i -
enm+®?l-$£?-®,
(13) where |w| 2 = w\ + wf. Therefore the kernel can be interpreted as a lowpass filter: it allows large structures (small u>) to pass the filter and hence to experience specular reflection. On the other hand, small structures are cut-offed, do not experience specular reflection and finally get in thermal equilibrium with the wall (that is, they enter in the count of the Maxwellian part of the scattering kernel). It is with this interpretation in mind that, in the next section, we shall suggest a different scattering kernel. 4. The nonlocal scattering kernel II In this section we pursue the idea of constructing a nonlocal scattering kernel that can act as low-pass filter. Instead of a power-law low-pass filter, as the one in Eq. (13), we propose a Gaussian filter, as it is common in turbulence modelling. Namely, for £i > 0, and with /3 > 7: R(C' - C. y' - y) = (1 - e/3) <5(Ci + Ci) <5«2 - <2) 5(Cs - £3) 6{y - y') (x2-x'9)2+(x3-x^)2
+ ej8(tx
+ Ci) «5(C2 - C2) KC3 - &)2 e ( / 3 - 7 ) ...
^ ,, .
_«i-uw<)2
.,,,
The first and the third term on the right hand side of (14) are the same as in the Maxwell kernel. The second term accounts for a small (0(e))
345
fraction of molecules which are nonlocally specularly reflected: particles that hit the wall at y ' are reflected from a inner layer of wall molecules and exit at y with Gaussian probability. The scattering kernel given by (14) acts as a Gaussian low-pass filter, whose filter width is A. The above scattering kernel satisfies the positivity condition, the conservation of mass and the reciprocity relation. We now consider the limit e —> 0. Following the same lines as in Sec.3, one finds the following boundary conditions for the fluid dynamic variables: «i = 0, 4\^li7rr
- P(ui-uiw)--y[uiw-G(\,X2,x3)*Ui]=0
(15) (i = 2,3)
10V^72J^+8/3(TWI-T)-97[T-G(A,T2,Z3)*T] = 0
(16) (17)
where with G(\,X2,xs) we have denoted the 2D Gaussian with standard deviation A, while * denotes the convolution in R 2 . Equations (15)-(16)(17) are the boundary conditions for the Navier-Stokes equations: (16) is the Robin boundary condition for the tangential component of the velocity plus an additive term, which is proportional to the difference between the wall velocity and a filtered flow. Notice that, if one takes the convolution of Eq. (16) with the gaussian kernel, one obtains the same boundary conditions of the near wall model 5 . Analogously, Eq. (17), prescribes a Robin condition for the temperature plus a nonlocal extra term. It is interesting to notice that, to second order, one gets the following BC for the mass flux: u^
= -^= [T~G(X,X2,X3)*T] 4V7T
(18)
The above condition prescribes (locally) a non zero mass flux at the boundary. This effect is due to the penetrative BC for the Boltzmann equation. If one integrates (18) on the whole boundary one gets that the total mass flux is zero. References 1. S. Ansumali, I.V. Karlin, and S. Succi, Physica A, 338 379, 2004. 2. R. E. Caflisch, M. C Lombardo, and M. Sammartino. in preparation, 2005. 3. C. Cercignani. The Boltzmann Equation and Its Applications. Springer-Verlag, New York, 1988. 4. G.P. Galdi and W.J. Layton. Math. Models and Meth. in Appl. Sciences, 10 343, 2000. 5. V. John, W.J. Layton, and N. Sahin. Comput. Math. Appl., 48 1135, 2004. 6. Y. Sone. Kinetic Theory and Fluid Dynamics. Birkhauser, Boston, 2002.
ON T H E SHIZUTA-KAWASHIMA COUPLING CONDITION FOR DISSIPATIVE H Y P E R B O L I C SYSTEMS A N D ACCELERATION WAVES
JIE LOU Department
of Mathematics, Shanghai University 99 Shangda Shanghai 200444, P. R. China E-mail: [email protected]
Road,
TOMMASO RUGGERI Department (CIRAM)
of Mathematics and Research Center of Applied Mathematics University of Bologna, Via Saragozza 8, 40123 Bologna, Italy E-mail: [email protected]
We consider dissipative hyperbolic systems of balance laws in which a block of equations are conservation laws. In this case, a coupling condition firstly introduced by Shizuta and Kawashima (K-condition) plays a fundamental role for the global existence of smooth solutions for small initial data and for the stability of constant states. Nevertheless the example by Zeng proves that the K-condition is only a sufficient condition. Using acceleration waves, we prove the necessity of t h e weak K-condition in which the condition is required only for the genuine non linear characteristic velocities and not for the linear degenerate ones.
1. Introduction The physical laws in continuum theories are balance laws: $ u + 0 ^ ( 1 1 ) = f(u)
(1)
where u, Fl and f are RN vectors, dt = d/dt; di = d/dxi and repeated indices mean summations from 1 to 3. The entropy principle require that any solutions of (1) are also solutions of the supplementary law: dth(u) + dih^u) = E(u) < 0
(2)
with /i(u) convex entropy function. In the general theory of hyperbolic conservation laws and hyperbolicparabolic conservation laws, the existence of a strictly convex entropy func346
347
tion, which is a generalization of the physical entropy, is a basic condition for the well-posedness (see for example K. 0 . Priedrichs and P. D. Lax 1 , S. Kawashima 2 ) . In fact, if the fluxes and the production are smooth enough in a suitable convex open set D £ Rn , it is well known that the Cauchy problem has a unique local (in time) smooth solution for smooth initial data. However, in the general case, even for arbitrarily small and smooth initial data, there is no global continuation for these smooth solutions, which may develop singularities, shocks or blowup, in finite time (see for instance Majda 3 and Dafermos 4 ) . On the other hand, in many physical examples, thanks to the interplay between the source term and the hyperbolicity global smooth solutions exist for a suitable set of initial data. For such a system the relaxation term induces a dissipative effect. This effect then competes with the hyperbolicity. If the dissipation is sufficiently strong to dominate the hyperbolicity, the system is dissipative, otherwise, if the dissipation and the hyperbolicity are equally important, only part of the perturbation diffuses. In the latter case the system is called by Zeng of composite type 5 . The production term f(u) in (1), represents the dissipation but, unfortunately, in the physical cases, not all the components of f are different from zero. In fact several examples of non equilibrium theories as the Extended Thermodynamics or the mixture of gases 6 are represented by systems (1) in which a block of equations are conservation laws (typically in Extended Thermodynamics: conservation of mass, momentum and energy), i.e. the production is:
2. Shizuta—Kawashima condition The coupling condition discovered by Shizuta and Kawashima (Recondition) 7 ' 2 states that the dissipation terms present in the second block of the equations has an effect also on the first block. This plays a very important role for global existence of smooth solutions. The K-condition reads: In the equilibrium manifold any characteristic eigenvector is not in the null space ofVf, i.e.:
348
Vf-d^
^0
V j = l,...,iV
(3)
E
where d represents the right-eigenvector of the hyperbolic system (1): ( A „ - A I ) d = 0,
An=Aini,
A* = VF 1 ;
(n = (m) : ||n|| = 1),
and -B stands for the equilibrium state, i.e.: f (u B ) - 0.
2.1. Global Existence
of Smooth
(4)
Solutions
If (1) is endowed with a convex entropy law (2) and the system (1) is dissipative (S vanishes in equilibrium and hence attains its maximum value 8 9 ' ), then the K-condition becomes a sufficient condition for the existence of global smooth solutions provided that the initial data are sufficiently smooth. Hanouzet and Natalini 10 in one-space dimension and Yong u in the multidimensional case, have proved the following theorem: Theorem 2.1. (Global existence of smooth solutions) Assume that the system (1) is strictly dissipative and the K-condition is satisfied. Then there exists 5 > 0 such that, if ||u(:r,0)||2 < 8, there is a unique global smooth solution, which verifies u G C° ([0, oo); H2(R)) n C1 ([0, oo); H1
Moreover Ruggeri and Serre 12 have proved in the one-dimensional case that the constant states are stable: Theorem 2.2. (Stability of Constant State) Under natural hypotheses of strongly convex entropy, strict dissipativeness, genuine coupling and "zero mass" initial for the perturbation of the equilibrium variables, the constant solution stabilizes
||u(t)||2 = o(r 1 / 2 )
349 In both theorems it plays an important role the possibility to put the original system (1) in a very special symmetric form thanks to the introduction of the main field u' = Vh introduced first by Boillat 13 in a classical context and by Ruggeri and Strumia in a covariant relativistic formulation 14
There are many examples of dissipative systems satisfying the Kcondition: the p-system with damping, the Suliciu model for the isothermal viscoelasticity, the Kerr-Debye model in non linear electromagnetism and the Jin-Xin relaxation model 10 . Moreover quite recently it was proved that the K-conditions is true also in the Extended Thermodynamics of gas 15 and in a binary mixture of Euler fluids in presence of chemical reactions 16>17. 2.2. The Zeng
example
On the other hand, there is an example due to Zeng 5 in which global existence holds without K-condition. The example comes from a simple model of a gas in a thermal non equilibrium state, and is governed by the one-dimensional equations: ' vt - ux = 0 Ut + Px = 0
(5)
' (e + V ) +(puh = 0 Q-q* V
T
with e = ei+q;
p=p(y,ei);
Q = Q(y,ei);
r = r(v,e 1 )
where v,u,p and e are respectively the specific volume, velocity, pressure and internal energy of the gas, while e\ and q represent the total specific translational and rotational internal energy and the internal vibrational energy of the gas. The system (5) has two characteristic eigenvalues: A^) = -Cf;
A(4) = cf;
cf = y/ppei
dp
dp
where
-Pv,
350
that are genuinely non linear, i.e. VA • d ^ 0
(6)
and two linear degenerate waves (exceptional waves): VA • d = 0 with velocities \W = X^ = 0. In the present case the Shizuta-Kawashima condition is violated for one of the eigenvectors corresponding to the zero eigenvalue, while Vf • d ^ ^ 0 E 4
and Vf • d^ ) ^ 0. Therefore it is proved that the K-condition is only a sufficient condition to have global existence of smooth solutions. 2.3. Weak
K-condition
Prom a physical point of view, it is important to find the minimal part of the K-conditions that is necessary for global existence of smooth solutions. Starting from the Zeng example, and with the aim to give a first contribution to this open problem, we consider the following weaker condition: Weak K-cond it ion:
Vf • d&
E^°
W
only for j G {!,..., N} for which VA(J) • d& Therefore in particular Vf • d
E
can be zero for linear degenerate waves, E
as in the Zeng paper. If there exists local exceptional waves, i.e. there exists some u such that VA(u) • d(u) = 0, than (7) must be verified only if U\E ^ u. In other word we impose the K-condition only for equilibrium fields for which the corresponding wave has genuine non linear character. We justify the necessity of the weaker K-condition in the class of solutions corresponding to the weak-discontinuity waves (acceleration waves).
351 3. Acceleration Waves For a generic system (1) it is possible to consider a particular class of solutions that characterizes the so-called weak discontinuity waves or - in the language of continuum mechanics - acceleration waves. There exists a moving surface (wave front) T of cartesian equation 4>('x.,t) = 0 that separates the space into two subspaces. Ahead of the wave front we have the known unperturbed field uo(x, t) and behind the unknown perturbed field u(x,t). Both the fields u 0 and u are supposed regular solutions of (1) and are continuous across the surface T, but are discontinuous in the normal derivative, i.e., du
[«] = 0,
= n ^ o,
(8)
where the square brackets indicates the jump 3
[•] = (-)*=o- ~ (•)*=(>+• We have the following well-known results 18>19>20: (1) The normal velocity V = —
is equal to a characteristic velocity
evaluated in Uo: V = A(u 0 ). (2) The jump vector I I is proportional to the right eigenvector d (corresponding to the eigenvalue A) evaluated in uo:
n = nd(uo). (3) The amplitude II satisfies a Bernoulli equation along the bicharacteristics curves: ^ + a(t)U2 + b(t)U = 0 (9) at where d/dt indicates the time derivative along the bicharacteristics curves, and a(t) and b(t) are known functions of the time through uoa
For simplicity we denote by g and go the values of a generic quantity g evaluated on F respectively for
352
For example in the case of one space dimension, we have
20
:
d dx — = dt + Xodx; — = Ao (characteristic); Ao = A(uo) dt dt a(i) = ^ ( V A - d ) 0
W = W duj ^ - diii ^ J ^dt + (l»>>(VA-d)-V(>.f).dl ^
+ (VA-ux)o^=0;
( 0)
'
<^(0) = 1.
where 1 indicates the left eigenvector of A = V F , which, in view of the hyperbolicity, may be chosen such that 1 • d = 1. The solution of (9) is n(t) =
v
° > 1 +11(0) & a(Oexp ( - J o C & m ) dC
(li)
If the considered wave satisfies genuine nonlinearity (6) the coefficient a(t) ^ 0 and may always be considered (with an appropriate choice of the right eigenvector) positive. In this case there exists in general a critical time tcr such that the denominator of (11) tends to zero and the discontinuity becomes unbounded. This instant usually corresponds to the arising of a strong discontinuity, i.e. , a shock wave, and the field itself presents a discontinuity across the wave front. The qualitative analysis of the Bernoulli equation (9) was studied by Chen 21 and Ruggeri 20 . In particular in 20 the stability of the zero solution of (9) (A-stability) was proved under the conditions: oc
-/6(C)dC
/ o(£)e o
3 a constant m such that
d£ = K < oo
(12)
/ b(£)d£ > m, V t > 0.
(13)
Jo
In fact if (12), (13) are fulfilled then if |n(0)|
IIcr = l/K,
(14)
the solution U(t) exists for all time and remains bounded. Moreover if
&(£)# = +oo Jo
(15)
353
then lim^oo |II(£)| = 0 and the zero solution is asymptotically stable. Choosing as unperturbed state uo = u # = constant, i. e. an equilibrium constant state: f (u E ) = 0, we obtain
a = (VA • d)E = const.,
b = - ( 1 • Vf • d)E = const.
(16)
and n c r = b/a. In this case the conditions (12) and (15) reduce to b = - ( 1 • Vf • d)E > 0
(17)
that is a necessary and sufficient condition such that the zero solution of the Bernoulli equation is asymptotically stable. Therefore for genuine non linear waves the K-condition must be satisfied, otherwise 6 = 0 and (17) is violated. If a = (VA • d)E = 0 (linear degenerate wave or locally exceptional) the Bernoulli equation becomes linear
and we do not have critical time also if b = 0 (Il(t) = 11(0)). Hence in this last case the K-condition is not necessary. Therefore, at least in the class of discontinuity waves, the weaker Kcondition together with dissipation condition (b > 0 if a ^ 0) is a necessary condition such that the discontinuity wave solution remains valid for all times for small initial perturbation. 4. Remark and Conclusions Of course this weaker K-condition is in general only necessary. In fact it is satisfied for completely linear degenerate systems or in the case of semi-linear systems but we know that in general also in these cases smooth solutions cannot exist for all time. Therefore we need to add more conditions to the weak K-condition to ensure global existence and the general problem is still a difficult and open problem! Nevertheless the knowledge of a necessary condition is very important in order to select admissible physical productions and also to decide if a problem in which K-condition is violated has possibility or not to have global smooth solutions.
354 For example in t h e case of a single t e m p e r a t u r e mixture of Euler fluids, it was proved 1 6 ' 1 7 t h a t the K-condition is violated. As the K-condition is only a sufficient condition, at t h a t time we did not know anything about global existence. Now we know t h a t it is impossible for smooth solutions t o exist, because the K-condition is violated for an eigenvector of a genuine non linear wave, therefore also the weak K-condition is not verified. As a consequence, global smooth solutions cannot exist for this model even if we choose small initial data. A more enlarged paper on this presentation is in press ( 2 2 ).
Acknowledgments This paper was supported in part (T.R.) by fondi M I U R Progetto di interesse Nazionale Problemi Matematici Non Lineari di Propagazione e Stabilita nei Modelli del Continuo (Coordinator: Prof. T . Ruggeri), by the GNFM-INDAM, and by the Istituto Nazionale di Fisica Nucleare (INFN).
References 1. K. O. Priedrichs, P. D. Lax, Systems of conservation equations with a convex extension. Proc. Nat. Acad. Sci. U.S.A., 68 (1971) 2. S. Kawashima, Large-time behavior of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edimburgh, 106A, 169 (1987) 3. A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Springer Verlag, New York (1984) 4. C. M. Dafermos, Hyperbolic conservation laws in continuum physics, Springer-Verlag, Berlin (2000) 5. Y. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal. 150 (3), 225 (1999) 6. I. Miiller, T. Ruggeri, Rational Extended Thermodynamics, 2nd ed., Springer Tracts in Natural Philosophy 37, Springer-Verlag, New York (1998) 7. Y. Shizuta, S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation., Hokkaido Math. J. 14, 249-275 (1985) 8. G. Boillat, T. Ruggeri, Hyperbolic principal subsystems: entropy convexity and subcharacteristic conditions, Arch. Rat. Mech. Anal. 137, 305-320 (1997) 9. G. Boillat, T. Ruggeri, On the shock structure problem for hyperbolic system of balance laws and convex entropy, Continuum Mech. Thermodyn. 10, 285292 (1998) 10. B. Hanouzet, R.Natalini, Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy,Aich. Rat. Mech. Anal. 169, 89-117 (2003)
355 11. W.-A Yong, Entropy and global existence for hyperbolic balance laws, Arch. Rat. Mech. Anal. 172 247-266 (2004) 12. T. Ruggeri, D. Serre, Stability of constant equilibrium state for dissipative balance laws system with a convex entropy, Quarterly of Applied Math. 62 (1), 163-179 (2003) 13. G. Boillat, Sur ['existence et la recherche dequations de conservation supplementaires pour les Systemes Hyperboliques, C. R. Acad. Sc. Paris 278A, 909 (1974); CIME Course, Recent Mathematical Methods in Nonlinear Wave Propagation, Lecture Notes in Mathematics 1640, 103-152, T. Ruggeri (Ed.), Springer-Verlag (1995) 14. T. Ruggeri, A. Strumia, Main field and convex covariant density for quasilinear hyperbolic systems. Relativistic fluid dynamics, Ann. Inst. H. Poincare 34A, 65-84 (1981) 15. T. Ruggeri, Global existence of smooth solutions and stability of the constant state for dissipative hyperbolic systems with applications to extended thermodynamics, Trends and Applications of Mathematics to Mechanics (STAMM 2002), Springer-Verlag, 215-224 (2005) 16. T. Ruggeri, Some Recent Mathematical Results in Mixtures Theory of Euler Fluids, Proceedings WASCOM 2003, R. Monaco, S. Pennisi, S. Rionero and T. Ruggeri (Eds.), World Scientific - Singapore, 441-454 (2004) 17. T. Ruggeri, Global Existence, Stability and Non Linear Wave Propagation in Binary Mixtures of Euler Fluids, New Trends in Mathematical Physics (Convegno in onore di S. Rionero, Napoli 2003), World Scientific - Singapore, 205-214 (2005) 18. G. Boillat, La Propagation des Ondes, Gauthier-Villars Paris (1965) 19. G. Boillat, T. Ruggeri, On the evolution law of the weak discontinuities for hyperbolic quasi-linear systems, Wave Motion, 1, (2), 149-152 (1979) 20. T. Ruggeri, Stability and discontinuity waves for symmetric hyperbolic systems, Nonlinear Wave Motion, A. Jeffrey (Ed.) Longman, NewYork, 148161 (1989) 21. P. Chen, Growth and decay of waves in solids, Mechanics of Solids III, Handbuch der Physik, 6 A (3), 303, Springer-Verlag (1973) 22. J. Lou, T. Ruggeri, Acceleration Waves and Weak Shizuta-Kawashima Condition, In press on Rend. Circ. Mat. Palermo. Special Volume in honor of G. Boillat. A. Greco & T. Ruggeri Eds. (2005)
P O I N T W I S E STABILITY OF SOLUTIONS OF T H E NAVIER-STOKES EQUATIONS
P. M A R E M O N T I Mathematical Department via Vivaldi, 43, Caserta, 1-81100, Italy E-mail: paolo.maremonti@unina2.
it
We prove some results of pointwise continuous dependence and of pointwise stability of steady solutions of the Navier-Stokes system. The perturbation at the initial instant is just assumed a continuous function, small in a such a way that for any t > 0 the perturbation is a smooth solution. The new results stated in this note are part of a forthcoming paper.
1. Introduction Let T be an incompressible, homogeneous, viscous fluid. The symbol f2 denotes a sufficiently smooth three-dimensional region of space. We assume that a motion mo of T in Q. is governed by the Navier-Stokes equations. The pair (w,p) denotes a steady motion of T:
w • Vw = —Vp + —Aw, it V • w = 0, in ft, f / a • nda = 0, Jan HQ, is unbounded, v —> e for |x| —> oo, w = a on dtt,
(1)
if £1 is bounded
v • Vu = Vk-Q^-v and R Reynolds number. The pair (u,n) denotes a perturbation to (w,p) generated by a perturbation to the kinetic field w(x) at the instant t = 0. It is well known that
356
357 the pair (u, IT) is a solution to the following initial boundary value problem: ut+u-
Vu + w • Vu + u • Vt» + V7r = — Aw, R
V - u = 0, u(x,t)=0,
i n f i x (0,T),
(2)
(x,t)£dnx{0,T),
u{x,0)=u0{x)
in fi,
if Q, is unbounded, M —> 0 per |a;| —> oo, where ut = -^u. The concept of stability of a basic motion mo is connected with the intuitive idea that slight perturbations evolve in such a way that mo is observed again. Mathematically we verify the stability of a motion by a measure of the perturbation. Physically, it is natural to assume as laboratory measure for the flow its maximum speedy or its kinetic energy (cf. [2]). As far as the stability of the steady motion mo, we start describing the measure of the energy of the perturbations. Shortly, we recall the chief ideas. Formally a solution of the system of the perturbations satisfies the so called energy relation: 2 T t « W l 2 + p|Vu(t)H = -
u-Vw
udx, t G (0,T).
(3)
Assuming as region of motion a bounded domain ft, Serrin (1959) [10] has introduced the variational formulation of the energy stability. Let us consider the following functional
J
u • Vto • udx &(u) = - Jn |Vu(t)| 22
u £ J1'2^),
where J x ' 2 (fi) = completion of %(Q) sup u^J1-2(Q)
a
in H^Q). If
&{u) — — < +oo and R < Rc, Re
from (3) we formally deduce W(t)\22 + 2 ( I - i - ) J
\Vu{r)\ldT < \u(s)\l
t>s>0,
which implies unconditional asymptotic stability: |w(i)|2 < \u0\2 for any t > 0 and lim |u(i)|2 = 0. t—i-OO
if0(n) = {u(x) e cg°(n) with v • u = o}.
358
Rc is called critical Reynolds number. The above formal considerations can be made rigorous by collecting the results of [10, 11, 13]. In several models of fluidynamics we find the applications of the variational energy method: [2, 4, 12, 16]. The study of the energy stability, introduced in the case of bounded regions of motions, can be Considered, in a suitable way, to study the stability of steady flows whose region of motion is an exterior domain. In this connection we recall the interesting case of the so called physically reseanable solutions. Now we introduce the pointwise stability. The idea is to give a measure of the perturbation u(x, t) which is not global with respect spatial variables, as it happens in the case of L2 norm (or \ • \p, p £ (1, oo]). As it will be clear in the sequel, at least in the case of unbounded domains, our analysis cannot be reduced to a measure of the perturbation with respect to the uniform norm (| • loo.) This reason infers to distinguish, in the analysis of pointwise stability, between unbounded regions of motion D, and bounded regions of motion Q. In the case of unbounded regions what do we mean by saying pointwise stability? In order to better explain it we introduce the following simple notations: M(Q) = {h(x) e C(ty : h(x) = 0 on dQ, V • h(x) = 0} , and, for \x > 0, Mf x| (fi) = {h(x) € M(Q) : \h(x)\ < H0(l + l i l ) - " , for any x £ Q} . Definition 1.1. Definition of Spatial Stability. Let /J > 0. A solution (w,p) of the Navier-Stokes equations is said spatial /x—stable if for any u0(x) £ M?xA£l) there exists an interval (0, T) and a function c(t) £ C([0,T)) such that \u(x,t)\ < c{t)U0(l + \x\)-'i,
for any (x,t) eflx
[0,T),
where u(x, t) is the perturbation corresponding to u0(x). The above definition ensures that the profile of u0 (x) is preserved during the motion of T, however not uniformly in t £ [0, T) : it can be T < +oo
and
lim c(t) = +oo.
Definition 1.2. Definition of Pointwise Stability. Let \i > 0. A solution (w,p) of the Navier-Stokes equations is said /i-stable in x and t if there
359 exists A € (0, +00] such that if u0(x) G M?,($l) with U0 < A, then the perturbation u(x,t) corresponding to u0{x) is defined for any t > 0 and moreover for any v £ [0,/x], \u(x,t)\
utjVPeC(0,T;C(a)). Results concerning the maximum modulus theorem for solutions to Stokes and Navier-Stokes system in half-space are obtained in [8, 14]. Of course in the case of Navier-Stokes system a priori the existence of the classical solution is only local in time. There are some papers related to the question about spatial and time behaviour of solutions of the Navier-Stokes equations in the case of an exterior domain. However the results of these papers do not reproduce the property stated in the definition of pointwise stability since there are extra assumptions on the initial data, see [7, 9, 3] Assume w — 0 (perturbation to the rest state) and Q, = R3^. In [1] it is proved the following Theorem 1.1. Foranyu0(x) e A4?,(R\),fi G (5,3), there exists a unique classical solution (u,p) of the Navier-Stokes system in some interval (0, T).
360
Moreover, ifT^< MXjt)\
<
(4bBU0) \ there is spatial stability: l
-
m
\
B U
°
BU
°
for
{xt)
g R3
(Q
j.y
if \x £ [l,n) and U0 < (46-B) -1 , then there is pointwise stability, that is T = oo and, for any v £ [0, /i], , , ,, l-3bBU0 BUa , „ r r — ;—7T v 1u(x,i) v n < ' ~ 1 - 46BC/0 (1 + \x\y~
1 , , m, r r rr-, /or any xL 0,oo). n (x,t) G Mi + y (1 +1) 2 '
Constants b and B are independent of u0. The theorem is completed with estimates concerning Vu,p and Vp. Thought it is known, we would like also to stress that an L p -theory a priori does not give results of pointwise stability. 2.
Pointwise stability in the case of bounded regions.
The above definitions of stability become: • for its a priori local character in time, the definition of spatial stability coincides with continuous dependence on the data with respect to the uniform norm; • the definition of ponitwise stability becomes a result of stability and asymptotic stability with respect to the uniform norm. Remark 2.1. The energy stability does not imply pointwise stability. Actually we have the following implication: 1) By the energy stability we are able to prove attractivity of the basic motion (w, ir) with respect to the uniform norm: there exists an instant To = T0(\uo\2,R,Q)
such that
u(x,t) £ C{Q) and \u(x,t)\ < C(|u 0 | 2 ,.R,u;,ft)e~ 7 t , for any t > T0. The existence of the perturbation is a consequence of Leray-Hopf theorem, the attractivity result is connected with the theorem de structure of a weak solution. 2) Assume that ua(x) £ C(U) n i 2 (fi), with rnax|u 0 (a;)| + |u 0 |2 < < e,
361 we do not known if the L 2 -theory ensures that u(x, t) exists as a classical solution for t > 0, in particular if |u(z,£)| < oo for any (x,t) £ fi x [0,7b). 3) An L-theory with |u013 < S, S sufficiently small, ensures the existence of a unique solution u(x, t) for any t > 0 and the attractivity for large t with respect to the uniform norm: lu(*)lc(n) ^ 0 f o r t - ^ 0 0 , but it does not imply stability, in particular no bounds for neighborhood of t = 0. The estimate is
|W(£)| C (Q\
in a
\u(x,t)\ < c(|it 0 | 3 ,i?)t"5|u 0 | 3 , t > 0. The above considerations leads us to understand that the problem of pointwise stability is still open. We attack the nonlinear problem by making use of some results concerning the initial boundary value problem for the Stokes system and of an assumption of smallness of the initial data in a suitable sense. We start recalling the following result for solutions of the Stokes problem: Theorem 2.1. Maximum Modulus Theorem. If(v,p) is a classical solution of the Stokes problem, then there exists a constant c such that \v(x,t)\ < cmax|i> 0 (x)|, for any (x,t) £ Q. x [0,00), with c independent of v. The above theorem has been proved in [15]. We introduce the space ^| 0 (fi) = {h(x) £ C(Tl) with V • h = 0 and h = 0 on dQ] , the symbol |0 means that a function has the null trace on dtt and we mean V • h = 0 in a weak sense, that is (h, Vrj)) = 0 for any £ C1(fi!). Moreover, we consider "4, (ft) = completion of %(Q) in C(H). Of course a priori ^(Cl) C ^j 0 (ft). We have Lemma 2.1. Let Q be a Cl'h-smooth
domain. Then ^(Q)
= ^j 0 (fi).
362
Remark 2.2. In the light of the minimum requirement on the measures of the initial value of the perturbation, to deduce stability with respect to the uniform norm, the above lemma appears essential in our approach. Indeed we first prove our result for perturbations which are smooth at the initial instant and then by density for elements of ^j0(f2). Another important auxiliary result is the following Lemma 2.2. Let us consider the problem V • h = g, K\QQ = 0, with g G C(Ii) and J 9(x)dx
= 0.Ifn
rs C^-smooth
domain, then there exists
h G C(Q.) solution of the problem such that max.\h(x)\ + max|V • h(x)\ < c m a x | / ( x ) | . Remark 2.3. The above lemma is the analogous of Bogowski's result for the Lv-theory, which consists in solving the problem with g G Lp(fl). The result of the L p -theory not only ensures the existence of a solution of the problem, but that any solution has V/i G LP(Q,) also. In Lemma 2 a priori we just have h G C(Q.) and V - / i £ C(f2). However this is a sharp result. Indeed, let us consider H(x) =^(x)(xiX2loglog|x| _ 1 ), where 4>{x) is a smooth function with compact support and 4>{x) = 1 in neighborhood of 0. Then h(x) = VH(x) is a continuous function together with V-/i, but Vh(x) is non smooth in 0. An improvement in the regularity of h, as for example h G C 1 (fi), is possible assuming, for some a G (0,1], g a-Holder continuous also. Now we are in position to state our chief result about the pointwise stability, Theorem 2.2. There exist 8%c > 0 and r\ = T](£l, R, W) such that if R<£%c and max \u0 (x) \ < 77,
n
then there exists a unique perturbation u(x,t) corresponding to u0(x) G ^j0(f2); which is a classical solution defined for any t > 0 and with \u(x,t)\
< c(u0,R, w,£l) max \u0(x)\e~7t,t n for some 7 > 0 independent of u0.
> 0,
363 R e m a r k 2 . 4 . A remark on t h e number g%c. It is well known what is the meaning of the critical Reynolds number Rc in t h e energy stability thery: if R < Rc, the energy is a monotonic decreasing function of t; this implies unconditional stability and asymptotic stability with respect to the L2—norm. Rc is critical not only for one flow, but for a family of flows corresponding to different parameters having the same Reynolds number R. In our result £$c is a sort of a critical Reynolds number for the pointwise stability. We start saying why 8&c is not a critical Reynolds number: • t h e existence of 3%c and t h e assumption of R < &c do not imply t h a t m a x \u(x, t)\ is a monotonic decreasing function of t; n • taking into account t h a t R = ~- b , if we consider fl fixed, then our result ensures t h a t £%c is critical for the family of flows corresponding t o different W and v in such a way t h a t R is unchanged; however this is not sufficient to apply the theory of similar motions. We think t o &c as critical Reynolds number since • the existence of 2%c and the assumption of R < £%c imply conditional stability in the norm of C(Q); • 2%c has the character of a critical Reynolds number because R < 2$c ensures asymptotic stability and because 2£c is equal t o the Rc of energy stability; • we guess t h a t if there exists a 2%c for t h e pointwise stability, then, as in Theorem 2.2, it cannot be 3%c > Rc. References 1. 2. 3. 4. 5. 6. 7.
F. Crispo and P. Maremonti, Zap. Nauch Sera. P O M / 3 1 8 , 147 (2004) P.G. Drazin and W.H.Reid, Cambridge University Press G.P. Galdi and H. Sohr, Arch, for Rat. Mech. and Analysis 172, 363 (2004). D.D. Joseph, Springer Verlag, 1976. G.H. Knightly, Arch. Rational Mech. Anal. 21, 211 (1966). G.H. Knightly, SIAM J. Math. Anal. 3, 506 (1972). Knightly G.H., , Approximations methods for Navier-Stokes problems, Lecture Notes in Math., Springer, Berlino 8. P. Maremonti and G. Starita, Zap. Nauch. Sem. POMI, 295, 118 (2003). 9. Mizumachi R., J.Math. Soc. Japan 36, 497 (1984). 10. J. Serrin, Arch. Rational Mech. Anal. 3, 1 (1959). 11. S.Rionero, Ann. Mat. Pura Appl. 78, 339 (1968). b Here L denotes a characteristic length, W a characteristic velocity and v the kinetic viscosity.
364 12. 13. 14. 15.
S.Rionero and J.N.Flavin, CRC Press (1996). D. H. Sattinger, Journal of Mathematics and Mechanics 19, 797 (1970). V.A. Solonnikov, J.Math. Sciences 114, 1726 (2003). 771, 287 (1980). Solonnikov V.A., Note in Pure and applied mathematics, M. Dekker 223, (2002). 16. B.Straughan, New York, Berlino, Springer-Verlag (1992)
BIFURCATION ANALYSIS FOR S Y M M E T R I C EQUILIBRIA W I T H LOCALISED M A G N E T I C SHEAR IN 2D R R M H D
L. MARGHERITI Department of Mathematics, University of Messina Salita Sperone 31, 98166 Messina, Italy E-mail: margheriti@mat520. unime. it C. T E B A L D I Department of Mathematics, Politecnico of Torino Corso Duca degli Abruzzi 24, 10129 Torino, Italy E-mail: claudio. tebaldi©polito. it
For reduced resistive magnetohydrodynamics we consider the class of symmetric equilibria ip"(x) = i>Q /cosh2ax with no motion and analyse the sequence of bifurcations varying the aspect ratio e of the domain as a control parameter of the magnetic shear. The stability threshold in the inviscid case eo scales linearly with a. Destabilization happens because of a symmetry-breaking bifurcation, occurring at ec < eo because of viscosity, which originates a stable equilibrium with a small magnetic island and vortices. For a = 1 the equilibrium disappears by tangent bifurcation at a critical value ep. Above ep, no stationary solutions with small island exist and the system very rapidly develops an island of macroscopic size. Differently, for a = 2, i.e. for higher magnetic shear at x = 0, the equilibrium with small island does not disappear, but becomes unstable at ep = 2.437. Below this value a stable equilibrium of weaker symmetry has been detected, with magnetic island of essentially the same size but velocity field with different topology and strongly enhanced. The phenomenology is observed for the first time in RRMHD and is of interest because connected with experimental evidences in fusion plasmas.
1. Introduction Solar flares, storms in the earth's magnetosphere and disruptions in laboratory fusion experiments are examples of large scale explosive events which occur in plasma systems, the magnetic field being the source of energy driving these phenomena. A mechanism by which magnetic energy can be released is magnetic reconnection. In ideal MHD regimes, plasma elements that are initially connected by a field line remain connected by the same field line as they move. This magnetic connection property forbids tran365
366
formations between configurations with the same total energy but different magnetic topology. Magnetic reconnection is a violation of this property and it can be imagined as a local cutting and re-knitting of field lines. This topological change in the magnetic field requires a breakdown in the ideal "frozen-in" flux condition, which occurs at small scales where magnetic field reverses direction and resistivity becomes relevant. Even though a local phenomenon, occuring in narrow layers, magnetic reconnection has global effects on the rearrangement of the magnetic field. A relevant process involved is the increasing of plasma kinetic energy at expense of the magnetic energy. We refer to Ref.1 for more details. The present work refers in particular to the problem of plasma confinement in tokamaks, where magnetic field lines are helical and lie on nested surfaces centered around the (toroidal) magnetic axis. Magnetic reconnection in tokamaks leads to a specific change in the field topology: a secondary helical magnetic axis appears and, depending on the strength of the perturbation, a portion of the magnetic field lies on helical magnetic surfaces centered around this secondary axis. One speaks in this case of magnetic islands, after the pattern generated by the intersection of the helical magnetic surfaces with the poloidal plane. The study of the generation of equilibria with magnetic islands is an area of practical importance for understanding the behavior of magnetic confinement machines. Often, the presence, inside the plasma, of a slowly growing magnetic island, originated by a symmetry breaking of a previously axisymmetric equilibrium, can be identified as a "precursor" of events which degrade the confinement of the devices. In this class of events we find the sawtooth oscillations, edge-localised modes and disruptions 5 . A theoretical analysis of this type of phenomenology in tokamaks shows in general that the excitations which break the symmetry of the initially axisymmetric state are of the "soft" type: for values of some control parameter p slightly above the stability threshold pc of the axisymmetric state, the system settles in a neighboring state of slightly broken symmetry, for example an equilibrium with a small saturated magnetic island. The difficulty in understanding the frequently observed "hard" excitations mentioned above,characterised by fast island growth 7 ' 10 ' 12 ' 17 and strong increase in kinetic energy, on the basis of perturbation analysis around the initially axisymmetric state has become known as the "trigger" problem 19 ). On this line, Thyagaraja 16 and Wesson et al. 18 , using methods based on an expansion in the parameter A' (A' is defined as the jump in the logarithmic derivative of the eigenfunctions of linear ideal MHD across the
367
singular layer where resistivity becomes important), found two saturated island solutions which coalesce and then disappear. The expansion, however, is only appropriate to treat the region around the symmetry breaking bifurcation and breaks down when approaching any subsequent bifurcation point. Thus, any conclusion, based on those methods, on the nature of a subsequent bifurcation are only speculative. A more adequate approach to the problem of sudden changes can be found in the conceptual framework of bifurcation theory in dynamical systems 3 . In particular, hard excitations can generically appear as a tangent bifurcation9 or a subcritical Hopf bifurcation. In fact, upon a slow variation of the control parameter, both bifurcations lead to abrupt ("catastrophic") changes in some observable quantity. The sequence of bifurcations in reduced resistive MHD (RRMHD) had been investigated by Saramito and Maschke11; their analysis is valid beyond the first bifurcation, but they considered only situations with constant slab aspect ratio, using the Lundquist number S as a bifurcation parameter. Parker et al. 8 studied the bifurcation sequence in RRMHD with an initial value code, using a variable aspect ratio, but at a constant Lundquist number S ~ 102. In both case, no tangent bifurcation of the non symmetric saturated state was detected, but we point out that both investigations were carried out by employing rigid boundary conditions (zero radial flows at the borders), which limit the island growth. Later, Tebaldi et al. 14 considered a slab model of RRMHD with periodic boundary conditions, by varying the aspect ratio of the slab e or, equivalently, the tearing mode stability parameter A' for S ~ 103. The main result was that, after a first bifurcation (at A' ~ 0) of the unreconnected equilibrium 4ie{x) = cosx with no motion, leading to an equilibrium with a small magnetic island and vortices, the system undergoes a tangent bifurcation at LA' ~ 1, where L is a macroscopic scale length. Above this value of A' no equilibrium with a small island exists and the system jumps to a state where the island width is of order of the system size. On the basis of that analysis it was proposed that fast reconnection events observed in laboratory plasmas would occur from a state of already broken symmetry when the second "hard" bifurcation takes place. This state can be identified as the precursor state in which a small stable island can in principle be observed. The disappearance of small size island because of a tangent bifurcation as a possible mechanism for "hard" excitations has been confirmed also in the case of island rotation 15 and in the presence of "error fields"6.
368
In this paper we address the phenomenology after symmetry-breaking for different equilibria of interest in experiments, taken in the class tp"{x) — ipQ /(cosh 2 ax), a G R+, with no motion, where a is a measure of the magnetic shear, localised at x = 0. 2. The M H D model and numerical techniques We consider a two-dimensional incompressible plasma obeying the reduced resistive magneto-hydrodynamics (RRMHD) equations 4
dt^ + [<j>,^] = -r1{J-Je)
K}
The equations are defined on a two-dimensional domain with coordinates x and y: with reference to the magnetic geometry of a tokamak, x and y can be thought as radial and poloidal coordinates respectively. The third direction is considered ignorable. The model equations describe the evolution of a plasma vorticity U — V2<^, where ^ is a stream function, and of the magnetic flux function I/J associated with the magnetic field in the plasma (a constant magnetic field is assumed in the ignorable direction) 15 . The other fields are the current density J — —X72ip and a driving current density J e , associated with equilibrium. Moreover for any two fields A and B, [A, B] = dxAdyB — dyAdxB, so that [
369 plete orthogonal set for the expansion. One has ((j>, ip) = /J(0k> V ; k)e' k ' x
k=(Z,me)
l,m
integers
(2)
k
We truncate the expansion to a finite set L of 2TV wave vectors ("modes") such that if k belongs to L also —k belongs to L. This gives 47V ordinary differential equations for 4TV real unknowns. Moreover, a 27V invariant subspace exists, characterised by imaginary amplitudes for the magnetic and velocity fields, which allows to reduce the system to 27V real unknowns. The set L is constructed, starting from a "ball" around the origin with N=364 and adding modes in a slab centered at m = 0. It has to be noted that because of the spatial localization the computational problem becomes much heavier than in Ref.15, in fact TV up to TV = 985 had to be considered. A suitable tool to find the equilibria is Newton's method, used in connection with the theorems of bifurcation theory 13 . It allows to find also unstable equilibria and it avoids the difficulty, encountered by the initial value approach, that the time scales can become extremely long, especially near the bifurcations. Finally this method is an efficient way to track the sequence of equilibria when parameters are varied, even if computational effort strongly increases with the dimensionality of the problem. Since the model equations are supplemented with periodic boundary conditions, a suitable initial value code is a spectral code, which advances in time the Fourier amplitudes of the relevant fields. In some cases a direct truncation of the model equations to the relevant degrees of freedom was used either to compute stable equilibria or to study transients. 3. Sinusoidal unreconnected equilibria We report here briefly the results obtained 14 when the magnetic flux is ipe(x) = cos a;
(3)
2
It is known that a symmetric magnetic configuration can be unstable to symmetry breaking perturbations under certain conditions. The stability threshold of tpe can be obtained solving the linearised equations related to Eqs. (1) by asymptotic matching. In the limit of large Lundquist number, dissipation is only important in a narrow region around the x = 0 and x = ±7r lines, where the magnetic field vanishes. By denoting ipout the solution far from this region ("outer solution"), the stability condition is usually expressed in terms of the parameter A' = lim dx log 4>out ~ lim dx log V w x—>0+
x^>0-
(4)
370
For the given equilibrium and for a given mode number m, there is a oneto-one correspondence between A' and the slab aspect ratio e, thus these two quantities can be used interchangeably as a control parameter. The stability boundary of the zero viscosity case is A 0 = 0 or e0 = 1, while in practice, for finite values of viscosity, this threshold is shifted to some e = ec < 1 or A' = A'c > 0 (Refs. 1 4 - 1 5 ). As regards the bifurcation diagram, the sequence of equilibria with magnetic islands was studied for 1 > e > 0.80 and S up to 1000 using the fixed point code. The main result is shown in Fig. 1, where the island width w for the different equilibria is plotted against e. The unreconnected equilibrium
1
f
Q * '
»t
w 0 1
Po
...Q' ''•.,,
s
.90
E
Figure 1. Normalised island width w for the equilibria P0, P, Q, Q* versus e for S = 1000 and Pr = 0.2. Solid lines denote stability, dotted lines instability.
Po with w = 0 becomes unstable at e = ec — 0.975 or A' = A'c = 0.19, in agreement with analytic estimate. The bifurcation is a pitchfork leading to a new stable equilibrium P with a small magnetic island and four vortices. When e = CQ = 0.977 a pair of equilibria, Q stable and Q* unstable, appears via tangent bifurcation. At a smaller value e = eP — 0.896 another tangent bifurcation occurs, characterised by the coalescence of Q* with the small island equilibrium P. Thus below eP the only stable solution is Q, with an island width of the order of the system size, which can account for the rapid growth from small island situation and strong increase in the kinetic energy. In Fig. 2 we show the contour plots of ip and <j> for the three equilibria P , Q* and Q at a value of e just above ep. One can see that in the case of P the magnetic island retains approximately its linear shape. It is less so for the velocity field, which however is still organised in four main convective cells. By comparison the island width of the Q-equilibrium is
371 comparable to the equilibrium scale length. The corresponding velocity field is more complicated, with four main elongated vortices aligned along the separatrices and substantially enhanced. In the e range under study, the bifurcation diagram is unchanged to a further increase of S.
Figure 2. Contour plots of ip and <j> at e slightly larger than ep and Pr = 0.2 for: P, a) and b), Q*, c) and d), and Q, e) and f).
A more detailed analysis of the parametric dependence of the first bifurcated equilibrium was done by analytic perturbation theory and numerical methods 15 . The model was also extended to include the diamagnetic effect through the equilibrium ion drift velocity v* under the assumption of constant (frozen) pressure gradients. The symmetric equilibrium becomes unstable because of a supercritical Hopf bifurcation, giving rise to a timeperiodic solution ("small rotating island"). The analysis of the bifurcations has confirmed qualitatively the bifurcation diagram in Ref.14 for the periodic solutions, up to a critical value of the magnetic velocity v*. Also in the presence of error fields, i.e. considering an equilibrium that for an aspect ratio e = 1 has already a small island, there is the occurrence of the tangent bifurcation leading to disappearance of the small island equilibrium 6 . Therefore, in the case of sinusoidal unreconnected equilibria and in different situations, a tangent bifurcation is the phenomenology able to explain "hard" excitations in magnetically confined plasmas 15 .
372
4. Unreconnected equilibria with localised magnetic shear The equilibrium ipe(x) = cosx gives a magnetic field with a maximum shear not only at x = 0, but also at x = ±7r. Since Eqs. (1) allow to specify symmetric equilibria where <j!>o = 0 and xfie is a function of x, in order to localise the magnetic shear around x = 0, the class of equilibria with no motion
(5)
a £ R+
C(z) = ff
>
cosh ax has been considered, where ip$ is a normalization constant, tp" retains the main qualitative characteristics of the flux ipe (x) = cos x, giving a zero of the magnetic field at the origin, similar monotonic properties and the shear at infinity goes to zero. Even if the domain [—ir,ir} is considered, the magnetic shear is very small in a neighbour, increasing with a, of the boundaries. Fig. 3 shows the magnetic field 5 " for different values of a. The stability boundary of the reference equilibrium can be obtained for an infinite domain by linear theory. Writing 4>(x,y,t) = tWe-M+^v [ ip(x,y, t) = ip«(x) + i!(x)e-iiot+iky ' where k = me, with m integer, the linearised version of Eq. (1) is given by uV24> = -k{dx^V2tp
- ^ V e ) + VV40
(7)
where V 2 = d2. — k2. The equilibrium becomes unstable when to2 goes through zero, then the stability boundary is obtained by solving the linearised equation for w2 = 0. For p, = 0 the linearised equation is 2asinax
[d2J + ( 8 a 2 - k2 - 12a2tanh2ax)j>] = 0 (8) cosh ax This equation can be solved by asymptotic matching. Far from the x = 0 line, where the magnetic field vanishes, the solution is approximated by the solution of the outer equation: <9xVw + (8a 2 - k2 - 12a 2 tanh 2 ax)^out
=0
The solution is given by 1+ + To
kf-4
*
-tanh a\x\ + , ., »
tanh a\x\ - t a n h a\x\
fci(fcf-4)
(9)
373
where k\ = k2/a2 + 4. As it's usually done, the stability condition is expressed in terms of A', the jump in the logarithmic derivative of V w - In this case
A
' = 2o (s$^>-* 1 )
<M>
Instability occurs when A' > 0, i.e. when k < a\fb. k = me, then instability occurs for m = 1 when e < e0 = a\/E. 4 . 1 . Bifurcation
diagram for a = 1
Equilibrium (5) has been considered first for a = 1 (see Fig.3, where the related magnetic field is represented) and has been approximated with its expansion in Fourier series truncated to six terms, which is found to be a good representation of i>\{x). This choice allows the use of periodic boundary conditions and therefore the use of the spectral codes described in Sec.2. Because of the localisation of the shear in the unreconnected equilibrium, the modal structure, N = 348 for the sinusoidal case, was considered up to N = 721 in order to check independence of the results from the truncation procedure. The bifurcation diagram is shown in Fig.3,
Figure 3. Bf for a = 1, solid line, a = 2, dash-dot line, a = 3, dotted line (left). Normalised island width w for the equilibria Po, P and Q* versus e at Pr = 0.2; solid lines denote stability, dotted lines instability (right).
where the island width w of the equilibria is plotted against e. The results are qualitatively similar to the one in Sec.3. At ec = 2.026 < eo = y/E, a symmetry breaking of the unreconnected equilibrium Po (with w = 0) gives rise to a stable equilibrium P with a small island. At CQ = 0.988 two equilibria with magnetic island, Q stable and Q* unstable, appear via
374
tangent bifurcation. Differently from the case ipe{x) = cosx, this tangent bifurcation happens for equilibria with island width comparable with the size of the domain. For ep = 0.865 a second tangent bifurcation takes place with coalescence of P and Q* and their subsequent disappearance. For values of e less than ep the system jumps to an equilibrium Q with a macroscopic island, which can account for the rapid growth from small island situations and relevant increase in kinetic energy. Also with this choice of ipe, when the control parameter e is near the threshold of the symmetry-breaking bifurcation (2.024 > e > 2.008), the island width has a square-root dependence on the departure from ec, while for 2.005 > e > 0.880, the dependence of w on e is essentially linear.
Figure 4. Contour plots of tp and cf> at Pr = 0.2, a = 1 and e = 0.89 for: P, a) and b), Q*, c) and d), and Q, e) and f).
As regards the velocity field, it is still organised in four vortices localised along the separatrices and enhanced. Fig. 4 shows the contour plots of ip and
375
4.2. Increased
magnetic
shear: the case a = 2
With the aim to investigate destabilization of unreconnected equilibria with higher magnetic shear, we have also considered ip" for a = 2. This equilibrium has been approximated with an expansion in Fourier series truncated to 21 terms because of the stronger localization of the magnetic shear at x = 0 and the modal structure has been taken up to N = 985, making the problem computationally heavier. The results presented here have been found unaffected by the truncation procedure. The unreconnected equilibrium Po becomes unstable at ec = 4.162 < eo = 2\/5. Also in this case the destabilization of the unreconnected equilibrium happens through a symmetry breaking that originates a stable equilibrium with a small magnetic island, P. However, further decreasing e the equilibrium P does not disappear by tangent bifurcation, but becomes unstable at ep = 2.437. Below this value, a stable equilibrium Q has been found, also with a small island but with remarkable differences with respect to P. In fact, while the magnetic field has essentially the same topology for P and Q, the differences appear very clearly in the velocity field, which for Q is not organised in four vortices as for P , but transport takes place in a region around the separatrices, aligned with them (see Fig. 5).
Figure 5. Contour plots of ip and <j> at Pr = 0.2, a = 2 and e = 2.44 for P (left) and at e = 2.435 for Q equilibria (right).
It is also quite evident that Q does not have the same symmetry properties of P. For all the detected equilibria ip(-x,-y,t)=ip(x,y,t),
cf>(-x,-y,t)
=
(11)
hold, which define an invariant subspace characterized by real amplitudes for the Fourier expansion of ip and
376
and (/> take the form V'fe*) = 2 y ^ VjjCosk -x, kez/
(x,i) = 2 Y^ ^ c o s k -x kgi/
(12)
where X' includes the vectors k = (kx, ky) with kx > 0 and k = (0,fcy)with ky > 0. ^ and
^
which define an invariant subspace included in the above one and characterised, in terms of Fourier amplitudes, by t h e conditions ^{kx,-kv) = ^{kx,kv)
V(fcs, fcj,), kx jk 0, ky ^ 0 y(kx,ky),kx^Q,ky^0 VfcX, ky
(14)
Such symmetry does not hold for t h e equilibrium Q. In particular t h e amplitudes related t o t h e modes (kx,0) and ( 0 , ^ ) are different from zero and in fact give a relevant contribution t o determine t h e new topology of the velocity, which is associated t o a strong increase in t h e kinetic energy. A further symmetry breaking of t h e reconnected equilibrium seems t o have taken place, with t h e most relevant consequences on t h e velocity field.
5. Conclusions and further developments In R R M H D t h e class of equilibria with no motion tp"(x) = ipo / cosh ax ( a G R+) h a s been considered as Fourier expansion on t h e domain [—7r, TT]. Instability occurs, in t h e inviscid case, by symmetry breaking bifurcation at e < ay/5, i.e. the stability threshold scales linearly with a. For a = 1 bifurcation diagram a n d topology of the magnetic and velocity fields are qualitatively t h e same as for tpe{x) = c o s x 1 4 : t h e stable small island equilibrium disappears by tangent bifurcation, which in this case is confirmed as a possible mechanism for "hard" excitation. For a = 2, however, t h e equilibrium with a small island does not disappear by a tangent bifurcation, b u t it becomes unstable at a critical value ep. Below this value a stable equilibrium exists, which h a s essentially t h e same magnetic island width b u t different symmetry properties. This is very relevant for t h e velocity field, which is not organised in convective cells, b u t becomes aligned with magnetic field. T h e increase of the magnetic shear
377 at x = 0 for the unreconnected equilibria has given rise to a new phenomenology, observed for t h e first time in R R M H D and requiring further investigations. T h e transition t o the weaker symmetry equilibrium has to be clarified in t e r m s of bifurcation theory, as well as the role of t h e physical parameters besides the magnetic shear, in particular viscosity, discussed. T h e study is of great interest because it allows t o make connections with experimental evidences in fusion plasmas.
References 1. D. Biskamp, Nonlinear magnetohydrodynamics, Cambridge University Press, Cambridge, 1993. 2. H.P. Furth et al, Phys. Fluids 6, 1963. 3. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1986. 4. B.B. Kadomtsev and O.P. Pogutse, Sov. Phys. JETP 38, 1974. 5. B.B. Kadomtsev, Tokamak Plasma: a Complex System, Institute of Physics Publishing, Bristol, 1992. 6. L. Margheriti and C. Tebaldi, in Waves and Stability in Continuous Media, World S c , Singapore, 2002. 7. E.N. Parker, J. Geophys. Res 62, 1957. 8. R.D. Parker et al, Phys. Fluids B 2 , 1990. 9. Y. Pomeau and P. Manneville, Comm. Math. Phys. 74, 1980. 10. P.H. Rutherford, Phys. Fluids 16, 1973. 11. B. Saramito and E.K. Maschke, in Magnetic Turbulence and Transport (P. Hennequin and M.A. Dubois eds.), Editions de Physique, Orsay, 1993. 12. P.A. Sweet, in Electromagnetic Phenomena in Cosmic Physics (edited by B. Lehnert), Cambridge University Press, Cambridge, 1958. 13. C. Tebaldi, in Nonlinear Dynamics, World Scientific, Singapore, 1989. 14. C. Tebaldi et al, Plasma Phys. and Contr. Fusion 38, 1996. 15. C. Tebaldi and M. Ottaviani, Plasma Physics 62, 1999. 16. A. Thyagaraja, Phys. Rev. Lett. 24, 1981. 17. F.L. Waelbroeck, Phys. Rev. Lett. 70, 1993. 18. J.A. Wesson et al, in Proc. of the Tenth Int. Conf. on Plasma Phys. and Controlled Nuclear Fusion Research, Vol.2, IAEA, 1985. 19. J.A. Wesson et al, Nucl. Fusion 3 1 , 1991.
SIMULATIONS AT KINETIC SCALE OF RELAXATION MODELS FOR SLOW A N D FAST CHEMICAL R E A C T I O N S
R. MONACO, M. PANDOLFI BIANCHI* Dipartimento di Matematica, Politecnico di Torino, Italy E-mail: [email protected], [email protected] A. J. SOARES^ Departamento de Matematica, Universidade do Minho, Portugal E-mail: [email protected]
Two BGK-type models for reactive gas mixtures, derived in a previous paper, are here considered. The former is related to slow reactions, whereas the latter takes into account fast chemical processes. The paper, through numerical simulations carried out at the kinetic level for both fast and slow bimolecular reactions, is mainly devoted to show the characteristic behavior and features of the two corresponding chemical regimes.
1. Introduction In the last years a great interest has been devoted to kinetic theory extended to reactive mixtures in order to improve the knowledge of macroscopic phenomena, as for instance those related to combustion or plasma physics, starting from the physical description of the system at the mesoscopic scale. For rather general chemical processes the reader could be addressed to the book 1 and papers 2 ' 3 . More in details for bimolecular reactions involving four chemical compounds Ai of molecular masses m*, with mi + iri2 = m 3 + 7714, undergoing *INDAM-GNFM and the National Research Project COFIN 2003 "Non linear mathematical problems of wave propagation and stability in models of continuous media" (Prof. T. Ruggeri). t Minho University Mathematics Centre and Portuguese Foundation for Science and Technology (CMAT-FCT) through the research programme POCTI.
378
379
the reaction Aj. + A2 ^
A3 + A4 ,
(1)
Rossani and Spiga 4 have derived the kinetic equations that can be rewritten in the following compact form, for i = 1 , . . . , 4,
M + z,. V/, = VilfiWiGilfiiv) - /<} + ai[/](tO{&Lfl(») - /*}, (2) that better puts in evidence the production and loss contributions. In Eq.(2), fi = fi(t,x,v), t 6 R + , x,v 6 M 3 , are the one-particle distribution functions such that / = {fi,- • • ,fi}; the dependence on (t,x) has been omitted for simplicity. Moreover, the first term on the r.h.s. of equation (2) refers to elastic scattering whereas the second one is related to reactive interactions, G» and Qi being the gain operators. Finally, Vi and di are the elastic and reactive collision frequencies, respectively, which do not depend only on the distribution / but also on the elastic and reactive interaction potential laws. In literature 2 , if Vi and a are of the same order of magnitude, the regime is called the kinetic chemical equilibrium regime (fast reactions). On the contrary, if Vi is of some orders greater than <jj the regime is referred to as the one proper of slow reactions. Discussion upon the form of kinetic equations and derivation of the corresponding fluid dynamic limit at Euler level for slow and fast reactions can be found in 5 . Nevertheless the mathematical complexity of the operators Gj and Qi has recently addressed researchers to derive simplified kinetic models of BGK-type. As well known, starting from the pioneer idea of Bhatnagar, Gross and Krook 6 , several BGK models for a one-component gas 7 ' 8,9 and for inert mixtures 1 0 ' n have been derived. Further extensions of such approaches to reactive mixtures have been performed in 2004 and 2005 in the papers " . " . I M B . All the BGK-type models of the last references share the same functional form of the kinetic equations (2) with the gain terms replaced by Gj[/] and <7j[/], where / is a suitable reference distribution function. In particular in paper 12 / is an equilibrium distribution whose components are true Maxwellians, i.e.
7i \ h{v)=n
(
\^f)
m
i
Y/2 eXP (
(
rni(v-u)2\
2^T-J'
(3)
where Tij denotes the number density of each gas-species, u and T the mean velocity and temperature of the whole mixture. The model of Ref.12 has been derived under rather strict mathematical assumptions which however
380
assure to the model conservation of mass, momentum and energy, and, at the same time, allow to provide a description of the collision mechanism of the chemical processes. Nevertheless, numerical simulations at both mesoscopic and macroscopic levels confirm a reasonable behavior with respect to the trend to thermodynamical equilibrium and entropy production (see also paper 1 6 ). Almost in the same period Groppi and Spiga 13 have derived, following the philosophy of Ref.11, a BGK model which considers as reference distributions the functions /» denned by
where nl, ul and Tl are parameters related to number densities, mean velocities and temperatures of each single species, to be determined in such a way that the kinetic equations assure conservation of the measurable physical quantities, as total mass, momentum and energy. The mentioned papers 12>13'16 are related to slow reactions. Conversely, a very recent BGK-type model for comparable values of elastic and reactive collision frequencies is the one proposed in Ref.14. In this paper the reference distribution functions, which are different when referred to either elastic or reactive collisions, are given by ~
/
/,/W^(l + 4 + B S -c + C ^ K ( ^ )
\ 3/2
/
2\
e x p(-^), (5)
where n* is the number density of i-species, c = v — u, u and T are the macroscopic variables of the whole mixture. The subscripts ij denote the chemical compounds, whereas the superscripts T = E and V = R are referred to the elastic and reactive mechanism, respectively. Moreover, A^j, Bfj and Cfj are coefficients to be determined by imposing that the production terms are the same in the exact and approximate models. This BGK approach, reported by the same authors in this book of Proceedings, seems to be very efficient to derive explicit expressions of the transport coefficients for a Navier-Stokes fluid. Finally paper 15 has improved the early ideas of Ref.12, removing the mathematical constraints therein imposed. Two BGK-type models for slow and fast reactions have been there derived in a rather different form from the one of Refs. 12 ' 16 , assuming as reference distributions the true Maxwellians (3). In particular the model related to slow reactions takes into account Maxwellians, hereinafter called f^\ representing mechanical equilibrium
381 only. Conversely the model relevant to fast chemical processes considers as reference distributions Maxwellians stating mechanical and chemical equilibrium at the same time. These Maxwellians, which will be indicated by f^2\ will be recovered in a way that will be specified better in the next section. In Section 2 the models of paper 15 will be briefly resumed in order to present in Section 3 some numerical simulations for slow and fast chemical reactions of the hydrogen-iodine-chlorine chain. 2. The BGK-type models for slow and fast reactions In this section the two aforementioned models will be presented without giving their rigorous derivation. All the details can be found in paper 15 , where also the exact kinetic equations (2) are given together with the explicit definition of the collision frequencies i/» and o-j. However, before writing the model equations of the next two subsections, it is convenient to recall some properties of the exact kinetic equations (2). First of all, the elastic operators {Gi[f](v) — fi} vanish if and only if the distributions fi are Maxwellians of the form (3). Moreover also the reactive terms {Gi[f](v) — fi} vanish if and only if the distributions fi are of the form (3) with the number densities n, satisfying the relation =M*exp
n3n4
— - ) ,
(6)
\kBTJ
which represents the well-known * mass-action law of a chemical reaction. In Eq.(6), M = mim2/(m3m4) is a reduced mass and AE = E3 + E4 — Ei — E-x is the binding energy difference, Ei being the heats of formation of each chemical compound. In the case that AE > 0 then the forward reaction A\ + Ai —> A3 + A4 is the endothermic one. As already said in the introduction, a Maxwellian distribution with arbitrary values of number densities will be indicated by f> . Conversely f± will mean that the distributions are Maxwellians (3) with number densities n, satisfying the equilibrium conditions (6), so that they fulfill thermodynamical (mechanical plus chemical) equilibrium. The family of Maxwellians f\ depends on eight parameters u, T, n;, i = 1 , . . . , 4, fixed by the eight independent mechanical collision invariants
382
^(1) = ^
(2)
ip^
=
vx{mi,m2>m3,m4} vy{m1,m2,m3,m4}
=wz{mi,m2,m3,m4}
i){i) = Y^m^2
+ Ex, ^m2v2 + E2, ~m3v2 + E3, -m4v2
+ EA
0 « = {1,0,0,0}, flW = {0,1,0,0}, 0<3> = {0,0,1,0}, 0 (4) = {0,0,0,1}. The invariants 6^k\ k = 1 , . . . ,4, are related to conservation of individual number densities n^ during elastic collisions only. Conversely, the family of Maxwellians /• depends on seven independent parameters, i.e. it, T and number densities n, satisfying the massaction law (6). Introducing the invariants ^(5) = 0(D + 0(3),
^,(6) = 0(D
+
0(4)}
^(7) = 0(2) +fl(4)_
joined to the conservation of partial number densities n\ + n3, m + n^, n2 +ri4, then the seven independent invariants of the system are V ( 1 ) , . . . , ^ ( 4 ) , V (5) = {1,0,1,0}, ^W = {1,0,0,1}, V (7) = {0,1,0,1}. 2.1. T/ie BGK
model for slow reactions
(Model
1)
Let v\ and cr^ denote the elastic and reactive frequencies computed by substituting the distributions / ; with the Maxwellians f- . The model proposed in this sub-section is based on the following assumptions: (i) The gas evolves sufficiently near equilibrium, so that after at least one collision particles may reach a local mechanical equilibrium (slow chemical process). Thus, the relaxation towards a Maxwellian implies that in the gain contributions, Gi and Qi, the distribution functions /* are substituted by the Maxwellians f> . (ii) The differences between the integrals defining v^ and v\ , and Oi and aI , are neglected, so that v\ — v%, a\ = <Ji. Moreover the elastic and reactive interaction potential laws are chosen as the ones of Maxwell-molecules form 17 , so that the collision frequencies will be v-independent. (iii) The parameters u, T and n* for the Maxwellians f> ' are fixed in terms of the mechanical invariants through the conditions
383 4
E
4
r
fc)
r
L,^ («)/i(v)
L^\v)fP{v)dv, fc = l,...,4
1=1 • / J K -
1=1 - ^
^
6f/,(„)<*„ = j ^ 0?>tf\v)dv
,1=1,...
A-
Thanks to these hypotheses and to the choice of the collision frequencies modelled as in Ref.18, model 1 has been obtained in the following form ^
+ v • V/< = 4TT £ > « « , • [fP(v)
- /*(«)] + ^
l
\ v ) - a^v), (7)
3=1
where 7i = An3n4/ni, o\ — Bn2,
-y2 = An3n4/n2,
73 = Bnin2/n3,
(J2 = Bn\,
fcBT
'
v y
^i =
03 = A714,
'
J
\MJ
5(T) = ^ y ^ | exp(-x) + (1 - erf VX) (
Bn1n2/ni
(T4 = A n 3
^ \ kBTj ' ^ - C
In particular, in the last equations, a^- are the constant elastic crosssections, G = m2,m4/{rni+m,2), £ is the threshold velocity of the exothermic reaction given by £ = (2Ea/Q)1^2, Ea being the activation energy of the exothermic reaction, and \ — ®£,2/(2kBT). Moreover, j3 is a scalar factor which is relevant to characterize the chemical regimes (slow and fast) since it is related to the reactive collision frequencies. In paper 15 it has been proven that model 1 verifies the properties of the true Boltzmann equation consistent with the physical laws of conservation of mass, momentum and total (mechanical plus chemical) energy; in addition, it satisfies also the indifferentiability principle n . Moreover, elastic collisions and chemical reactions contribute to increase the entropy of the system according to the following result. Proposition Let Ji be proportional to the kinetic entropy of the system and 4>-n be its diffusive flux defined by H(x,t) = Y , l
fi log (JL) dv ,
Mx,t) = J2 f
h log
—~ ) vdv . TO? /
384
Then — + div 4>n < 0, provided that the distributions of products of the forward reaction satisfy the inequality 4
where K™\£,l™](v)
r _ > g ( ^ ] 7 ^ U / = ^[fWmifW]
W
] ( ^ < 0 ,
(8)
- /,}. (2)
Moreover dTC/dt + div^-n = 0 if and only if fi = f{ . Observe that the proof of the proposition depends on the constraint (8) that has only a mathematical meaning. Nevertheless in paper 15 it has been verified that this inequality is always fulfilled in the numerical simulations there performed. 2.2. The BGK model for fast reactions
(Model
Similarly to the previous sub-section, let v\
and a\
2) denote the elastic (2)
and reactive frequencies computed in terms of the Maxwellians f£ . The model is then based on the following assumptions: (i') Particles reach a local thermodynamical equilibrium after one collision (fast chemical process). Thus, gain contributions Gi and Qi (2)
are re-written in terms of the Maxwellians /> . (ii') The differences between the integrals defining J/J and v\(2) , and cr, and a\ , are neglected, so that v\ ' = vi: a\ = Oi . The assumptions on the elastic and reactive cross sections are the same as those in (ii) for model 1. (iii') The seven independent parameters for the Maxwellians f\ ' are fixed in terms of the system invariants through the conditions 4
4
k
E
L34 \v)Mv)dv J]R
-
i=i
= J2 I jlk\v)fl2\v)dv, »=i
k = l,...,7.
J
>&
With these hypotheses model 2 can be deduced in the following form ^
+ v • VA = I 4TT J2 <xijnj + a% j \ff\v)
- fi(v)],
(9)
385
where again the a^- and a, are the same as those of model 1. The determination of the distributions f> , which are referred to mechanical and chemical equilibrium as stated in item (i'), is performed through the following steps: (a) after the numerical integration at each time step of equation (9) the values of the number densities n^ are computed through the moments of /^; (b) through the mass-action-law (6) the equilibrium temperature T is obtained by the formula nin 2 n3n4
T =
/J_\3/2 \MJ
(10)
(c) finally the parameters n* and T, provided by items (a) and (b), are inserted into the equilibrium distributions (3) in order to recover the correct expressions of f} . It has been shown 15 that model 2, for what concerns the conservation laws and the indifferentiability principle, has the same properties as model 1 and, in addition, has the advantage that entropy growth can be proven removing the mathematical constraint (8). 3. Numerical simulations of slow and fast reactions The aim of the numerical simulations consists in showing, at a mesoscopic level, the trend to thermodynamical equilibrium of both models. More in particular, in order to compare qualitatively their behavior, the numerical experiments are performed starting from the same initial data, given by asymmetric bimodal distributions. In such conditions the functions / i , . . . , ft represent a mixture in an arbitrary mechanical and chemical state of disequilibrium and their shapes are those reported in the upper frames of both figure 1 (slow chemistry) and figure 2 (fast chemistry) at time t = 0. Accordingly, the slow reaction 19 ICl + H2 ^ and the fast reaction
HI + HCl
19
h
+ HCl — ICl + HI
will be considered using model 1 and model 2, respectively.
386 T h e purpose of the experiments consists then in checking how quickly the distributions reach a Maxwellian state, represented by the characteristic Gaussian shape. For the slow and fast reactions t h e following d a t a have been assumed, respectively Ea = 145000, £ a = 120000,
AE = 83495.15, AE = 73892.81,
atj = 1, a»j = 1,
/3 = 0.0066, /3 = 2.3.
For w h a t concerns t h e initial densities of t h e chemical species for b o t h reactions t h e following values ny = 0.4,
ri2 — 0.3,
773 = 0.2,
714 = 0.1
have been chosen. Conversely, to obtain the same shapes of the distributions at time t = 0, the t e m p e r a t u r e s have been fixed equal t o 1400 and 2475 for the slow and fast reaction, respectively. T h e evolution is well described in figures 1 and 2 for the slow and fast chemistry, respectively, where in the upper frames the shapes of the distributions are shown, versus v, at times t = 0.000, t = 0.015 and t = 0.025; in t h e lower frames the shapes are those reached at times t = 0.05, t — 0.1 and t = 0.2. It can be observed t h a t , as expected, the fast reaction provides distribution shapes close t o a Gaussian since time t = 0.1 (lower frame of figure 2). On the contrary at this time for the slow reaction, the distributions have not yet reached a Gaussian form, and even at t = 0.2 the distribution fi is far from a Maxwellian (lower frame of figure 1). Moreover time t = 0.2 can be considered t h e final time of the chemical process for the fast reaction since no more production occurs, whereas, after t h a t time the slow reaction still presents changes in reactant and product concentrations. References 1. V. Giovangigli, Multicomponent Flow Modeling, Boston, USA, Birkhauser, 1999. 2. A. Ern and V. Giovangigli, The kinetic chemical equilibrium regime, Physica A, 260, 49-72, 1998. 3. A. Ern and V. Giovangigli, Kinetic theory of reactive gas mixtures with application to combustion, Transp. Theor. Stat. Phys., 32, 657-677, 2003. 4. A. Rossani, G. Spiga, A note on the kinetic theory of chemically reacting gases, J. Stat. Phys., 272, 563-573, 1999. 5. F. Conforto, M. Groppi, R. Monaco, G. Spiga, Steady detonation problems for slow and fast chemical reactions, in Modelling and Numerics of Kinetic Dissipative Systems, Ed. L. Pareschi et a l , Nova Science, New York, 113-126, 2005.
387 6. P.L. Bhatnagar, E.P. Gross, K. Krook, A model for collision processes in gases, Phys. Rev., 94, 511-524, 1954. 7. C.D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83, 1021-1065, 1996. 8. P. Andries, P. Le Tallec, J.P. Perlat, B. Perthame, The Gaussian-BGK model of Boltzmann equation with small Prandtl number, European J. Mechanics (B Fluids), 813-830, 2000. 9. L. Mieussens, H. Struchtrup, Numerical comparison of Bhatnagar-GrossKrook models with proper Prandtl number, Phys. Fluids, 16, 2797-2813, 2004. 10. V. Garzo, A. Santos, J.J. Brey, A kinetic model for a multicomponent gas, Phys. Fluids, 1, 380-383, 1989. 11. P. Andries, K. Aoki, P. Perthame, A consistent BGK-type model for gas mixtures, J. Stat. Phys., 106, 993-1017, 2002. 12. R. Monaco, M, Pandolfi Bianchi, A BGK-type model for a gas mixture with reversible reactions, in New Trends in Mathematical Physics, Ed. P. Fergola et AL, World Scientific, Singapore, 107-120, 2004. 13. M. Groppi, G. Spiga, A BGK-type type approach for chemically reacting gas mixture, Phys. Fluids, 16, 4273-4284, 2004. 14. G. Kremer, M. Pandolfi Bianchi, A.J. Soares, A relaxation kinetic model for transport phenomena in a reactive flow, submitted, 2005. 15. R. Monaco, M. Pandolfi Bianchi, A.J. Soares, BGK-type models in strong reaction and kinetic chemical equilibrium regimes, J. Phys. A: Math. Gen., 38, 10413-10431, 2005. 16. R. Monaco, M. Pandolfi Bianchi, A.J. Soares, A reactive BGK-type model: influence of elastic collisions and chemical interactions, in Rarefied Gas Dynamics, Ed. M. Capitelli, American Inst, of Phys., New York, 70-75, 2005. 17. C. Cercignani, Rarefied Gas Dynamics, Cambridge University Press, Cambridge, 2000. 18. F. Conforto, R. Monaco, F. Schiirrer, I. Ziegler, Steady detonation waves via the Boltzmann equation for a reacting mixture, J. Phys. A: Math. Gen., 36, 5381-5398, 2003. 19. I. Prigogine, R. Defay, D.H. Everett, Chemical Thermodynamics, Longman, London, 1973.
388
f
fj (ICl)
,
(H 2 )
t=B.BB8 t=B.815 t=B.B25
fA (HCl)
f3 (HI)
^S3r-- t=B.BBB ^//fe^t=B.B15 / C_\-t=8.B25
f! (ICl)
f,
(H 2 )
f\
(HCl)
/ t=B.B5B /.t-B.IBB ^/t=B.2BB
^ f3 (HI)
3 C \ / t=B.B5B "^\v)/,t=B.lBB W^t=B.28B
Figure 1. times
Slow reaction ICl + Hi
^
H / + # C £ : distributions versus u at different
389
f, (h)
f, (HCl)
f 3 (ICl)
f4
fl
(ffl>
f2 (HCl)
d2)
/^v V \1 \
.t=B.85B X^t=B.108 At=8.2BB
^\
^t=B.B58
J > ^ > t=B.lB0 < O V ^ t=B.2B8
^J f 4 (HI)
^t^.Bsa -^-^^--t'B.iBa ' J ^ ~ ~ ~ ^ t=B.2BB
Figure 2. times
Fast reaction I2 4- HC7 ^
7C7 4- i J / : distributions versus u at different
A N O P E R A T I V E M E T H O D TO D E F I N E G E N E R A L I Z E D - E N E R G Y F U N C T I O N A L S IN P D E s A N D IN CONVECTION PROBLEMS
G. M U L O N E Dipartimento di Matematica e Informatica Citta Universitaria, Viale Andrea Doria 6, 95125 Catania, ITALY E-mail: [email protected] We give an operative method to define optimal Lyapunov functionals to studying stability of solutions of a class of PDEs systems which includes reaction-diffusion systems and convection problems. By using the results of linearized instability analysis and the classical eigenvalue-eigenvectors method, we obtain new field variables and transform the system in an equivalent one in terms of the new fields. Then, we use the classical energy to define an optimal Lyapunov functional and we obtain the coincidence of the critical parameters reached by the linearized instability analysis Rc and with the Lyapunov method RE-
1. Introduction Let us consider a dynamical system U = fR(U),
U(0) = U0
(1)
where U(t) is an element of a Hilbert space H for any t > 0, and Uo G H. {R : Hi C H —> H is a function defined in a subset H\ of H and R is a positive real number which depends on the "physics" of the problem (sometimes R is a "set of parameters"). In many applications H — L2(Q), where Q is the space-domain of motion, or H is a subspace of L2(Q.) (with scalar product (•, •) and norm || • ||). Assume that U = U is a steady solution of (1), i.e., {R(U) = 0, E/(0) = U. In order to study the stability-instability problem of U the main classical methods are: a) the method of the linearized instability: it provides a critical value Rc such that for any R > Rc the basic solution is unstable, if
390
391 R < Rc the solution U is linearly stable (stable against infinitesimal perturbations); b) the Lyapunov method (with a Lyapunov functional E): it provides a critical value RE below which the basic solution is linearly, and nonlinearly (conditionally or globally) stable whenever the nonlinear terms can be controlled. In general, we have RE
< Rc.
If, in particular RE — Rc, one obtains necessary and sufficient of linear and nonlinear (conditional or global) stability.
conditions
In many physical applications the "classical energy" E0{t) = \\\U\\2 is used as a good Lyapunov functional to control the stability, Serrin 26 , Rionero 22 . In these cases generally one obtains < Rein order to improve the critical value of nonlinear stability RE0, especially in presence of some stabilizing physical effects, a generalized energy E{t) different from the classical energy Eo(t) may be introduced (see Joseph 8 , Galdi and Straughan 7 , Mulone and Rionero 15 , Galdi and Padula 5 , Flavin and Rionero 3 , Straughan 28 ). In particular, RE0
E(t)=E1(t)
+ E2(t),
(2)
where E\ (t) - introduced in a heuristical way - is a Lyapunov functional for the linearized problem and E2(t) controls the nonlinearities. E\{t) gives via a variational problem - the critical Lyapunov stability parameter REIn some applications the dynamical system (1) is given by U' = LU + NU,
U(0) = U0
(3)
where L represents a linear operator and N a nonlinear operator (with iVO = 0) which satisfy suitable hypotheses (typically L is an operator with compact resolvent, the bilinear form associated with L is defined on a subspace H*, which is compactly embedded in H, and (NU,U)
<0,OT(NU,U)
see Galdi and Rionero 6 , Straughan 28 ).
392
If L is symmetric or simmetrizable the critical values of linearized instability and of nonlinear (energy) stability coincide (see Davis 2 , Galdi and Straughan 7 ). The operator L is in general non-symmetric, although it allows a decomposition into two parts L\ and Li such that: L — L\ + Li, with L\ symmetric and Li skewsymmetric. From a physical point of view, a skew-symmetric linear operator Li represents a stabilizing effect that gets lost if we use the classical energy, EQ. In fact, from (3), and (LiU, U) = 0, we have the energy identity lWf
=
{LlU,U)
+ (NU,U).
(4)
Moreover, whenever Li = 0, RE0 = Re and we have necessary and sufficient stability conditions. In general, if the operator L 2 J^ 0, we have RE0
< Rc-
We assume here that L\ is a symmetric uniformly elliptic operator. The aim of this paper is to give an operative method to define an optimal Lyapunov functional (see Sec. 2) to studying stability of the zero solution of a class of PDEs systems which includes many reaction-diffusion systems and convection problems (see Mulone 1 2 _ 1 4 ) Lombardo and Mulone 9 , Mulone and Straughan 17 , Lombardo, Mulone and Trovato 10 - 11 in the cases of reaction-diffusion systems and/or Benard problems in fluid dynamics, magnetofluid dynamics and double diffusive convection in porous media). The plan of the paper is as follows: in Sec. 2 we give the operative method to define generalized-energy functionals which are optimal, in Sec. 3 we apply the method to an easy ordinary differential system, in Sec. 4 optimal Lyapunov functionals are obtained for some nonlinear reactiondiffusion systems. 2. Operative method to define optimal Lyapunov functionals Here we first define an optimal (generalized-energy) Lyapunov function, then we give the most important steps of its construction. The main idea is to use the classical eigenvalue-eigenvectors method and the results of the linearized instability (which is related to the first eigenvalue of the particular elliptic operator).
393 Definition 2.1. A Lyapunov function E is optimal if the associated critical parameters Rc (obtained with the classical linearized instability theory) and RE (obtained with the Lyapunov function E) coincide: in this case, we have necessary and sufficient stability conditions. The main steps are the following: (1) Given the dynamical system Ut = LU + NU,
(2)
(3) (4)
(5)
U(Q) = UQ,
(5)
first, we consider the associated linearized system Ut = LU and find the critical instability parameter Rc, we introduce a linear ordinary system X = AX. with matrix A, associated to L via the principal eigenvalues of the uniformly elliptic operators of the particular problems; we compute the eigenvalues of A; we introduce a matrix S (modal matrix) of eigenvectors (and generalized eigenvectors in the case of multiple eigenvalue with different geometric and algebraic multiplicity) of A and its inverse S_1 (S is given by an n by n array such that the jth column is the jth eigenvector corresponding to the jth eigenvalue. If the jth eigenvalue is complex, the jth column and the (j + l)th column in the eigenvectors array are the real and imaginary parts corresponding to the jth eigenvalue); we introduce the new field variables Y = S"_1X and write the new linear system Y = BY = S~1ASY (topologically equivalent to the initial ordinary differential system) in terms of these new field, the optimal Lyapunov function for this problem is - | Y | 2 (the matrices B and A are similar);
(6) we write the PDE system in terms of the canonical new field V (which is in a one-to-one correspondence with the field U) connected with Y(x, t) = (t/i(x, t),..., 2/n(x, t))T by the principal eigenvalues of the elliptic operators: Vt = LV + NV, L associated to 5 _ 1 A 5 ;
V(0) = V0,
(6)
394 (7) we introduce the Lyapunov function
m) = \\\v(t)\\2 for the new linearized system and study the stability of the zero solution, we write the energy equation of Ei(t) and solve the related variational minimum or maximum problems (see 10 ~~ n) to obtain the critical stability Lyapunov parameter RE and show that it coincides with Rc; (8) we consider the PDE nonlinear system written in terms of new variables V^x, t) and define the Lyapunov function E(t) = E1(t) + E2(t), (E2 controls the nonlinearities) and write the energy identity for E. We shall obtain the coincidence of the linear and nonlinear (whenever we are able to control the nonlinearities) stability boundaries for any values of the physical parameters of the basic motion. In the following sections some applications are given to an ODE system (in this case some of the previous steps can be skipped) and two reaction diffusion systems. Other applications to fluid-dynamics systems and Benard problems can be seen in n ~ 14 . 3. A n ODE s y s t e m Let us consider the ordinary differential system: ±i = -xi + ni(x1,x2,x3) x2 =-2x2-3x3 + n2(xi,x2,x3) x3 — ax\ + 3x2 - 2x3 + n-i(xi,x2,x3)
(7)
where a is a given real number and n = (ni,n2,n3)T represents the nonlinear contribution. We assume that n* are (nonlinear and sufficiently smooth) functions defined in K3 and n(0) = Dn(0) = 0, where Dn(0) is the Jacobian of n evaluated in 0. Thus the system admits the zero solution. The linearized system near the origin can be written in the form X = AX with matrix A given by
395 The eigenvalues of the matrix A are Ai = - 1 ,
A2 = - 2 + 3i,
A3
-2 — 3i
and we have linear (and local nonlinear) stability for any a G If we use the classical Lyapunov energy
we obtain sufficient stability conditions whenever |o| < 2\/2. By using the eigenvalue-eigenvector method, it can be proved that there exist an invertible matrix S and a similar matrix B = S~XAS given by
10 0 0N
-3a 0 1 1 , 5 a
1 0,
-1
-10 0 £= I 0-2 3 0 -3 -2.
—a 10
Vio
l
°J
such that system (7) is transformed in the equivalent system y = By + m, where m = 5 _ 1 n . An optimal Lyapunov function is given by
E:=\{yl + y22 + yi). With this Lyapunov functional we have nonlinear (local) stability for any real number a. Moreover, we have also global stability if the nonlinear terms satisfy the inequality 5 _ 1 b(5'y) • y < 0; for example, if n = 10^3Xi(-10,3a,-a)T, we easily obtain E = -(y21+2yl
+
2y23)-y$
and we have global stability for any real number a.
396
4. Nonlinear reaction-diffusion systems Nonlinear reaction-diffusion systems are well suited to model a wide range of physical, chemical and biological pattern formation processes (see for instance 18 ~ 2 0 ) . Set a nonlinear reaction-diffusion system U' = AU + DAU + NU,
U(0) = Uo,
(8)
where U is a vector with m components Ui(x), x £ M™, n = 1,2,3, A and D are m x m matrices with real coefficients, A is the n-dimensional Laplacian and JV is a nonlinear operator with N(0) = 0. In many applications D = diag(£>i, D2,. •. ,Dm) with Dt > 0. For example (1) Field-Noyes equations (see 4 , 27 ) U = («, v, w)T, D = diag(r>i, D2, £>3) fa
a
0
0 a"1 a_17
\5
0
f ~af3u2
\ ,
N(U) =
-S J
x
-a
\
\
uv
0
with a, /?, 7, S positive parameters. (2) Equations introduced by Schenk et. al., 24 U = (u, v, w)T, D = diagCA, D2/T, D3/9)
f\-3u2 1/r
-1 -k3 \ -1/r 0
0
( -3uu2 - u3' ,N(U)
-1/0/
0 \ / with u = v •=• w, a constant basic solution, and A, r, 0 and £3 positive parameters.
V I/*
Here we first consider an application of the method to a two-component model (the Prigogine - Brusselator model, see 21 , 25 , which is an example of an autocatalytic chemical reaction). The perturbation equations to the stationary solution (A, B/A) are given by j ut — Bu — u + A2v + DiAu + ni(u,v) \ vt — -Bu - A2v + D2Av + n2(u, v),
(9)
397
in ft x (0,oo). ft is a bounded domain of E n (n = 1,2,3). A, B, L>i, £>2 are positive parameters, A and B are the fixed concentrations of two initial chemical products. D\ and D2 are the diffusion coefficients and n* (i = 1,2) are the (cubic) nonlinearities. To the system (9) we add the initial (small enough to guarantee the global existence, see 28 )
u(x, 0) = uo(x),
v(x, 0) =
v0(x),
and the Neumann boundary conditions du _ dv _ dn dn on the boundary and the average conditions Jn u dx = 0, JQ v dx = 0 or the Dirichlet boundary conditions u = 0,
v = 0.
For the sake of the simplicity here we consider the one-dimensional case, x = x £ [0,1], and the Dirichlet boundary conditions. If we consider the classical energy
we have (in the case of the linearized system)
E0 = (S- i ) H 2 - 4dN 2 - (AIMI 2 + £ 2 |M 2 ) <[B-1-
T T ^ I H
2
- | - ( A 2 + 7r2)||.;||2
(10)
-D1\\uxP-D2\\vx\)2. If B
:=1 + TT2D1,
we have linear Lyapunov stability (with respect to the classical energy EQ). It can be proved that we have also (conditional) nonlinear stability if we use the Lyapunov function EQ + E with suitable E. Now we apply the previous method. By linearizing system (9) and by searching solutions of the form u = uoext smn-KX, v = voext sinn7rx, we
398 easily obtain the eigenvalues A„. It is easy to verify that all the eigenvalues have negative real parts (i.e. linear stability) whenever B < Bc = minmin(l + £„-Di
-,l+A'+UDi+D2)),
ZnD2
(11)
where £„ = n27r2. From (11) we see that A2 has a stabilizing effect. This effect has not been reached with the energy EQ. For simplicity here (in the general case, see 10 ) we suppose A* = B,
4n2 B < n^D
D1=D2=D,
Taking into account that the first eigenvalue of the operator A in [0,1] with zero boundary conditions is — n2, we easily obtain the matrix A
-A2 - w2D2
-B
Its eigenvalues are given by -l-2n2D±Vl-4B Ai,
Let us assume that B > 1/4 (the case 0 < B < 1/4 can be studied in the same way). A matrix of vectors S and its inverse are given by l_ S = | 2§ ~
1
0
V4B - 1 2B
2B V4B-1
0
2B - 1 VlB-1
From the expression of the matrix S and S"1 we have the new canonical fields 4> and ip = v 2B
V> =
V4B-1 and we easily obtain the new system 1 , VAB which is equivalent to (9).
u H—,
y/AB-1
-
(12)
2B-1 „
=v.
V4B-1
ip + D(j)xx +n1{
I
I
399 We introduce the Lyapunov function
Em = \
(14)
+
We easily obtain Ex = -Ei - D(\\(t>x\\2 + \\ijx\\2) + (#,«!) + (V,n 2 )
(15)
2
and therefore, for any A we have linear-Lyapunov stability. In order to study nonlinear stability we introduce the Lyapunov function E = Ei(t) + E2{t), with E2 a suitable complementary energy to control the nonlinearities. For example as E2 we can choose that proposed by Straughan 28 :
Thus we are able (see
10
) to obtain conditional nonlinear stability for any
B > 0 and we reach the linear stability results. Remark. Since we are dealing with a 2 x 2 system, the previous results can be obtained also with the method of Rionero 23 . Now we consider another reaction-diffusion system (8), a 3 x 3 system, the Schenk et. al. model in two-dimensional spatial case: U = (u, v, w)T, D = diag(£>i, D2/T, D3/0) (
A=
A-3u2 1/r
V 1/6*
( —3uu2 — u3 \
-h\ -1/T
0
0
0
,N(U)
V
-1/0/
0
/ with u = v = w, a constant basic solution, and A, r, 9 and £3 positive parameters. By using the classical energy norm
^ = ^[W 2 + r|H|2 + MIHI 2 ], we have stability (linear and conditional nonlinear) whenever 2
a := A — 6u
2TT^DI
—z— < 0.
We note that this condition does not hold (at least) for the physical case considered by Schenk: fci = -6.92, k3 = 10.5, A = 2, u = -0.69, Dx = 10" 3 , D2 = 1.25 x 10" 3 , D3 = 0.064, r = 48, 9 = 1, L = 1.3.
(16)
400
In fact, here a = 0.546. In this case, the matrix A of the associated ODE system X = AX has the eigenvalues A1>2 = -0.59 ± 3.03 i, A3 = -0.025 and the new PDE system we obtain is fa = -0.587 - 3.03^ + 1O" 3 A0 + m ( 0 , V, x) ipt = 3.03<£ - 0.584^ + 1.25 x 10" 3 A^ + n2{4>, i>, x)
(17)
Xt = 0.722X + 0.064A X + n 3 (0, ip, x), where n* are cubic nonlinearities. By using the optimal Lyapunov function
Ei = \{w\\2 + \\n2 + \\x\\2) (for the linearized system) we have Ex = -O.587H0!!2 - 0 . 5 8 4 M 2 + 0.722|| x || 2 -(10-3||V||2 + 1.25 x 10— 3 jiW|| 2 + 0.064||Vx|| 2 ).
(18)
With E = E\ + Ei (Ei a suitable complementary energy to control the nonlinearities) we can find also (see 10 ) nonlinear (conditional) stability with a known radius of attractivity for the initial data. Acknowledgments This work is supported by the Italian Ministry for University and Scientific Research, PRIN: Problemi matematici non lineari di propagazione e stability nei modelli del continuo (Coordinator Prof. T. Ruggeri), by Gruppo Nazionale della Fisica Matematica of the Istituto Nazionale di Alta Matematica, and by the University of Catania. References 1. S. Chandrasekhar, Hydrodynamic and hydromagnetic stability. Clarendon Press (1961). 2. S.H. Davis, Proc. Roy. Soc. London A, 310, 341 (1969).
Oxford:
401 3. J. Flavin and S. Rionero, Qualitative estimates for partial differential equations. An introduction. Boca Raton, Florida: CRC Press (1996). 4. R.J. Field and R. M. Noyes, J.Chem.Phys. 60,1877 (1974). 5. G.P. Galdi and M. Padula, Arch. Rational Mech. Anal. 110, 187 (1990). 6. G.P. Galdi and S. Rionero, Weighted energy methods in fluid dynamics and elsticity, LNM 1134, Springer, Berlin (1985). 7. G.P. Galdi and B. Straughan, Arch. Rational Mech. Anal. 89, 211 (1985). 8. D.D. Joseph, Stability of fluid motions, Springer Tracts in Natural Philosophy, vols. 27 and 28 Berlin: Springer (1976). 9. S. Lombardo and G. Mulone, Nonlinear Anal., doi:10.1016/j.na.2004.09.003 (2004). 10. S. Lombardo, G. Mulone and M. Trovato, A general analytical procedure to obtain optimal Lyapunov functions in reaction-diffusion systems, Rend. Circolo Mat. Palermo, (to appear), (2005). 11. S. Lombardo, G. Mulone and M. Trovato, An operative method to define Lyapunov functionals in PDEs and in fluid dynamics problems, (in preparation), (2006). 12. G. Mulone, Proc. "New Trends in Mathematica-Physics" Naples, January 2003, World Scientific, 121 (2004). 13. G. Mulone, Proc. Waves and Stability in Continuous Media, WASCOM 2003, Villasimius, June 2003, R. Monaco, S. Pennisi, S. Rionero, T. Ruggeri Eds., World Scientific, Singapore, 352 (2004). 14. G. Mulone, Far East J. Appl. Math. 15, n.2, 117 (2004). 15. G. Mulone and S. Rionero, J. Mat. Anal. App. 144, 109 (1989). 16. G. Mulone and S. Rionero, Arch. Rational Mech. Anal., 166 no. 3, 197 (2003). 17. G. Mulone and B. Straughan, An operative method to obtain necessary and sufficient stability conditions for double diffusive convection in porous media, (submitted), (2005). 18. J.D. Murray , Nonlinear Differential Equation Models in Biology, Clarendon Press, Oxford, (1977). 19. J.D. Murray , Journal of Theoretical Biology, 98, 143-163, (1982). 20. J.D. Murray , Mathematical Biology, 2nd Corrected Edition, Springer, (1993). 21. I. Prigogine and R. Lefever, J. Chem. Phys. 48, n. 4, 1695 (1968). 22. S. Rionero, Ann. Mat. Pura Appl. 76, 75 (1967). 23. S. Rionero, J. Mat. Anal Appl. doi:10.1016/j.jmaa.2005.05.059 (2005). 24. C.P. Schenk, M. Or-Guil, M. Bode, and H. G. Purwins, Phys. Rev. Lett, 78, n. 19, 3781 (1997). 25. L. A. Segel and J.L. Jackson, J. Theor. Biol. 37, 545 (1972) . 26. J. Serrin, Arch. Rational Mech. Anal, 3, 1 (1959). 27. J. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, New York, 2nd Ed. (1994). 28. B. Straughan, The Energy Method, Stability, and Nonlinear Convection, Springer-Verlag: Ser. in Appl. Math. Sci., 9 1 , New-York, 2nd Ed. (2004).
O N T H E EXISTENCE OF H I G H - E N E R G Y TAILS FOR T H E BOLTZMANN T R A N S P O R T EQUATION IN SEMICONDUCTORS
ORAZIO MUSCATO Dipartimento di Matematica ed Informatica, Universita di Catania, Viale A. Doria 6- 95125 Catania, Italy E-mail: [email protected] We prove analytically the existence of high-energy tails for the Boltzmann transport equation for silicon semiconductors, in the stationary and homogeneous regime, in the quasi-parabolic band approximation, and acoustic and optical phonons scattering.
1. Basic Equations Hot carrier effects play an important role in many semiconductor devices. In order to predict the performance of advanced devices it is therefore necessary to be able to model the electron energy distribution function (EED). In this paper we shall tackle this problem in the Boltzmann transport equation (BTE) framework, which writes 1,z ^ + v ( k ) - V x / - | E - V k / = C[/]
,
(1)
where the unknown function /(i,x, k) represents the probability density of finding an electron at time t in the position x = (x\, X2, £3) with the wavevector k = (fci, &2i ks), and q is the absolute value of the electron charge. In the neighborhood of the band minimum the dispersion relation can be considered approximately quasi-parabolic: h2V.2
e(k)[l + a e ( k ) ] = 7 = — ,
(2)
where m* denotes the effective electron mass, which is 0.32 m e (free electron mass) in silicon, a the non parabolicity factor (0.5 eV^ -1 ),and h the Planck constant divided by 2TT. Consequently, the electron group velocity v =
402
403 (vi,v2,v3)
writes v(k) = - V k e =
The electric field E(t,x) = {Ei,E2,Ez)
.—
.
(3)
satisfies the Poisson equation
A(e
(4)
E = -Vx<£ where (f>(t,x) is the electric potential, No and NA are respectively the donor and acceptor densities (which are positive functions), e the dielectric constant, n the electron density. Concerning to the scattering mechanisms, we shall consider the electron-phonon interaction (acoustic and non polar optical) as modelled in 2 . 2. Exponential tails In the following we shall draw our attention to the asymptotic behaviour of the steady state distribution function of the BTE /0O(k)= lim/(t,k) t—*oo
for large |k|, i.e. the high-energy tails. We expect that the behaviour of this solution is /oo(k) ~ exp [-r(2e)*]
,
|k|-> oo
(5)
where r and s are some positive constants. We introduce the functionals FrAf)=
f
JR3
/(k)exp[r(2e)«]
3
(6)
and study the values of s and r for which these functionals are positive and finite. We can give the following definition: Definition. We say that the function / has an exponential tail of order s > 0, if the following supremum rj=sup{r>0 is positive and finite.
] Fr,,(/)<+oo}
(7)
404
The functionals (6) can be represented by using the symmetric moments of the distribution function, i.e. mp=
f(k)(2e)2pdk,
f
PeR+
.
(8)
In fact by expanding the exponential function in (6) into Taylor series we obtain (formally): «
k
/ oo
n
\
oo
/
n=0
^(/) = /R./( ) E ^ ) * * = E ^ B • JM
\n=0
-
'
o»
In order the expansion (9) have a sense, we shall suppose that the moments of all orders are finite. Then the value r* can be interpreted as the radius of convergence of the series (9), and the order of the tail s is therefore the value for which the series has a positive and finite radius of convergence. For investigating the summability of the series (9) we look for estimates of the sequence of moments {mp}, with p = 4p, n = 0,1,2.. , and study the dependence of the estimate on s > 0. We shall be interested in the situation when the sequence of the coefficients satisfies ma —f-
1
lim r / _ 2 _ = — n—oo y n! r* and from the estimate (10) we have
r:>^>o .
(ID
To achieve the estimate (10), we shall study the moments equations obtained by integrating against (2e)2p the BTE, in the stationary and homogeneous regime : Qp + Gp - 0
(12)
f C[f](2e)2pdk Jm3
(13)
where Qp=
GP = l I n JR
3
E-V k /(2e) 2 ^k
(14)
405
We underline that in the stationary and homogeneous regime, the electric field E is a constant. Now we can formulate the main result 4 : Theorem 1. Let /(k) be a nonnegative s o l u t i o n of t h e BTE (1) in t h e s t a t i o n a r y and homogeneous regime, t h a t has f i n i t e moments of a l l o r d e r s . Then the supremum r* defined i n (11) i s f i n i t e for some s > O.and in t h e energy range hw < e < £o, Ve0 > 0.
To the best of our knowledge this is the first theorem which proves the existence of energy tails for the BTE in the semiconductor framework. 3. Simulation results The numerical existence of high-energy tails for the BTE, has been widely proved in literature by MC simulations with parabolic bands 5 ' 6 , as well as in the full band approximation 7 ' 8 . MC experiments show that, in the parabolic band approximation a and acoustic and optical phonons scattering, the tail is a global maxwellian, i.e. the order of the tail in eq.(5) is s = 2. Theorem 1 proves the existence of such tails in the quasi-parabolic band approximation, but no information is given about the order s and the radius of convergence r* of the series (9), which should be determined numerically. Then we run our Monte Carlo code for bulk silicon, where an homogeneous electric field is frozen in the material. The histogram of the numerical steady solution is then obtained as a function of e, discretizing the whole energy space in a system of concentric shells with increasing radius pn=nh,
n = l,..., AT,
h
= —
and by counting the number of particles which are in the corresponding shells / i = {#particles : e < px} fn = { # particles : pn-\ <e
= { # particles : R < e)
n = 2,..., N
.
In order to obtain the parameters in (5), we assume: /„ = c e x p [ - r ( £ „ - £ „ 0 ) * ] a
,n>n0
T h e parabolic band approximation is obtained from eq.(2) for a = 0 , and eq.(5) reduces to/oo(k)~exp[-r|v|s]
406
and plot the pairs Xn
=
^V^-n
(xn,yn)
E-no)
ln(ln/„0-ln/„)
> Vn
,n = n0 + l,—,N
Thus we expect = lnr+
-xn
(15)
almost linear plot, which slope will show the exponent s. In figure 1 we show the simulation results obtained with an electric field of 80 KV/cm. The solid curve shows the MC data y^ (computed with n 0 =1), while the solid straight line is y = 2.2342 + x
.
(16)
Thus the asymptotic /oo ~ exp(—re)
, |k| —• oo
,with s=2
is clearly indicated also in the quasi-parabolic band approximation. In eq.(16) the value logr= 2.2342 corresponds to a temperature of 1243 °K. Similar results are obtained for other values of the electric field.
-4
-3 -2 x (units log(e))
-1
Figure 1. Logarithmic plot of the EED obtained with MC simulation (solid curve), and with eq.(15) (dashed straight line), E = 80 kV/cm.
407
References 1. A. Markowich, C.A. Ringhofer and C. Schmeiser, Semiconductor equations , Springer-Verlag, Wien, (1990) 2. C. Jacoboni and L. Reggiani,fiei;. Mod. Phys., 55 , 645-705, (1983) 3. A.V. Bobylev, I.M. Gamba and V. Panferov, J. Stat. Phys.,116, 1651-1682 , (2004) 4. O. Muscato, "Moment inequalities and high-energy tails for the Boltzmann transport equation with analytic bands for semiconductors", in preparation 5. A. Abramo, C. Fiegna, J. Appl. Phys., 80 , 889-893, (1996) 6. C.C.C. Leung, P.A. Cbilds,Appl. Phys. Lett, 66(2), 162-164, (1995) 7. J.E. Chung, M.-C. Jeng, J.E. Moon,P.-K. Ko, and C .Hu, IEEE Tran. Elec. Dev., ED-37, 1651, (1990) 8. M.V. Fischetti, S.E. Laux, IEDM Tech. Dig., 305-308 , (1995)
STABILITY OF SOLITONS OF THE ZAKHAROV-RUBENCHIK EQUATION *
FILIPE OLIVEIRA Departamento de Matemdtica, Faculdade de Ciencias e Tecnologia Universidade Nova de Lisboa, Portugal, E-mail: [email protected] Centro de Matemdtica CMAT, Universidade do Minho, Portugal
We prove the global well-posedness of the one-dimensional Zakharov-Rubenchik equation in the space H2(R) X H1(R) x if 1 (E). We also prove the existence and the orbital stability of solitary wave solutions to this model.
1. I n t r o d u c t i o n In order t o describe the dynamics of small amplitude Alfven waves propagating in a plasma, C h a m p e a u x & al. 2 consider the Hall-MHD equations, written in non-dimensional units :
(
' dtpM + V.(/9 M u) = 0 pM(dtu + u . V u ) = - f V ( p ^ ) + (V x b) x b 3 ( b = V x ( u x b ) - -jf-V x ( ^ - ( V x b) x b ) V . b = 0,
where dissipative processes have been neglected. Here, b denotes the magnetic field, pM the density of mass, u the fluid speed, Ri the non-dimensional ion-cyclotron frequency, 7 the polytropic gas constant and /? the squared ratio of the sonic to the Alfven speed. Assuming the existence of a strong ambient magnetic field along the x-axis
"This work is supported by Minho University Mathematics Centre and Portuguese Foundation for Science and Technology (CMAT-FCT) hrough the research programme POCTI.
408
409
and variations of the fields in this direction only, the authors get 2 dtPM + dx(pMUx) = 0 pM(dtux +uxdxux) = -dx(^pM + ±\b\2) PM(dtv + uxdxv) = dxb dtb + dx(uxb-v) = --^dx(^b), where u = (ux,uy,uz), notation
(1)
b = (bx, by, bz) and where we have used the complex b — by + ibz and v = uy + iuz
to describe the transverse directions of the fields. The longitudinal component bx of b remains constant and has been absorbed in the ambient magnetic field, taken as unity. The system (1) possesses the exact solutions b = - £ v = B 0 e i ( f c x ~ w t ) , ux = 0 , p M = 1, k where the frequency UJ and the wave number k are related by the dispersion equation k2
, L
~k?~
This solutions correspond to a monochromatic circulary polarized Alfven wave propagating along the x-axis. Also, by a classical multi-scale analysis 2 , the authors look for small amplitude solutions to (1) which are slow modulations (in time and space) of the exact solutions described above : For a small parameter 6 > 0, we define the slow variables X = 0x, T = 6t, and expand ( b = eei{-kx-^
(Bx + 6B2 + • • •)
i cx ut 2 + ...) v = e V - '>(6vi+8 V2 2 3 p M = i + e Pl + e p2 + ...
(2)
\,ux = e2m + e3u2 + ....
By inserting these quantities in (1), equating the coefficients of {6n}neN to 0, and putting B = B\ + 6B2, p = Pi and u = u\, one gets the system ( i(dT + vgdx)B + etudxxB - 6k(u - v-fp)B = 0 < drp + dxu = 0 {dTU + dx(Pp+±\B\2)=0,
(3)
410
where the group velocity vg is given by vg = fc,fcl" ^a-,. Then, changing the system of coordinates to a frame moving at speed vg, (3) is reduced to iBT + LOB^ - k(u - ^fp)B = 0 9pT + di(u-vgp)=0 9uT + di{(3p-vgu+\\B\2)=Q,
(4)
Note that by neglecting the terms in 9, one gets the Nonlinear Schrodinger Equation with cubic nonlinearity :
This model clearly breaks down at the resonance (3 — v2 = 0 . Far from the resonance, by some simple computations, one obtains the ZakharovRubenchik Equation iBt + OJBXX - k(u - \p + q\B\2)B = 0 Opt + (u - vp)x =-k\B\l .0ut + (J3p-vu)x = %v\B\2x,
(5)
where q = k + l^Tyl] • For this system, we obtained the following existence theorem for the Initial Value Problem associated to the system (5): Theorem 1.1. Let B0 G H2(R) et p0,u0£ iJ^R). Then there exists a unique strong solution (B, p, u) for the system (5), and {B,p,u)
£ nj=oAC3(R+;H2-2j'(R))
x
nj=0,iCj(R+;ff1-J'(R))x
xr\1i=0C(R+;H1Proof: First, a fixed-point technique is used, in order to obtain local (in time) solutions. Then, these solutions are globalized by using the following (formal) invariants for the flow of (5): (see the extended version4 for details)
h(t) - / \B\2, Jm.
«o-iXi«.i'^^+^,.->,.+|/;w.+
411
2 J R up
4 Ju and
I3{t) =9 I up+l-
[(BK-BXB).
2. Stability of the solitary waves In this section we prove the existence and the orbital stability of the solitary wave solutions to (5). 2.1. Existence
of solitary
waves
Let c > 0. We look for solutions to the system (5), of the form Q = (
(6)
(a, 6) 6 R2. By (5,b) and (5,c), we get
' = H0 = ^ i L a = a(c) =
a n d
(7)
k(-p+%(c0 + v)) (3-{c6 + vf
Moreover, by (5,a) wA" -icA'
-\A
+ q)\A\2A.
(8)
+ q)R2R = 0,
(9)
= k{a--b
Finally, setting
R(x) =
e^Aix),
we obtain R" + -(^--X)R--{a-h
which is the well-known stationary equation associated to the cubic NLS equation. A simple computation shows that if 2
| - - A > 0 or a(c) - \{c) 4LLU
2
+ q > 0,
412
then (9) has no solutions. Otherwise, if c2
, „ , , , A < 0 and a(c)
v
b(c) + q < 0,
then there exists a unique positive solution to (9), which is even and exponentially decreasing. We can therefore state the following result : Lemma 1. Let c > 0. Assume that E := ^(A — | ^ ) > 0 and that a(c) — |6(c)+g<0. Then Qii(x,t) = (cj)i(x,t),(f>2(x,t),
a(c)-^(c)+^-i^y<0. Remark 2. The expression of R is known : ^(X' 2.2. Orbital
E)
V k(a(c) - 26(c) + q) cos\i{jEx)
'
(10)
stability
We now state the orbital stability of the solitary waves described earlier. For a proof, see the extended version4, where essencially, we have used the method developed in previous works by Bona 1 , Laurengot 3 and Weinstein5 among others, which consists in using the invariants h to build a suitable Lyapunov functional. We introduce the natural orbit 0(B, u, p) = {eiaB(. + x0),u(. + x0), p{. + x0)/a € R, x0 e R}.
(11)
Note that if (B, u, p) is a solution to the system (5), then all the elements in its orbit 0(B, u, p) remain solutions. We can now state our main theorem:
413 T h e o r e m 2 . 1 . Let w > 0, v > 0, 0 < 9 < 1 and j3 - v2 > 0. There A > 0 such that for all (A, c) € R + x]0; j { satisfying c2
1 ^fte solitary
exists
wave
QR(X, t) = (<j)i(x, t), 4>2(x, t), c(>3(x, t)) = = {eiXte^R{x
- ct), a{c)R2{x
- ct), b(c)R2(x
-
ct))
is orbitally stable, i.e. : There exists t\ > 0 such that for all ei > e > 0 and for all (B0, u0, p0) £ H2(R) x H^R) x f f a ( R ) , there exists 6{e) > 0 such that if ||J30-e^iJ||Hi{K)
<5(e),
\\u0-a{c)R2\\L2{m
<5{e),
\\po-b(c)R2\\L2m<5{e), then for all t G R + ,
where (B,u,p) (B0,u0,p0).
/ n / a , , J e i a £ ( . + a:o,t)-&(.,t)||tfi(R) < e
(12)
^ / a : „ | | w ( - + a;o,i)-
(13)
/ n / x „ l b ( . + a:o,t)-^3(.,«)llL»(H) < C
(I4)
is the solution
of (5) corresponding
to the initial
data
References 1. J.Bona, On the stability theory of solitary waves, Proc. R. Soc. Lond., Vol. A344, 363-374 (1975). 2. S.Champeaux, D.Laveder, T.Passot and P.L.Sulem : Remarks on the parallel propagation of small-amplitude dispersive Alfven waves, Nonlinear Processes in Geophysics, Vol. 6, 169-178 (1999). 3. P.Lauren$ot, On a nonlinear Schrodinger equation arising in the theory of water waves, Nonlinear Analysis T.M.A, Vol. 4, 509-527 (1995). 4. F.Oliveira, Stability of the solitons for the one-dimensional ZakharovRubenchik Equation, Physica D, Vol. 175, 220-240(2003). 5. M.Weinstein, Modulational stability of ground states of nonlinear Schrodinger equations, SIAM J. Math. Anal., Vol. 16, 472-491 (1986).
GLOBAL SOLUTIONS OF B O L T Z M A N N - T Y P E EQUATIONS W I T H T H R E E - B O D Y CHEMICAL INTERACTIONS *
FILIPE OLIVEIRA Departamento
de Matemdtica, Faculdade de Ciencias e Universidade Nova de Lisboa, Portugal, E-mail: [email protected]
Tecnologia
A. J. S O A R E S Departamento
de Matemdtica, Universidade do Minho, E-mail: [email protected]
Portugal
In this paper we consider the discrete Boltzmann equation modelling a diatomic gas whose particles undergo elastic multiple collisions and chemical reactions of autocatalytic type. We state that the one-dimensional initial-boundary value problem in the half-space possesses a global solution for small initial data in B^_ n L 1 .
1. Introduction Kinetic models of the Boltzmann equation extended to chemically reacting gases have been investigated by several researchers 1 ' 2 ' 3,4 , due to their application to a wide range of engineering problems, as detonation 5 ' 6 and laser induction 7 . In the case of discrete models of the Boltzmann equation (DBE), the gas particles can attain only a finite number of prescribed velocities and a system of hyperbolic semi-linear partial differential equations describes the time-space evolution of the gas 8 . The simple mathematical structure of the DBE is liable to implement different numerical simulations with the aim of studying several complex problems of fluid dynamics. In particular, the steady detonation problem has been studied and the sound propagation investigated 6 by means of a particular model for a diatomic gas undergoing ' T h i s work is supported by Minho University Mathematics Centre and Portuguese Foundation for Science and Technology (CMAT-FCT) hrough the research programme POCTI.
414
415 triple elastic collisions and autocatalytic chemical reactions 4 . Some numerical studies have also been performed9 for the initial-boundary value problem associated to the above said model, without any support on existence and uniqueness results. In fact, only in a very recent paper 10 , the global existence of solutions is proven for the half-space problem, considering suitable smallness assumptions on the initial data. This will be the argument of the present paper, where we will state the main ideas concerning the referred existence result 10 . Some previous pertinent papers 11 ' 12,13 , in the context of the global wellposedness for the DBE, refer to quite different problems. In the first one 11 , a strong global result is obtained for a model with only binary elastic collisions. In the second one 12 , the considered model refers to an inert gas too, but triple collisions are also admitted; the existence result is global in time but it refers to the pure initial value problem. Finally, in the context of the reacting gases, the last paper 13 refers to the initial-boundary value problem and provides an existence result on bounded time intervals whose size depends on the initial data. 2. Equations of the model Let us consider a diatomic gas constituted by atoms A, accelerated atoms A* and molecules A2, whose particles undergo binary and triple elastic collisions, as well as autocatalytic chemical reaction of type A2 + M^A
+ A* + M,
M = A,A*,A2,
(1)
where M is the catalyst of the reaction. We suppose that the particles of each M-species move in the space with four selected velocities defined in such a way that 4 , in one-space dimension, there exist only three independent velocities for M = A, A2 and two for M = A*. We introduce the vector N = (Nf,N?,N£,Nf, Nf, Nf2, N22, N^3) of independent number densities N^r(x,t), associated to the prescribed velocities. Kinetic equations. The time-space evolution of the number densities is given by the kinetic equations of the model, which can be written in the form 4 ' 10 ^-Nfix^+Vi-^Nfix^^FfiN&t)), lNlA2(x,t) ^Nf
+
V
f-^Nf2(x,t)
ie{l;2;3}, = FlA*(N(x,t)),
(x, t) + Vi | - J V f (x, t) = Ff (N(x, t)),
t€{l;2;3}, i € {1; 3},
(2)
416 where (x,t)€ [0;+oo[x[0; +oo[ and (v\,V2,vs) = (c,0, —c) with c a positive constant denned in terms of the atom mass m and molecular bond energy e of the diatomic molecule A?, through the relation c = i/(4e)/(5m). For sake of simplicity, we will assume m = 1. Moreover, the collision terms i^ M on the r.h.s. of Eqs.(2) describe the elastic scattering and chemical mechanism among particles and are defined as polynomials of the form F?(N)
= (P™{N)
+ F*%(N)) - JVf (Q\%(N)
+ Q{;]M(N))
,
where P±M, Q\ M are homogeneous polynomials with positive coefficients. In particular, Pi^ and P\ ^ are of degree 2 and 3, respectively, and refer to creation of particles due to inert or reactive collisions of binary and triple type, respectively. Conversely, polynomials Q\ M and Q\ M are of degree 2 and 1, respectively, and are such that the terms N^Q^M and N^Q^ describe the disappearance of particles due to inert or reactive collisions of triple and binary type, respectively. Let us observe that inert collisions involve the same number of incoming and outgoing particles, i.e. two or three particles in correspondence of binary or triple collisions, respectively. On the other hand, collisions with chemical reaction (1) lead to the formation of two atoms A, A* and one M-particle when a molecule A2 dissociates in the presence of the catalist M. Therefore, a triple reactive source term corresponds to a binary reactive sink term, and vice-versa. The inclusion of multiple collisions with chemical reactions represents an additional difficulty, since cubic terms appear in the collision operator and the conservation laws differ from the ones of classical inert gases. Conservation equations The evolution equations (2) possess the following conservation laws4
D ^ 1 + ^ 1 ')J+^(E«'(^ + ^ ) J =° XX2 + E NA + | (tr^A2+E t=l
d_ dt
i=l,3
)
,1=1
\i=\
(3«)
«*A =° m i=l,3
J
i=l,3
^ ( E ^ ^ + ^2)+E^"*]=0' ' :1
«=1,3
417 which refer to the conservation of partial A - A2 and A* — A2 number densities, and total momentum of the system in the x-direction, respectively.
3. Initial-boundary value problem In order to study an initial-boundary value problem for Eqs.(2) in the one dimensional region 0 < x < oo, we prescribe the following initial data (4) where Nio are prescribed functions of x > 0, and boundary conditions
Nf(0,t) Nf(0,t)
fii fit fii, fit fii: fit .fit fii2- fit
2
X (0,*),
W>0,
(5)
at the plane wall x = 0, with (5M = 1 if M = A, A* and 8A2 = 1/2)
£>&
fi&+P£+\fiti<6M.
(6)
Conditions (5) mean that the contribution to the incoming flux of each species A,A*,A2 at x = 0 is given by a fraction of outgoing particles of the same species that have been reflected back from the boundary, and a fraction of molecules of other species that have been reacted giving rise to particles of M-species. Conditions (6) assure that the macroscopic flow of the gas is not inward to the boundary 10 . Compatibility conditions Finally, we assume the following compatibility conditions up to order one for the initial and boundary data
fii fii- fii2 fii' fii^ fit .fit fit- fit / d
(7)
\
fii fit fit fit fit fit Jt fit fit
(8)
418 4. Existence and uniqueness results We begin this section with some useful notations. For X C M+ x R+ and N(x, t) = i.Ni(x, < ) , . . . , A^ 2 (x, t) j , we introduce the space B\X)
= { / G C\X) / / , f , f
e L°°(X)}
and denote by B+(X) its positive cone. Also, we define the upper bound £ ( T ) = max lM
'
\NtM(x,t)\,
sup
Eo = E(0),
(x,t)€K+x[0;T]
for T > 0, N G L°°( [0; +oo[x [0; T]), and the initial mass of the system as /•+0O /
mo = / Jo
3
\
a
£ « + 2< ) + E < | (*) d^x)' ^ e L '( R + ) • \i=l
<=1,3
/
Let Rz = Ff3,
Ri = Ff + Ff, R3 = F2A + i ^ 2 ,
fl4
=^
2
,
ife = F3A + Ff,
and (W1,W2,W3,W4,W5) = [c,-,0,--,-CJ
.
For models for which Rj can be put in the form Rj(N) 1
= NiiR^-Rf\
(9)
k
where v^ ^ Wj and R^ \ k = 1,2 are polynomials with positive coefficients, we prove the following: Theorem 1. £e< No G S+([0;+oo[) nL 1 ([0;+oo[) satisfying the compatibility conditions (7), (8). Then there exists e, e' > 0 such that, if mo < e and E0 = max sup \Nt (x)\ < e', then the mixed problem (2),(4),(5) has a unique solution N G B+( [0; +oo[x[0; +oo[).
•
For sake of brevity, the detailed proof of theorem 1 is omitted here. First, a local existence result is proven via a fixed point technique. Then the following two key lemmas 10 , give an a priori estimate for the local solution, and Theorem 1 can be obtained using a standard continuation argument. Lemma 1. For all t G [0;T0], the upper bound E(T) satisfies the condition E{t) < CEo + C'm0{E(t) + E(t)2), where the constants C, C are independent of TQ and NQ . •
419 L e m m a 2 . There exists e > 0, e ' > 0 such that, if m 0 < e and E0 < e', then there exists a constant K = K(No) > 0 depending exclusively on the initial data N0 such that E(t) < K(N0), for all t G [0;T 0 ]. • References 1. V. Giovangigli, Multicomponent Flow Modeling, Boston, USA: Birkhauser, 1999. 2. A. Rossani and G. Spiga, "A note on the kinetic theory of chemically reacting gases", Physica A, Vol. 272, pp. 563-573, 1999. 3. R. Monaco, M. Pandolfi Bianchi and A. Rossani, "Chapman-Enskog expansion of a discrete velocity model with bi-molecular reactions", Math. Models Meth. Appl. Sci., Vol. 4, pp. 355-370, 1994. 4. M. Pandolfi Bianchi, A. J. Soares, "The discrete Boltzmann equation for gases with autocatalytic reversible reactions", Commun. Appl. Nonlinear. Anal, Vol. 1, pp. 25-48, 1994. 5. A. Em, V. Giovangigli, "Kinetic theory of reactive gas mixtures with application to combustion", Transp. Theor. Stat. Phys. Vol. 32, pp. 657-677, 2003. 6. M. Pandolfi Bianchi, A. J. Soares, "A kinetic model for a reacting gas flow: steady detonation and speeds of sound", Phys. Fluids, Vol. 8, pp. 3423-3432, 1996. 7. F. Hanser, W. Roller and F. Schiirrer, "Treatment of laser induced thermal acoustics in the framework of discrete kinetic theory", Phys. Rev. E, Vol. 61, pp. 2065-2073, 2000. 8. R. Gatignol, Theorie cinetique des gaz a repartition discrete des vitesses, Lect. Notes in Phys., 36, Berlin (Springer-Verlag), 1975. 9. R. Monaco, M. Pandolfi Bianchi, A. J. Soares, "Numerical Simulations of a Boltzmann Model for Reacting Gases", Appl. Math, and Comput, Vol. 85, pp. 61-85, 1997. 10. Filipe Oliveira, A. J. Soares, "On the global well-posedness of discrete Boltzmann systems with chemical reaction", Math. Meth. Appl. Sci., Vol. 28, pp. 1491-1506, 2005. 11. S. Kawashima, "Global solutions to the initial-boundary value problems for the discrete Boltzmann equation", Nonlinear analysis TMA, Vol. 17, pp. 577597, 1991. 12. H. Cabannes, "Global solution of the discrete Boltzmann equation", Eur. J. Mech. B-Fluids, Vol. 11, pp. 415-437, 1992. 13. A. J. Soares, "Initial-boundary value problem for the Broadwell model of a gas mixture with bimolecular reaction", Math. Meth. Appl. Sci., Vol. 21, pp. 501-517, 1998.
ON A N I N V E R S E PROBLEM IN G R O U P ANALYSIS OF PDE'S: L I E - R E M A R K A B L E EQUATIONS
FRANCESCO OLIVERI Department of Mathematics, University of Messina Salita Sperone 31, 98166 Messina, Italy E-mail: [email protected] GIOVANNI MANNO, R A F F A E L E V I T O L O Department
of Mathematics "E. De Giorgi", University of Lecce via per Arnesano, 73100 Lecce, Italy E-mail: [email protected]; [email protected]
Within the framework of inverse Lie problems we give some non-trivial examples of Lie-remarkable equations, i.e., classes of partial differential equations that are in one-to-one correspondence with their Lie point symmetries. In particular, we prove that the second order Monge-Ampere equation in two independent variables is Lie-remarkable. The same property is shared by some classes of second order Monge-Ampere equations involving more than two independent variables, as well as by some classes of higher order Monge-Ampere equations in two independent variables. In closing, also the minimal surface equation in R 3 is considered.
1. Introduction Lie group analysis (see Refs. i.2.3.4-5.6.7) j s a formidable tool for investigating differential equations in a general framework without using ad hoc methods: it may be used for determining the admitted symmetries useful for finding invariant solutions to differential equations, reducing the order of ordinary differential equations, transforming differential equations in more convenient forms 4 ' 8 , etc. Roughly speaking, in dealing with Lie group analysis of partial differential equations (PDE's), either a direct problem or an inverse one may be considered. In the direct problem, starting with a system of PDE's A (x,u,uW)=0,
420
(1)
421 where A is an assigned function of the independent variables x e l " , the dependent variables u e Rm, and u^fc' (the set of all partial derivatives of the u's with respect to the x's up to the order k), one is interested to find the admitted group of Lie symmetries. Let us consider a one-parameter (e) Lie group of point transformations x*=X(x,u;e),
u* = U(x,u; e),
(2)
(the transformation (2) for e = 0 reduces to identity). The system (1) is invariant with respect to (2) if A ( x , u , u ( f e ) ) = A(x*,u*,u* ( f c ) ) = 0 .
(3)
By expanding transformation (2) around e = 0, one gets the infinitesimal transformation
x* = x + -r x ( x , u ; 0
+ 0 ( e 2 ) = x + e£(x,u) + 0(e 2 ),
u* = u + — U(x,u;e) ae
+ 0(e 2 ) = u + eT7(x,u) + 0(e 2 ), e=0
to which it corresponds the infinitesimal operator (vector field)
U
dXi
ti
dUA
The straightforward Lie algorithm 3 ' 4 allows us to derive the general form of infinitesimal generators & and TJA of the Lie group admitted by (1) by solving an overdetermined system of linear PDE's arising from d ^A(x*,u*,u* ( f c ) ) ae
= 2 ( f e ) A(x,u,u( f c ) ) e=0,A(x,u,uCO)=0
= 0, A(x,u,u( fc >)=0
where E^ is the fc-th prolongation of H up to the order fe; E^ is obtained by using some recurrence relations accounting for the transformations of derivatives. The integration of the determining equations gives the infinitesimal operators Ej admitted by the system (1); these vector fields span a Lie algebra that can be finite or infinite dimensional. The above constructions admit a standard geometrical interpretation. Eq. (1) can be seen as a submanifold in the space with coordinates (x, u, u'fc)) (the jet space 3 ); then the operator S(fc) is just a vector field on the jet space, and is a symmetry if it is tangent to the submanifold (1). On the contrary, in the inverse problem one chooses a Lie group of symmetries and determines the most general system (having an assigned structure) admitting it 9 .
422
By imposing the invariance of an unspecified system of partial differential equations with respect to an assigned Lie group of symmetries one obtains a system of linear partial differential equations whose solution leads to consider the differential invariants of the Lie group 10 ' 5 . Definition 1.1. (see Ref.
n
) The function
I(x,u,u{k)) is called a differential invariant of order k for the Lie algebra C spanned by the vector fields H* (i = 1 , . . . ,p) if H « J(x, u, uW) = Ai(x, u, U W ) / ( x , u, u « ) ,
(t = 1 , . . . ,p),
fe
where Aj(x, u, u^ ') are some functions of the indicated arguments. If all the functions \ t are vanishing, / is said an absolute differential invariant, whereas if some function A, ^ 0, / is said a relative differential invariant.
• When the inverse Lie problem is considered, a maximal set of functionally independent differential invariants for the Lie algebra £ is required 5 . The most general equation left invariant with respect to C will be given as an arbitrary function of the differential invariants. Within this context an interesting question may arise whether there exist non-trivial equations which are in one-to-one correspondence with their invariance groups (see Refs. 1 2 ' 1 3 ). To this purpose, let us give the following definition14. Definition 1.2. Suppose we have a (system of) equation(s) A(x,u,uW)=0,
(5)
with A assigned function of its arguments, and determine the infinitesimal operators of its Lie symmetries, say Hi,
H2,
...,
Hp.
(6)
In general, if we consider a general system of PDE's A(x,u,uW)=0,
(7)
with A unspecified function of its arguments, and, by requiring that (7) has the Lie point symmetries (6), we find that A = A, then we call (5) a Lie-remarkable (system of) equation(s).
(8) •
423
In Ref. 15 this definition has been formulated from a geometric point of view, and the further subdivision of Lie-remarkable equations into strong and weak Lie-remarkable equations has been considered. 2. Second order Monge-Ampere equations The second order Monge-Ampere equation in two independent variables, introduced by Ampere 16 in 1815, has the form Hutt + 2Kutx + Luxx +M + N(uttuxx
- u2tx) = 0,
(9)
where the coefficients H, K, L, M, N (TV ^ 0) depend on t, x, u, ut, ux. In 1968, Boillat 17 discovered that (9) is the only second order equation possessing the property of complete exceptionality in the Lax 18 sense. The property of complete exceptionality has been used to derive MongeAmpere equations involving more than two independent variables (see Refs. 19-20>21). Given an unknown field u(xo,xi, • • • ,xn) (XQ denoting the time), and its associated Hessian matrix, the most general second order PDE being completely exceptional (and called Monge-Ampere equation) is provided by a linear combination of all minors extracted from the Hessian matrix, with coefficients depending at most o n i a , u and first order derivatives of u. Here we want to stress that Monge-Ampere equations, in addition to the complete exceptionality, possess another remarkable property, that of being uniquely characterized by their Lie point symmetries. We first analyze the classical Monge-Ampere equation in two independent variables, then we will consider generalizations to three independent variables and to higher order equations. Theorem 2.1. The classical Monge-Ampere equation written in the form «1 (uttUxx - U2X) + K2Utt + K3Utx + K4UXX + K5 = 0,
(10)
where the coefficients «:» (K\ ^ 0) are constant, is Lie-remarkable. Proof. In fact, Eq. (10) is invariant with respect to a Lie group of point transformations spanning a 9-dimensional Lie Algebra £ generated by the following vector fields: d
d
d
424
„
d
d
,
o
QN
- 6 = K I * ^ i - « I ^ T - + («2Z - M
S 7 = 2mt— „
+ (K 3 r - 2K2Xt) — , 9
C 8 = 2KiX—
s9 =
i jn:
o
2
+ (-K2X
9 2K x
d
)^-,
+ K4t
2 K X
+( 3
9 +2KIU)
—
,
^
-
2K4xt)—.
If we want to look for the second order partial differential equation &(t,X,U,Ut,Ux,UtuUtx,Uxx) =0, (11) where A is an unspecified function of the indicated arguments, admitting the Lie algebra £, we need to find the associated second order differential invariants. Prom a geometric point of view, a scalar second order partial differential equation in two independent variables characterizes a submanifold of dimension 7 in the 8-dimensional jet space whose coordinates are (t, x, u, ut,ux, utt,utx, uxx). The second order prolongations of the admitted vector fields give rise to a distribution of rank 7, and this implies that a single second order differential invariant exists, say I = K\{uttUxx
- U$x) + K2Utt + K3Utx + K4UXX.
(12)
Therefore, the most general second order partial differential equation left invariant by the Lie algebra of point transformations of Eq. (10) has the form A(/) = 0,
(13)
K = constant;
(14)
whereupon it follows
thus, we have an equation belonging to the same class as (10); by choosing K — —K$, we recover exactly the Monge-Ampere equation (10). D Let us now consider the second order Monge-Ampere equation in three independent variables 19 - U2xy) + Utx(UtyUXy
Kl[Utt(UXXUyy + K2{UXXUyy + K5(UttUyy +K8Utt
- Uxy) + K3(utyUXy 2
- U ty) + Ke(utxUty
+ K9Utx
+ KWUty
~ UtXUyy + Uty(UtxUXy ~ UtxUyy)
- UuUxy)
+ KllUxx
+ Ki{utXUXy + K7(uttUxx
+ K%2Uxy + Ki3Uyy
~ ~
UtyUXX)} UtyUXX)
2
- U X) + K44 = 0,
(15)
425 where K» (i = 1 , . . . , 14) are taken constant. The explicit determination of the infinitesimal generators of the admitted Lie group results quite complicated and the use of Computer Algebra packages7 reveals extremely memory consuming since the expression of the infinitesimals involves thousands of terms. Without loss of generality, it is possible to introduce the substitution u —> u + a\t2 + aitx + otzty + a^x2 + a$xy + a^y2, where on are suitable constants, and reduce Eq. (15) to an equivalent form where the linear terms in the derivatives disappear, that is, we may consider Eq. (15) with Kg = KQ = KIO = Mi = K i2 = «i3 = 0. Since the introduced transformation is invertible, both equations admit the same point symmetries. In general, the Lie algebra of point symmetries of Eq. (15) (with K 8 = K 9 = KIO = Kn = K12 = K13 = 0) is 11-dimensional; since Eq. (15) represents a 12-dimensional submanifold in the 13-dimensional jet space with coordinates (t,x, y, u,ut,ux,uy,utt, utx, uty,uxx,uxy, uyy), a 12-dimensional distribution generated by point symmetries is needed in order to determine a single second order differential invariant. Consequently, Eq. (15) can not be in general Lie-remarkable. Nevertheless, the following theorem can be proved. Theorem 2.2. Equation (15) (with K% = K 9 = KW = KU = Ki2 — K13 = 0), when the coefficients are such that Ki = 1,
K 2 « 6 ~ « 3 « 4 « 6 + K4K5 — ( 4 K 2 « 5 — K | ) « 7 = 0,
(16)
is Lie-remarkable. Proof. When the conditions (16) are satisfied, the Lie algebra admitted by Eq. (15) is 13-dimensional and is spanned by the following vector fields: Ei E5
d ~ dt d du' d
= (x +
d
_ d
„
H4 =
dx d
I_J<
^8
Eg
E2 =
d
6=X
d + dx
d 5M'
a 7=V
du-' d
du-
1
d n y—+2u--.,\ dy du
K3K4 - 2 K 2 K 6
\
d
y 1— - 4/t2«5 / dt (2n2t + K3X + K4y)(n2x + K3n4y Kg -
2(Ǥ-
- 2K2(2 ',K$X + K6y)) d du 4K2K5)
426
+ (2K2X
Hio = ( 2 K 4 K 5 - K3K6)y—
+ K3K4y
- 2K2(4K5X
+ ((K§ - 4K2K5)y — + («3 - 4 K 2 K 5 ) ( 2 U + K2t2 dy —2(K3K4KSX + K2{2KiKS,t / 2 2\ 2\ O ~(K4K5 - K2K6)y ) — ,
— K$(K3t +
+ n6y))
—
K5X2)
2n5x)))y
H l l = - 2 ( K 3 K 4 - 2K2«6)(-2ft4«5 + K3K6)y — + 2 ( « 3 - 4K2K5)(K3K4 +2(K2
- 2K2KQ)X
- 4K2K5)(K2X
+ (~(K3
+ 2n3K4y
—
- 4K2(«5^ +
~~ 4 K 2 « S ) ( - 2 ( / C 3 « 4 - 2 K 2 K 6 ) ( 2 U +
+2K2(—2K4KS
+ n3K6)tx
~2(K2(-K3K4
+ 2K2K6)(-2K4K5 2
+ ( 4 K 2 K 4 K | + K3K4{K
+K2(-KI
+ 8K2K5)K6
+ 8K2K5)K,l)x)y
- K6(K3K4
H i 2 = - 2 ( - 2 K 4 K 5 + K3K6)2y—
K3K6)t
- 2K2K6)2y2)
+ K2n3t2
—(K3 — 4K2K5)(—2KQ(2K3U
—2(K3K4
+
K4y))—-
2n2K5tx)
— 2K 5 (K 2 ft 6 £ + K4K5X)
K5K6x))y
+K6(-K3K4 + 2 K 2 K 6 ) ( - 2 K 4 « 5 + H13 = 2 ( - 2 K 4 « 5 +
+
n3K6)x~-
2K3Kstx))
— 2K2Ke)(K3lKet
+K3(~K4K5t
— ,
+ 2(«3 - 4 K 2 K 5 ) ( - 2 K 4 K 5 +
+ 2 ( K § - 4 K 2 K 5 ) ( « 3 * + 2 « 3 « 6 y - 4fv 5 (K 2 i +
+Ki(&KsU + K3t2 +
K2t2)
4K2K5)K,6X2)
+ (KI —
-
K6y))—-
K3K&)y2
du
2
K3K6) y—
2 ( « | — 4K2K5)(K3K4t
— 2K2K(,t + 4K4KSX
- 2 ( K § - 4K2K5)(~2K4K5y
—
2K3K,6X)~
+ K2K6y) —
dy ~(K3
— 4 K 2 K 5 ) ( 2 K 6 ( 2 K 3 W — 2n2K^tx
+K4(«Jf2 + 2K 5 (-4W - 2 ( ( K 4 « 5 ( - K 3 + 8K2K5) +K5(K3K4
-K4(~2K4K5
- 2K2KQ)(2K4,K5
+
— K3K5X2)
2K 2 £ 2
2
+ K3tx + 2K$X ))) + K 3 K 4 (K3 - 8 K 2 / « 5 ) K 6 + 4K22KcsK%)t -
it3K6)2y2—.
du
K3K6)x)y
427
The second order prolongations of these vector fields give rise to a distribution of rank 12, and we obtain the following second order differential invariant: I = {UttUXXUyy T K2\UxxUyy
- UttUly u
xy)
- ulxUyy
+ 2UtxUtyUXy
K3\Utxuyy
-
U^yUXX)
U±yUxy ) + K4(ut x^xy
+ K5(uttUyy
- Uty)
- K6(uttUxy
- UtxUty) + K7(uttUxx
'U'ty'U'xx) - V%x). (17)
Therefore, the most general second order partial differential equation in three independent variables left invariant with respect to the Lie symmetries of Eq. (15), along with the conditions (16), has the form (18)
A(/)=0,
where A is an arbitrary function of its argument. Similar reasonings as before allow us to say that Eq. (15), when the conditions (16) hold, is Lie—remarkable. • More generally, it is possible to prove that second order Monge-Ampere equations involving more than three independent variables, when some conditions involving their coefficients are satisfied, are instances of Lieremarkable equations. 3. Higher order Monge-Ampere equations The property of complete exceptionality has been used by Boillat 22 to determine higher order Monge-Ampere equations for the unknown u(t,x). By considering an equation of order N > 2, it is necessary to distinguish the case where N is even from the case where N is odd. If N = 2M, the most general nonlinear completely exceptional equation is given by a linear combination of all minors, including the determinant, of the following Hankel matrix: -X0 Xx
X!
x2
x2 X3
. ..
XM-I
. • • XM
XM XM+I
(19)
H = XM.XM
dNi
i XM
XM+I
• • • X2M-1
XM+I
XM+2
• • • X2M-1
X2M-1 X2M
n the r.a se where where Xj — „ .n .,\ .. .T~*—~~— , . . ^ i ^ .N, =- 1M _*„ - 1. ^, we 1. — -.„-—. the linear combination of all minors extracted from (19) where the last row has been removed. In both cases the coefficients of the linear combination
428
are functions of £, x, u and its derivatives up to the order N — 1. In the following we shall limit ourselves to the case where these coefficients are constant. By considering the third order Monge-Ampere equation TZliuttxUxxx
- Utxx)
+^4Uttt
+ 7i2(UtttUxxx
- UttxUtxx)
+ K3(utttUtxx
~
u
ttx)
+ K5Uttx + TieUtxx + 'KlUxxx + «8 = 0,
(20)
the substitution u —> u + a\t3 + a2t2x + a3tx2 + a^x3 provides the equation Kl(UttxUxxx-Utxx)
+ K2(utttUxxx-UttxUtxx)+K3{UtttUtxx-Uttx)
= K. (21)
The Lie algebra of point symmetries of Eq. (21) is 10-dimensional; since this equation is a 11-dimensional submanifold in the 12-dimensional jet space with coordinates (t,x,u,ut,ux,uu,Utx,uxx,uttt,Uttx,utxx,uxxx), a 11-dimensional distribution generated by point symmetries is needed in order to determine a single second order differential invariant. Therefore, Eq. (21) is not in general Lie-remarkable. Nevertheless, the following theorem can be proved. Theorem 3.1. The equation (UttxUxxx
~ U2Xx) + KUttt"U.xxx
+ \2(umUtxx
- UttxUtxx)
~ U2fx) = fl,
(22)
where _ (X —
. _ K3 A — , K2
KK3 —j-, «2
K2
obtained from (21) by choosing K\ = —, is Lie-remarkable. Proof. The Lie algebra of point symmetries admitted by (22) is infinitedimensional and is spanned by the vector fields
"
1_
d dt'
E4 = * d ot - 9 — 1 1?-, OU
~12 =
_, d 2 ~ ~~dx~'
z
"
(2t-3Xx)^-x£, d ox
n
Z-10=tx
F(t-Xx)l,
d ou — , OU
d 5u'
E5 = X2xft
+
(2t-
a
d
- 7 = *-£-, OU - 1 1 =X
-*>s
-8
2d
— ,
ou
^du' 7.
(23)
429
where F is an arbitrary function of (t — \x). The third order prolongations of these operators give rise to a distribution of rank 11; therefore, we are able to determine the unique third order differential invariant I = {uttxuxxx
- u2txx) + X(utttuxxx
+ \2{umutxx
- uttxutxx)
- u2ttx), (24)
and this completes the proof.
D
We may also consider fourth order Monge-Ampere equations and state, for instance, the following theorem. Theorem 3.2. The fourth order Monge-Ampere equation Utttt\uttxxuxxxx
~ utxxx) ^~ ^utttxuttxxutxxx
~~ uttxx ~ utttxuxxxx
=
K
i (25)
where K is constant, is Lie-remarkable. Proof. In fact, the Lie algebra of point symmetries admitted by Eq. (25) is 16-dimensional and is spanned by the vector fields n
_d_
„
_
d_
„
*2~dx'
^~dt' d
*4=tdi-Xdx-> S7 = t—, ox .2 ° —io = t — , au -13 = ^ ,
d
_d_
*3~ du „
d
d
n
„
*5 = Xfc+2udu->
d
*6 = Xdl'
H8 = t — , E9=x—, ou au j. ~ 2 ° s n — tx—, ci2 — x — , ou ou
(26)
a
-14 =t2X-,
^S
= tX-,
- 1 6 - X - .
Their fourth order prolongations give rise to a distribution of rank 16, whereupon we have uniquely the fourth order differential invariant I = Utttt{uttxxuxxxx
~~ utxxx) + <^utttxuttxxutxxx
~ uttxx ~ utttxuxxxxi
and this enables us to say that Eq. (25) is Lie-remarkable.
(27)
•
By similar arguments, it is possible to prove that the property of being uniquely characterized by their Lie point symmetries is also shared by some Monge-Ampere equations of order higher than the fourth.
430
4. Minimal surface equation in ]R The study of minimal surface equations in M 3 dates back to Lagrange (1762) who posed the problem of determining a graph over an open set £1 C R 2 with the least possible area among all surfaces assuming given values on dQ. Meusnier (1776) gave a geometric interpretation of the minimal graph equation recognizing that their mean curvature vanishes. Also, in the middle of 19-th century Plateau observed that minimal surfaces can be physically realized as soap films. In IR3, at least locally, a minimal surface can be represented in the form z = u(x, y), where the function u satisfies a quasilinear elliptic second order partial differential equation:
i.e., (1 + U2y)UXX - 2UXUyUXy + (1 + Ux)Uyy = 0,
(28)
that, through the substitution y —> it, provides the hyperbolic 2D BornInfeld equation (1 - u2)uxx + 2uxutuxt
- (1 + u2x)utt = 0.
(29)
Minimal surface equation is invariant with respect to a Lie group of point transformations whose vector fields d 1
—'1
d ?
*—'2
d ox d
d ay d
OX
OU
o
OX
r\
d 5
1- l
"3
*-v 5
OIL
OX
d ou
d ox d du
= 5 = y-K~ -
d dy
d ay
(30)
span a 7-dimensional Lie Algebra. Since the second order prolongations of (30) give rise to a distribution of rank 7, we have only one second order differential invariant a , say: ( ( l + Uy)UXX - 2UXUyUXy + ( l + l+Ux+ul)(UxxUyy-
yjX^xx^yy
R. Tracina, private communication.
UpUyy)
ut "-xy
(31)
431 T h e most general equation which is invariant with respect t o the given Lie group must be a function of this invariant. If we want t o have an equation in normal form we have t o set 1 = 0, so recovering exactly the minimal surface equation. Acknowledgments Work supported by P R I N 2003-2005 "Nonlinear Mathematical Problems of Wave Propagation and Stability in Models of Continuous Media." References 1. L.V. Ovsiannikov, Group analysis of differential equations, Academic Press, New York (1982). 2. N.H. Ibragimov, Transformation groups applied to mathematical physics, D. Reidel Publishing Company, Dordrecht (1985). 3. P.J. Olver, Applications of Lie groups to differential equations, Springer, New York (1986). 4. G.W. Bluman, S. Kumei, Symmetries and differential equations, Springer, New York (1989). 5. P.J. Olver, Equivalence. Invariants, and Symmetry, Cambridge University Press (1995). 6. N.H. Ibragimov, Handbook of Lie group analysis of differential equations (3 volumes), CRC Press, Boca Raton (1994, 1995, 1996). 7. G. Baumann, Symmetry Analysis of Differential Equations with Mathematica, Springer, New York (2000). 8. A. Donato, F. Oliveri, Transp. Th. Stat. Phys., 25, 303 (1996). 9. G. W. Bluman, J.D. Cole, Similarity methods for differential equations, Springer, Berlin (1974). 10. A.R. Tresse, Acta Mathematica, 18, 1 (1893). 11. W.I. Pushchych, LA. Yegorchenko, Acta Appl. Math., 28, 69 (1992). 12. V. Rosenhaus, Algebras, Groups and Geometries, 3, 148 (1986). 13. V. Rosenhaus, Algebras, Groups and Geometries, 5, 137 (1988). 14. F. Oliveri, Note di Matematica, 23, 195 (2004). 15. G. Manno, F. Oliveri, R. Vitolo, Preprint (2005). 16. A.-M. Ampere, Journal de I'Ecole Polytechnique, 10, n. 17, 549 (1815). 17. G. Boillat G., Det Kgl. Norske Vid. Selsk. Forth., 4 1 , 78 (1968). 18. P.D. Lax, Contribution to the theory of partial differential equations, Princeton Univ. Press, Princeton (1954). 19. T. Ruggeri, Rend. Accad. Naz. Lincei, 55, 445 (1973). 20. A. Donato, U. Ramgulam, C. Rogers, Meccanica, 27, 257 (1992). 21. G. Boillat, C. R. Acad. Sci. Paris Ser. I. Math., 313, 805 (1991). 22. G. Boillat, C. R. Acad. Sci. Paris Ser. I. Math., 315, 1211 (1992).
O N T H E STABILITY OF T H E M A G N E T I C A N I S O T R O P I C B E N A R D P R O B L E M W I T H HALL A N D ION-SLIP CURRENTS*
L. P A L E S E University Campus, Mathematics Department, Via E. Orabona, 4 70125 Bari, Italy. E-mail: [email protected]
We study, by the Lyapunov direct method, the Lyapunov stability of the conduction-diffusion solution of an anisotropic magnetic Benard problem, for a partially ionized fluid. We determine the critical hypersurfaces ensuring linear Lyapunov stability recovering those obtained by solving the eigenvalue problem governing the linear instability. We show that, if the conduction diffusion solution is linearly stable, it is asymptotically non linearly stable.
1.
Introduction
The stability of the conduction-diffusion solution of the Benard problem, relevant in many astrophysical and geophysical applications, has been largely investigated in the hydrodynamic as well as in the magnetohydromagnetic case. For very high magnetic fields we must assume the generalized Ohm's law for the density current vector and, for a partially ionized fluid, the Hall and ion-slip currents must be considered 7 . In this note we consider a magnetic anisotropic Benard problem for a partially ionized thermoelectrically conducting fluid in a horizontal layer, we determine a sufficient condition of linear global asymptotical Lyapunov stability with respect to normal modes perturbations, recovering the critical curves of the linear instability. Successively, we study the nonlinear stability with respect to normal mode perturbations showing that the same condition ensures the nonlinear (conditional) asymptotic stability of the thermodiffusive equilibrium state.
*Work performed under the auspices of the italian G. N.F. M.-C. N. R. and supported by the italian M.U.R.S.T.
432
433
2.
Mathematical problem
Let us consider an homogeneous thermoelectrically conducting fluid, in a horizontal layer S, in the presence of a constant vertical adverse temperature gradient, and of a uniform imposed magnetic field Ho normal to the layer. In an orthonormal reference frame { 0 , i , j , k } , with k upwards positive, the layer is bounded by the planes 7ro : z = 0 and m : z = 1, both stress-free, thermally conducting but electrically non conducting. The dimensionless equations governing the perturbation u, h, 9, p of the thermodiffusive equilibrium mo = { U = 0, Ho = Hok, T = —fiz + To, po = Po(z)} are 4
^
= - u • Vu - Vp + Au + M 2 (H 0 + h) • Vh +
f)h P ^ = V x [u x (Ho + h)] + - ^ A h +
P fe^V
ydk,
x [(Ho + h ) x
V x h] + / ? / ^ V x {(Ho + h) x [(Ho + h) x V x h]},
(1)
V - u = 0, [ V - h = 0, where u is the velocity field, h is the magnetic field, 6 is the temperature, p is the pressure. The positive coefficients Pr, Pm, M2 and TZ are the Prandtl, Prandtl magnetic, Hartmann and Rayleigh numbers respectively. The Hall and ion-slip coefficients were denoted by (3H and /?/ respectively. The boundary conditions at z = 0,1 read 1 u - n = 0 = O n x D n = 0, h = 0 t>0, where D is the strain tensor and n is the external normal to the layer boundary. Assume that the perturbation fields are functions doubly periodic in x and y, denote by Q the periodicity cell, and use the variables w = k • u, h'3 = dzk • h, C' = dzk • V x u, j = k • V x h 3 .
3.
Linear Lyapunov stability
We study, by the Lyapunov direct method, the linear stability of mo- We consider the evolution equations in the variables Aw, £', h'z, j , linearized
434
around the equilibrium solution, and (1)3, i.e. —— = AAw + M2Ati3 dt
^dty 3 dt d
M2dzzj,
+AC +
dt =
d^w + ^ l
A C + ^Aj
dt
m_
- — A16 Pr
+ MAh^-fafg-dtd
(2)
+ fa^Ah'z + Pj^dzd,
- u • W + u • k + ^-A9,
\ dt where Ai = d2/dx2 w
+ d2/dy2,
with the boundary conditions on z = 0,1
= dzzw = h'3 = Ati3 =j = Aj = c = e = dzze = Adzze = o, (3)
and choose as a Lyapunov function the expression8: El(t) = (Vw,Vw)+d1(C\?)
+ d2tiJ)
+ d3{h^h'3) +
where (/, g) = fQ fgdfl and the parameters di are, hithertoo, arbitrary. In order to evaluate the temporal derivative of (4), along the solution of (2), we must estimate: dEj, •Ji-T>i, dt
(5)
where Ji = -M2(w,
Ati3) - ( — + du)(w, AtB) + d.M2^',
8zzj)+ (6)
+ d3{%,dzzw)
+
pH^
V, = (Aw, Aw) + diW, VC) + d2-^(Vj,
Vj)+ (7)
d20i-^(j',j')
+ d3^(l
+ pI)(Vh'3,Vh,3)
+
d4—(VV19,VV19).
By using imbedding theorems of Poincare type , from the energy relation (5) we obtain
§**<*-»•
(8)
435
where
A
=
max^,
M
=
{(w,C,j,h'3,6)
£
(C°°(fi)) 5
|
(w, C, j , h'3,6) satisfy (2)}. Whence the condition A < 1 is sufficient to ensure the linear asymptotical exponential stability of the diffusion solution. Assume that w,(,j,h3 and 6 are normal modes, where their factors functions of z, W, Z, J, H3, Q are real. Then equations (1)4,5 show that h3 and (' are purely imaginary. Let us introduce the normal forms in (2) to obtain that E,(V,V*), V = (W,Z,J,H3,0), V* = (W*,Z*, J*,H 3 *,e*)is a sesquilinear Hermitian functional, therefore, the quadratic functional Ei(V) = Ei(V, V) is real and positive definite iff d\, d3 < 0, ^2,^4 > 0. This is our Lyapunov function. From the Euler-Lagrange equations associated with the previous maximum problem, if we expand W(z) in a Fourier series upon the set of functions {sin(nirz)} £ L 2 (0,1), maximizing with respect to di i = 1, • • • , 4 2 the condition A < 1, in terms of physical parameters, reads
Ra2 < RHi(n,x,Pr,Pm,M2,
fa,Pi) 2 2
=
[Bn(Bn + fan * ) + M n 7 r 2 ^ ] M 2 n 2 7 r 2 ^ B „ n
2
2
[Bn(Bn + /? / n 2 7 r 2 ) + M 2 n 2 7 r 2 ^ ] ( l + /?7) + /? 2 f n 2 7 r 2 J B„'
Bn = n2ir2 + a2, a2 = a2 + @2. Its minimum value is obtained for n = 1. We obtained the following Theorem. If R < RHi(l,x,Pr,Pm,M2,fa,0i), where x = ^ , then, in the class of normal modes perturbations, the thermodiffusive equilibrium mo is globally linearly exponentially asymptotically stable. From the previous theorem we recover the conditions of the linear instability 1 , 4 , of the thermodiffusive equilibrium in the hydrodynamic, isotropic magnetohydrodynamic and anisotropic magnetohydrodynamic case, respectively. In 9 numerical results concerning critical stability bounds for the thermal equilibrium state mo are provided.
4. Nonlinear conditional Lyapunov stability In order to evaluate the effect of the non linear terms on the stability of the conduction diffusion solution we consider the Lyapunov function 6 1 0 n V(t) = Ei(t) + bEn{t), b > 0, where £7 is the Lyapunov function of the
436
linear case, 6 is a suitable positive parameter we determine succesively, and En(t) = ^{(Vdzu,Wdzu)+d5(VC,VC) d7(Vj', Vj')+d5(Adzw,
+
d7(Ah'3,Ah'3)+
Adzw) + d10(A6, A9) + d n ( V A i 0 z u , V A ^ u )
+di 2 (VAi0 z h, VAjSzh) + d 1 3 (VV x dzh, VV x 9 2 h)}, where the coefficients di i>5 are, up to now, arbitrary. The energy relation for V(t) is the following: ^•=Xi-Vi+M
+ b(In-T>n+Arn)
(9)
where It and Vi are given by (6), (7), and Vn = (Adzu,Adzu)+d5{AC',A^)+ d7^(VAh'3,VAh'3)+fr^(Vdzj' ,Vdzj')+d7fy(Aj' ,Aj')+d5(VAdzw, S7Adzw) + d10^;(VAe,VAe)+dn(AA1dzu,AA1dzii)+dl2^:{AA1dzh, AAi5 2 h) + d i 3 ^ - ( A V x dzh, AV x dzh). We omit the expression of TV/, 2"n A/"„ for the sake of brevity. We remark that, in the class of normal mode perturbations, for the positivity of E\ and En,we must assume 8 d\, dz, d$, d7, < 0. If d7 = d$M2, d12 = M2dn, d^K^jj-^dw, it follows that In<{A0 + ^^JbVn + ^AbDi, where £ is positive parameter and Ao A\ are some positive constants. In order to estimate now the A// terms, by using the inequality for the supremum of a function, 12 if we choose d± = dio, since d2 — M 2 , | d3 |= M 2 ( ^ + ° 2 ) , it follows that Afi
^w^M
, , A,b
A,e
u,
—z~
- 1 + — - ) + bVn{A0 + — 1)+ A (A2 + -¥-)VnV* + ^Vtvi + bMn + BiVnV,
(10)
Assuming | d5 \ >j2 + ^-, it follows T>i < Vn. If we choose e2 = ^ ^ - , £2
=
^ i L r , the inequality (10) becomes ^^-<-CVn
+ BVnV^+BiVnV
+ bMn,
(11)
where B and C are positive for a suitably choice of the parameter b. Moreover it follows that 2 5 1 2 Nn< £ l = 4 A{DnV\ +B'IVnV, with At and Bj positive constants. Therefore: dV{t) -<-CVn(l-AV2 dt
-AiV\
(12)
437
where A, Ai > 0. We can now prove the following conditional non linear stability theorem 5 : Theorem In the hypothesis 0 < A < 1 (i.e. R < RHI), V(0) < A where A ~ — 2^4~+\/(2X~)2 "^ JT~' ^ e conduction diffusion solution mo is conditionally non linearly asymptotically stable with respect to the Lyapunov function V(t). We note that initial perturbations are of finite amplitude, and the norm involved in the Lyapunov function V(t) implies a pointwise non linear asymptotic stability. Acknowledgments The author is grateful to Prof. Salvatore Rionero for his valuable comments and improvements regarding this paper. References 1. S. Chandrasekhar, Hydrodynamic and hydromagnetic stability, Clarendon, Oxford, 1968. 2. D.D. Joseph, Arch. Rational Mech. Anal. 36 285 (1970). 3. D.D. Joseph, Stability of fluid motions, vols. I and II, Springer, Berlin, 1976. 4. M. Maiellaro, L. Palese, Int. J. Engng. Sc. 22 (4) 411 (1984) . 5. G. Mulone, S. Rionero, J. Math.Anal. Appl. 144 , 109(1989). 6. G. Mulone, S. Rionero, Bull. Tech. Univ. Istanbul 47 202 (1994). 7. Sh. I. Pai, Magnetohydrodynamics and plasma dynamics, Springer, Berlin, 1962. 8. L. Palese, A. Georgescu, Math. Sc. Res. 8 7 196 (2004). 9. L. Palese, A. Georgescu,D. Pasca, D. Bonea, Rev. Roum.Mec. AplAI 3-4 , 279(1997). 10. S. Rionero, Atti Meeting Energy stability and convection, Capri (Italy), 21-28 maggio 1986. 11. S. Rionero, G. Mulone, Arch. Rational Mech. Anal. 103 347(1988) . 12. B. Straughan, The energy method, Stability and non linear convection, Springer Verlag, New York, 1992.
H I G H E R - O R D E R SHALLOW WATER EQUATIONS, EXPLICIT SOLUTIONS A N D T H E CAMASSA-HOLM EQUATION
D. F. P A R K E R School of Mathematics, University of Edinburgh, The King's Buildings, May field Road, Edinburgh, EH9 3JZ, U.K. E-mail: [email protected] This paper extends, to the case of variable depth, the recent derivation of the logical link between the Camassa-Holm (C-H) equation and shallow water theory. While solitons of the C-H equation have peak curvature adjustable independently of amplitude, the derivation emphasizes that 'peakons' are inappropriate to shallow water. Explicit solutions for C-H periodic and solitary travelling waves define equivalent shallow water profiles (and the associated fluid velocity field). A preliminary outline of modulation theory for variable depth is given.
1. Introduction The Camassa-Holm (C-H) equation (1993)1 is in some sense a generalization of the Korteweg-deVries (KdV) equation and, likewise, is completely integrable. Even though the 'peakon' solitary wave solutions discovered in Ref.1 are non-differentiable, the equation is often considered to be a higher-order model for water waves. It continues to generate a substantial literature 2 ' 3,4 ' 5 ' 6 ' 7 ' 8 ' 9 , focused primarily on its integrability properties. This paper aims to clarify the links between the C-H equation qr + 2ujqy + 3qqy - qyyT = 2qyqyy + qqyyy
(1)
and water wave theory, through emphasis on the Kodama-Helmholtz 4 transformation linking the surface elevation rj(x,t) to the variable q(y,r) which satisfies the C-H equation. It relates some recently found explicit soliton5 and periodic solutions 9 of Eq. (1) to solitary and periodic water wave profiles, so allowing the velocity fields at all depths to be found. It indicates, also, how these solutions and flow fields are part of a description of waves of appreciable amplitude travelling into regions of variable depth.
438
439 2. Linking the C-H Equation t o Shallow Water Theory Two-dimensional, irrotational, inviscid flow of an incompressible fluid is considered in the region —hb(x) < z < ehfj(x,t), where x and z are horizontal and vertical cartesian coordinates, t is time and h is a typical depth (so that b — 0(1)) . If the pressure is p, the density is p and the velocity has components (u, w), the Euler equations of motion are du
_du
_ du
dp
dw
_ dw
_ dw
_, dp
while the boundary conditions are p = 0 and w = eh(dfj/dt + udf]/dx) on the free surface z = ehfj(x,t) with w = —hudb/dx on the rigid, impermeable ocean bed z = —hb(x). If L is a typical length scale of waves, suitable rescalings are x = Lx, z = 5Lz, t = L(gh)-1/2t;
u = e(gh)~1/2u,
e5{gh)~1/2w,
w=
where 5 = h/L and e are parameters to be treated as small. The incompressibility and irrotationality conditions become du/dx + dw/dz = 0 and du/dz = S2dw/dx. A velocity potential (x, z, t) and stream function ip(x,z,t) may be introduced so giving w = 5~~24>z = -ipx,
u = <j>x=ipz ,
(2)
where subscripts denote partial differentiation. Then, when the pressure is written as p = pgh(erj — z + 52ir(x,z,t)), where r](x,t) = fj(x,i), the governing system reduces to the dynamic Bernoulli equation <j>t + 627r(x,z,t) + ±(u2 + 52w2) = 0,
(3)
with 7r = 0,w — r]t = Eurjx on z = erj and ip = 0 on z = — b{x) (= —b(x)). Formal expansion of <j>xx + S~2
<j>(x,z,t) = ${x,t)-52zQx ip(x, z, t) = Q{x, t) + zU-
z
z
-52—$xx
+ 5i—Qxxx
52^QXX
- S2^UXX +
z + 54—$xxxx-...
,
S4^QXXXX Z5
_i_A 4 _rr I
U
-I
_
V XXXX
' • • 1
where U(x,t) = $x(x,t) and Q is a volume flux. Then, Eq.(3) with ir(x,rf,t) = 0 gives, after differentiation, the dynamic boundary condition r)x + Ut + eUUx+e52{\Qx2 - r)Qxt}x + e262 [rj(QxUx - UQxx) - \r,2Uxt]
= 0(e362),
(4)
440
while the kinematic boundary condition m + Q* + e(vU)x - \e252{r?Qxx)x
= 0{eH2)
(5)
is merely an approximation to the statement rjt + [ip(x,£r),t)]x = 0 that volume is conserved. The condition at the rigid bed becomes Q - 6 U - | « 2 b 2 Q M + 5* 2 6 3 Uxx+^* 4 6 4 Ox ra *-T3o* 4 b B U'xxxx = 0(S6),
(6)
which does not involve e and may be rearranged as Q = bU+ \52b2[2bUxx + 6bxUx + 3bxxU] + ^54b4[4bUxxxx
+
20bxllxxx
+ 30bxxUxx + 20bxxxUx + 5bxxxxU] +
0(56).
For right-running waves (rjt « —b1^2^), Marchant and Smyth 10 derived (correctly, though with intermediate typographical errors, taking 6 = 1) the extended Korteweg-de Vries (eKdV) equation Vt + Vx+^eWx + \52r]xxx - \e2r\2r]x =
\24^/x^?xx ~r "^^^/xxxj i 360^ ^Ixxxxx
U.
\* )
4
From this, Dullin et al. showed that the near-identity transformation
r,(x, t) = ((*, t) + e{U2 ~ U ^ Q
+ §U2Cxx
delivers (correct to o(e2,S4)) an equation which, in new coordinates, is the C-H equation (1). Here, <9-1£ = J C(^i*) dx. An analogous procedure, for gradually varying depth b(x) = b(x) = B{^x) (7 -C 1), now follows. The linearization of Eqs. (4) and (5) motivates the introduction of new variables fj, v through rj = B~1^4fj(x,t), u = B~3/4r)(x,t) + v(x,t), where the powers of B~1'4 arise as an 'adiabatic' approximation, while v(x,t) ( = o ( l ) ) allows for both back-scattering and nonlinearity. Taking 7
- ±52B5'%X
x
- ± 7 £ _ 7 / 4 £ ' ( 7 z ) © + v{x, t), _1
2
(8)
2
where 0 = d~ fi (so that 0 t w - £ / 0 x ) and v = o(e,S ,'y). As might be expected, the term in efj2 is a nonlinear correction, that in 82fjxx is a dispersive correction, while that in •jB'Q is due to the slope of the bed. Continuing this procedure by substituting expression (8) and working to higher order gives further perturbation terms which determine v as
v = \e2B~^4f
+ e52(^l
+ l^B-^B'd^if,)2
+ \fffjxx) - I^B^B'fi,
lQ54B^%xxx + 0{e\ 56, e \ 7<54,72) • (9)
441 With these expressions for v(x,t) and v(x,i), Eqs. (4) and (5) are then consistent provided that f)(x, t) satisfies the one-way wave equation r)t + B1'2^
+ l6S2B^2f,xxx
+ leB-^m*
+ e52Bb/\fif]xflxx
-
§e2B~2f,2fjx
+ &m*xx) + M6S4B^2f,xxxxx
-e7B-^B'(§V2
^52B^B'fixx
+
+ 1*7*0) = 0 ( £ 3 , * W , 7 2 * 2 ) .
(10)
Eq. (10) reduces to the eKdV equation for JB'(JX) = 0 (e.g. for B = 1) and to the KdV equation at 0(s, 52). To relate it to the C-H equation, the Kodama (near-identity) transformation of Ref.1 is modified as f, = C(x,t) + eB-^ia^2
+ a 2 C*d _1 C) + 62a3B2(xx
+ ja^B"1
B'd"1 (, (11)
in which the 'coefficients' a\,..., a± may depend upon the 'slow variables' X = jx, T = 7 t . Consequent expressions 6 =d~1C + £ B - 5 / 4 [ ( a ! - a2)d~1(;2 + c ^ d ^ C ) + 62a3B2(x + 1aAB-1B'd~2C
+ 0(£7,7<52,72),
f,x =Cx + £B- 5 / 4 [(2a! + a2)CCx + a2Cxx0_1C] + 52a3B2(xxx + £7-B-
5/4
l
2
{(a 1 , x - \B- B'ai)(
2 2
+ y5 B (a3tX
+
X
1
+ [a2,x - \B~
l
B
^B^B'Q a2)Cxd~\}
2
- \B- B'a3)^xx
+ 0(7 )
(and equivalent expressions for fjxx, rjxxx and fjxxxxx) into Eq. (10). By exploiting the fact that Ct + 5 1 / 2 C* = - ! e £ r 3 / 4 C C * - \52B5'\XXX
are then substituted
+ (D(s2,5\e
(12)
and related expressions involving £xt + -B Cxx, Cxxt + B1'2QXXX a n d ( 9 - 1 0 t + B 1 / 2 C). terms such as 2ai£-B _5/4 (Ct + £ 1/2 Cx)C are determined with error o(e2,54,e/y,^52), so yielding Ct + £ 1 / 2 C* + leB-^CCx
+ 182B5/%XX
+ le2B-2(ai
+ \a2 - A)C2C*
+ e52B*'4[{fl + Q l + \a2 - 3a3)CxCxx + ( £ + |a2)CCxxx] +
£7i?-
3/4
{{ai,x +B~1/2a1,T
- ( f + fax -
+ [ a 2 , x + B-l'2a2,T
-{\
^B^B'K2
+ \a2 - | a 4 ) B - 1 B ' ] C x 9 - 1 c }
+ 7 <5 2 £ 5 / 2 [a 3 , x + B - 1 / 2 a 3 , r + ( ^ + 2a 3 )B- 1 B']Cxx + ^ ^ B
9
/
2
^ , , = 0( £ 3 ,«5 6 , £ 7 2 , 7 2 ,5 2 ). (13)
442
As in Ref.1, the term in Cxxxxx is removed from Eq. (13) by subtracting the equation (differentiate Eq. (12) twice) OQS B jCxxt + B ' (xxx + 32sB~ ' (3^xCxx + CCxxx) + 16<5 B ' Cxxxxxj
which additionally in Eq. (13) modifies only the coefficients of Cxxt, Cxxx, sxsxx ana L,L,XXXMaking two choices a i + \a2 — \ = 0 and a\ + \a2 — 3a 3 — jg = 2 ( | a 2 — j|o) then eliminates the term in C2Cx a n d creates the grouping 2CxCxx + CCxxx analogous to the right-hand side of Eq. (1). This delivers the generalization of the C-H equation Ct + fclCx + foiCCx - hCxxt
+ h(xxx
+ /c 5 (2CxCxx + CCxxx)
2
+ e7(fc6C + krQd-'Q
+ -y52ksCxx = 0
(14)
which has error 0(e252,S6,e12,1252), with h = B 1 / 2 , k2 = f e B " 3 / 4 , 2 2 2 b2 k3 = ^6 B , k4 = -^5 B ' , k5 = (I/? - JL-)£62B5/\ where aua2 and a.z are written parametrically as Qi = | - | / 3 , a2 = ft, a3 = — ^ — ^/3. The coefficients arising due to depth variation are k6 = -\B-*'A[(3X k7 = B-3'*\J3X k8 = -\B5'2[(3X
+ B-1'2!* + B-"2pT
+ (f| - | / ? + | a 4 ) B - 1 B ' ] , -{\
+ B-Wfr
+ \p-
+ (2/? -
|a4)B-1B'], f0)B-lB>].
For uniform depth (fc6 = fc7 = fc8 = 0), a further restriction k5 = — \k2kz (= —j^e52B5''i) sets /? = — A and yields an equation which, after a change to new variables C ( ^ ) = B1/2q{y,r), y = yf^B-15-1{x
- ^Bl'2t),
r =
<J^B~5/4s5~1t,
takes the standard form (1) of the C-H
equation, with parameter UJ = ^e~1B3^4. It is for Eq. (1), or an equivalent, that many authors 1 ' 4 , 5 ' 9 exploit the integrability properties or construct specific solutions. However, the mapping from (x, t) to (y,r) coordinates indicates difficulties - 0 ( 1 ) dependence upon y corresponds to dependence upon x/5, which violates the initial scaling assumptions. Also, the loci of constant y have speed merely ^ times the speed of linearized long waves. Both these facts suggest that mapping to the C-H equation is inappropriate. Nevertheless, explicit solutions 5,9 ' 11 for Eq. (1) are useful in developing a modulational description in regions of variable depth B(X).
443
3. Modulation due t o D e p t h Variation For a specified constant value of depth B, Eq. (14) has a iV-parameter family of explicit travelling wave solutions written as C, = B1/2q(y;pi,... ,PN), with yx and yt each constant. Moreover, q is periodic in y or describes a solitary peak (see Sec. 4). Thus, to describe modulation due to X-dependence of B, it is appropriate to allow each of the parameters pj (like (3 and 04) and also yx, yt to depend upon X and T, by writing y = ~{~lZ(X,T), so that yx = Zx{X,T) and yt = ZT(X,T) vary only gradually. Solutions are sought as C = Bl'\X)[q{y,Pj)
+1Q(y,X,T)]
,
Pj=Pj(X,T)
.
(15)
2
Omitting terms involving 7 gives a partial derivative as
where dots denote ^-differentiation. Similarly (omitting 0{-y2) terms) expressions for Cx, . . . , C,xxt are obtained and inserted into Eq. (14) to give (ZT + B^Zx)q
+ \eB-V*Zxqq
- BH\ZX)\^B^ZX
+
2 7
+
+ ^{ZT + Bl'2Zx)Q
(l{3-TL.)eS B /\zxf(2qq
+ \eB-V*Zx{qQ
+
+
frB-^B'q 2
2
+ e 7 (f B -
- 15 {fQ{2B ZxZXT
2
+ 2
B ' + k6B^ )q
+ BZT(BZX)X)
qq)
+ qQ)
-BH\Zx?{±B^Zx 5/4
%ZT}q
+
+
fQZT}Q} ^k^Zxfq
3 2
^B / [B{Zx)2]x}q
+ ^ E ( ^ + (*1/2 + ^ - 1 / 4 * ) f | ) £ + nkrB^qQ
(16) with Qy = q. The strategy for solving this is to take q is taken as a solution to the ODE describing travelling waves and to take Zx = Jf§B-1 with ZT+BXI2ZX as small, so that the terms on the first two lines of Eq. (16) sum to something as small as the remaining terms. Applying a compatibility condition to the resulting linear ODE for Q, then yields the evolution equations for the parameters pj.
444
4. Explicit Solutions for the eKdV Equation While explicit solutions to the C-H equation have recently been given in Refs. 8 ' 5,9 , the 0{e~l) size of the 'background' parameter u suggests that the C-H equation (1) is inappropriate to water waves. It is better to amend the choice of y and r to V = \[ilB~lx
- # - 1 / 2 ( l + eB-3'4v)t]
,
^B~5'H,
T=
thereby replacing Eq. (14) by qT + e(3q - 2v)qy - 62qyyT + 2 (j§.B 3 / 4 + eu) 62qyyy = AeS2(2qyqyy
+ qqyyy)
(17)
with A = (7 - 60/3)/19. Although this reduces to the C-H equation only for v = — j^B3/4e~1 and A = 1 (i.e. (3 = — | ) , travelling wave solutions may be constructed for all P and v (which defines a small perturbation from the speed of long, linearized waves). The substitution 3q-2v = %t"x^U(y) reduces Eq. (17) to the ODE 2UU{y) + (B3^ + $ev)U(y)
= %A52 [2U(y)U(y) + UU{v)] ,
(18)
(where dots denote ^-differentiation) which may be integrated twice to give lL{U)2+U3+11U
+ l2 = A52u(u)
,
(19)
where L = (B3^ + $ev). Since Eq. (19) is an ODE of the form 2 [U(y)] = G(U), every bounded solution describes either a periodic or a solitary wave solution for q = q(y). For the C-H Eq. (1), general solitary waves are constructed by A. Parker 5 , while various possibilities for periodic solutions were described recently by Parkes & Vahnenko 9 . However, to identify solutions relevant to shallow water it is best to solve Eqn. (19) directly. In the limit 5 —> 0, the general solution is the familiar cnoidal wave (for the KdV equation) U = A {6« 2 cn 2 ([AILf'2{y
- y0), K) + 2(1 - 2K2)] ,
where cn(^,«) is the Jacobian elliptic cosine function with modulus K. More generally, Eq. (19) is expressed as ( | L - A52U){U)2 = -(U -
Cl)(U
- c2){U - c3) = F{U)
(20)
445
with Ci 4- C2 + C3 = 0 and labels chosen so that c\ < c 2 < C3. Since 1^3/4 _|_ | 7 £ - _ j\pif must remain positive (when £ = 0(1)), the graph of G{U) = F{U)I{\L - AS2U) has an asymptote at U = c 4 = ( A ^ 2 ) " 1 ^ and zeros at U = c\,c2 and C3. Thus, bounded solutions to Eq. (19) must correspond to Ci < c2 < U < C3
A,? fdYY
- Mdi-
Ad
~
\dy)
d2
(21)
Y
in which Y > 1 with K = (d2- d3)/d3 > 0 and 0 < M = Kd1/{d1 - d2) < K. As in Ref.5, an auxiliary variable 9 is introduced, satisfying ay
2<W2
This converts Eq. (21) to the form (dY/dO)2 = 4{Y + M)Y(Y-1), showing that y + | ( M — 1 ) is a Weierstrass P-function of 9. Using standard results 12 , this may be written in various ways in terms of Jacobian elliptic functions, the form appropriate for Y > 1 being Y = cn~2 (r9, K) which is consistent with Eqs. (21) and (22) when r = (1 - K 2 ) " 1 / 2 and M = K 2 / ( 1 - K2). Then, U = c 4 - w and w = d2Y/{Y + K) = d2[l + Ken2(r9, K)}'1. Introducing as parameter the peak-to-trough amplitude A = C3 — c2, so that 62A A 52Aj 2 2 ~ \L - A<S c2 - 5 A ~ Ad2 - A ~ | A - 52A1 ' where the algebra has been simplified by introducing A\ through A/Ai A~lL - I<52c2, leads to Acn2(r9,K.) U — c2 + — 2 2 2 A-lS AlSn (r9,K)
=
'
Then, with %jj = r9 — 0/Vl — ft2, the travelling wave solutions are given by e( = eB1'2
B en2 (ip,K)
l(v+c) + — eB\sn2(ip,K)
(23)
in which eAB = §62A, eA#i = \b2Ax and ec = |§<52c2 , so that B/Bx ^62d2 = f § A _ 1 L — §ec. Solving these relations together with 2__
M __ M +l
AB
(L + \Aec\ 2AB+\c\L~\Aec)
446
gives a quadratic equation defining c in terms of AS, K, A lL and e. Its 0(1) root is c=
C(AB,K,
A~lL; e) = \{2e~1ArlL
- (4 +
K~2)AB
- e~ 1 [4(A- 1 L) 2 + 2(4 - bK~2)eBL + (4 + K~2)e2 A2B2)XI2} , so determining the 'background value' v + c = v + c(AB,K,A~1L;e) in terms of the perturbation ev in wave propagation speed. If V were the travelling wave coordinate (linear in x and t), profiles (23) would be just mild perturbations of cnoidal waves (due to the term eB\ in the denominator). However, there is also a mapping from ijj to y = J^B"1 [x - Bl/2(l + B~3/4ev)t] found by integrating Eq. (22) in the form dy_ #
=
252d2A1'2 V T ^ A^Ci = y/(di - d2)52dz 1 + Kcn2{tp, K) ~ 1 - sABlSn2(ip,
K) '
[
'
where d is found (after using r = y/2{AB + c)/[(l - eASi)(AS + 2c)]) as C\ = £ _ 1 / 2 5 A / 2 S / [ S I ( A S + 2c)]. Thus, using standard definitions, it is found that y = y0 + A 1 / 2 C 1 n(amV',£AS 1 ,K),
(25)
where II is the incomplete elliptic integral of the third kind 12 . Expressions (23) and (25) (with c = c) now give a parametric representation for the relevant travelling wave solutions C = B1/2q{y) — Bll2(^v + ^e~l52U{y)) (of the Camassa-Holm equation), where U{y) satisfies LA-XU
- p2(2UU
+ UU) + 2A~XUU = 0.
(26)
Apart from e, 6 and the undisturbed depth B, it contains the parameters A S = pi, K = p2, v = ps, A _ 1 L = p4, A and the 'phase' j/o = P5 (since ASi = Pi/(f§P4 - fee)). (Recall that A = (7 - 60/3)/19, in terms of the disposable parameter (3 in the Kodama-Helmholtz transformation). The limit K —> 1, in which CV?{-4>,K) —> sech ip, gives solitary waves. The denominator of Eq. (24) is reponsible for the sharpening of profile peaks illustrated in Refs. 5 ' 6 ' 8 , when compared to cnoidal waves. However, the derivation of the eKdV and C-H equations requires that E « 1 . Thus, the limit eABi —> 1 which produces peakons or periodic peaked waves is far beyond the validity of the derivation from shallow water theory. However, by varying pi,... ,p 4 , a considerable range of periodic and solitary wave solutions is described.
447
The travelling shallow water waves in regions of uniform depth B are derived from Eqs. (23), (25) and (11) as er, = I T ^ e C + B^l2{{\
- ±/3)(<) 2 + (3e2{(y) J \dy)
-§(/? + ^ i r V ^ )
(27) (28)
and have the (non-dimensional) speed f? 1//2 (l + eB~3/4v). Moreover, the representations of velocity components (u, w) in terms of f) and its derivatives allow appropriate approximations of flow field, polynomial in depth, to be identified. The expressions in Eq. (27) may be evaluated explicitly, though details are complicated, as for example dt
nz,i/2f^/2\
„
x / 6 i ( A g + g)sn(V;,K)cn(y>,K)dnO/,,K)
- = -2B' ^ _KnJ B ( i - e Bpl ) y — j ^
i_ £SlSn2 (^ K)
•
C(y) is a considerably more complicated expression, while the term J Qdy is found, after considerable manipulation using results in Ref.12, as a linear combination of ip, the incomplete elliptic integrals n ( a m ^ , e B i , K ) , E(amtp,K) and the expression in ((y). In regions where depth varies, the parameters pj and A (equivalently /?) are forced to vary, due to the terms in ^B'{X). Details have yet to be completed, but the strategy outlined in Sec. 3 is expected to yield the required parameter-modulation equations (showing, for example, that as K varies, solitary waves become undular). References 1. R. Camassa and D.D. Holm, Phys. Rev. Lett. 71, 1661 (1993). 2. R.S. Johnson, J. Fluid Mech. 455, 63 (2002). 3. A. Constantin, Adv. Appl. Mech. 457, 953 (2001). 4. H.R. Dullin, G.A. Gottwald and D.D. Holm, Physica D 190, 1 (2004). 5. A. Parker, Proc. Roy. Soc. A 460, 2929 (2004). 6. A. Parker, to appear in Proc. Roy. Soc. A. 7. H. Kalisch and J. Lenells, Chaos, Solitons & Fractals 25, 287 (2005). 8. R.S. Johnson, Proc. R. Soc. Lond. A 459, 1687 (2003). 9. E.J. Parkes and V.O. Vahnenko, Chaos, Solitons and Fractals 26, 1309 (2005). 10. T.R. Marchant and N.F. Smyth, J. Fluid Mech. 221, 263 (1990). 11. D.F. Parker, in preparation. 12. P.F. Byrd and M.D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Springer-Verlag, Berlin (1954).
T H E GALILEAN RELATIVITY P R I N C I P L E AS NON-RELATIVISTIC LIMIT OF EINSTEIN'S ONE I N EXTENDED THERMODYNAMICS
S. PENNISI, M . C . CARRISI, A. S C A N U Dipartimento di Matematica ed Informatica Universita di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy e-mail: [email protected];[email protected];[email protected]
It is well known that, in t h e relativistic context the relativity principle isn't imposed by separating variables into convective and non convective parts, but by imposing that the costitutive functions satisfy particular conditions; likely to this, the present considerations show that the same results are obtained also in the classical context. The result is achieved by taking the non-relativistic limit of Einstein' s Relativity Principle. This fact furnishes further arguments on the naturalness of the work "A new method t o exploit the Entropy Principle and Galilean invariance in the macroscopic approach of Extended Thermodynamics" by Pennisi and Ruggeri.
1. Introduction This paper is strongly based on the work *, so that it cannot be understood without reading it; we remaind to it also for the notation, for the sake of brevity. There one starts from the relativistic balance equations
Iff"'0"', daBaa2-aM
daAaa2-°" proceeding to
=l2i'"aM
+ dkF^1"-*' = P*1"-*' dtFJ}''"'• 4r 1 1 r-
then
(1)
for 0 < s < N -
+ dkFfi -* = PJP - for 0 < r < M -
and, from these, to Qtpii-U dtF
+Qkpii--isk
=
-^-2"
ph-i, ""^~5
+ dkFn
t
'-'ceiei
= Q* 1
%
r (3)
for 0 < s < N-l and 0 < r < M-1. Notice that eqs. (1) -(3) are all in the relativistic context, even if (2) -(3) are expressed in the 3-dimensional formalism. The advantage of eqs. (3) is that they are convergent for the light velocity c going to infinity and that their limits give independent equations. In section 2 we will exploit the consequences of the Einstein's relativity principle for the entropy-(entropy flux) tensor ha and for Aaa2-aN, Baa2-aM; we will see how they translate into the Galilean's relativity principle when c
448
449
goes to infinity. In this way we obtain the results of the work 2 , but generalized (because they haven't considered eqs. (3)2)- So we find another proof, from a physical point of view, of the mathematical method described in 3 to approach this problem, always restricted to eqs. (3)i. The comparison with 3 shows also how our results are general at all: even if they have been obtained in Galilean equivalent frames, nothing changes if v_ is assumed to depend also on position and time, because all the treatment isn't affected by the dependence on x and t of each independent variable. Moreover, we obtain the advantages above exposed in the abstract. It is also true that, if the above mentioned decomposition is applied subsequently to the results of the present paper, these become exactly those of the well known paper 4 but this is natural too, because with the previous method correct passages have been used, even if leading to more complicated calculations. The bigger difficulty in the traditional approach is due to the fact that it faces two problems in a single step, i.e., 1) The Galilean's relativity principle, 2) The solution of the implicit function ^ = 0. With the new approach instead we face only the difficulties of the first of them, leaving the solution of the second, but noticing that it involves only mathematical calculations without the necessity of proving the existence and uniqueness problem. In section 3 we will see how the matter becomes simpler if we take the lagrange multipliers as independent variables. 2. The classical limit of Einstein's relativity principle Let us denote with lA a set of independent variables for the system (3) and with IB(IA) their expressions after a Lorentz transformation. The Einstein's relativity principle for ha imposes that both ways in the following diagram give the same result, pa
IA
I'BVA)
ha
ha a
h (lA)
P2h"'[l'B{lA)] i.e.
pa
ha(lA) = Pa'ha [l'B(U)]
h«'[l'B(lA)} or, for a = 0,1,2,3
(4)
450
mth{lA) = 1mi0ti{l'B{lA)]+1^mA0(t, m; fcj>\l
A)
B{lA)}
= ^mZh'll'BilAK+ri-ft
' 0 J 21;
(IA) =
c
3 j / li[l' r/'
m ^-4>2[I (IA)} B
^(l
,
HI'BVA) A
) =
^p[l'B(lA)}
(see in J the grounds for this 3-dimensional decomposition). Moreover, for the sake of simplicity, the particular Lorentz transformation P£ = ( 7 ^ 0 0\ — 7 00 - 3 - 3 ^ has been used. If we multiply these relations by m 0 , m 0 c,
Vo o o i / UIQ c, m,Q c respectively, and then take their limits as c —> oo, we obtain h(lA) = h'[l'B(lA)],
^k(lA)
= ^'k\l'B{lA)}
with^ f e = / - t o f c .
(5)
We have now to impose the Einstein's relativity principle also for the functions Aaoc2aN and Baa20lM; for the first of them it imposes that both ways in the following diagram give the same result I'BVA)
IA j^aa2...ccN
Aaa*-a»
Aa
(lA)
pa
ps>p:i-pzAaa2-aNV'B(iA)]
that is
Aaa2aN(lA)
=
a
2---aN
A"'a*-<{lB(lA)}
P£P^P°NAa'a*a'"[l'B(lA)}
or, with aa^ctjv = ii...i r 0...0, and using the identities
PZ = Poa&+P?X',
72
n = l~
and
P}1=5^
vl1 Vjx
+- ^ ^ ,
(6)
451
Ai\...irO...O
(2
72
7 — 5^ + **> + - i - r «r
c
\^ J r
u'fUj
5^,
^2
„ U A ^ +
£U
Ja + l ,
\
+2^+1#+1
75°,
+ 1
7
^ i r „ ( n N-r
U»l
7-^+f^ + ^ - ^ ^ V -
(U) =
/ i \
t,
7+ 1
",+1^+.
3r
7 + 1 c2
N-r-b ,. 1,. /l'O...Oja + l---jVO...OjT. + () + 1 ...jjv Jr+6+1 "-"J N^ 1
from which (see 1 for the expressions of F^' in terms of A ")
^- ir (u) = E 1 Sir)
,
JV
7
2 i-! 72 ^ r > + ! c2
\
o + J V - r „l (l n
„i
V . . . u * °0
/ r»a+l +1 1
,
<5° + Ja+1
7 7
+
^
l
^Ja + 1
(7)
c2
J V _ r
w-
£
-2iV+2r+26„
3a + l---3r3r+b + l---3N Vi,,F, L Jr+b+l--uJNN
u
We can notice that the exponent of c is even and is —2(iV — r — b) < 0; so that, at the limit as c —> oo, only the terms with b=N-r remain, i.e.
Fr^(iA) = J2 (r) v{ii-v^F'^-^ or,fors=r-a
F^(lA)
=£
(r)^ ^ { 1 ' ^ ) ^ . . . v ^
. (8)
This can be written also as r
Fi^^(lA)
= Y,X}l.::ys(^F'^---^{l'B(lA)}
for r = 0,---,N-l
(9)
••S^vi'+1...vir)
(10)
s=0
with
X^Zl
Similarly for Baa*-aM
TX
5{il •
we find (eq. (7) with M instead of AT)
K^VA) = £ (ra) r+M-^..,>° ($£ + ^ T ^ ^ M—r
5^) + jf_^3^\ i
j r
7+ 1
V
c
2
E 6=0
(11)
3a+l---3r3r+b+l---3M fM - ^ „-2M+2r+2f,r Vi.,F, M "]T+b+l—"3M'v 6
452
Let us now consider eqs.
(13) of paper
eq. (7) and (11) finding F F'ja+i-jrjr+b+i-JN a n d F'ja+i-JrJr+<,+i-3M_
1
, into which we substitute
2
2
m
terms of expression,
In the resulting
we substitute ir^"+i->>+"+i-JM firstly obtained from eqs. (13) of paper written in £ ; m this way we find r Of Fjj
+
JrJr+b+l
JN
j
an(
Q£
2
p
2
j
1
m terms .
2
finaUV)
Wg
do the limit as c —> 00 and we find F
»1-«re1e1-ew+M-i-arejr+M-i-2r
j - v ^ • • • i r ^ p ' h -in
=
(12)
+
/i=0
M-l
53 ^;:::i;(^i
, ?J1
'" J " e i e i "" ; w + M "" 2 , ' e " + M "'' 2 '' for r = o,..„ M-I,
?7=r
l^-J
m/{/i,r}
with
«
=
E
E
fc=sup{0,/i-JV-M
k)
Pl\{h
G'l ' ' ' J t U
+ l+2r}
p1=sup{o,/i-fc-^±Mfi^^}
- k - 2pi)! (^+M-i-2r
""' ^
r
^jjfc + i ' ' " Vjh~2P1
{v2)Z±Mfl^r+Pl-h+k
+ p i
_
ft +
°jh-2P1+ljh-2P1+2
ky
' ' '
"jh-ljh)
^
(13)
[Jv+M-l-n-fc
K ."t=
E
E
fc=sup{0,2r-?7}
Pl=snp{0,
W
+ ^ - ' -q-fc+r}
(M ,k 2"+"-i-,-*-*.
'JV+M-l-2r\; 2
pi!(M + N-l-r)-k0
Ul
°3kV
6, u
3N + M-r,-2P13N
2pi)! ( - r + p i + jfc - ^+M=i V
V
V
^ +l
•
JN +
•••<$• + M-n-2P1
+l
U
+
^1
M-l-n-2P1
• .tv2)-r+Pl+k-^±f^+r,
]j)-l3v)\
'
(u) '
V"/
For the sake of brevity we leave the transformation of the productions which, on the other hand, is obvious. We notice that as independent variables I A we can take F ! l " , ! ' and F 2 r 2 r; in this case eqs. (9) and (12) give I A = IA{1'B) while the same equations with s+1 and r+1 instead of s and r give conditions on the costitutive functions, besides that for h and i[)k. Alternatively, we can take as independent variables
453
the lagrange multipliers defined by AT-1
M - l il...tj-eiei...e
dh=T/X*H-isdFil-is+Eri1-irdF s=0
it+M - l - 2 r
eN+M-l-2r
r=0 M - l
N-l n ,h
= E^-i^
-
2
+E^ . . . ^
ft = 0
2
(15)
T) = 0
iV-1
w h
M - l
* C ^ = E K-^tX + E
inf{h,r}
vjk+i...vjs+j^ r=0
J2
I
- k - 2pO! (
r - U P * + 1 ...VPr U*
'—
JV+M 1 2
' - ' - + P 1 -ft+
,- •
p +I
fc)!
'
--*0l--Jt
I" N +
<** 4..j^i^...^tz=i r = 0
c
2h-k-2tn
(I)
N+M-l-2r\i
±—, Pl !(h
E k=sup{0,h-N-M+l+2r}pi=sup^0th_k_N+M-l-2r]!
r = 0
E fc=»«p{0,2r-,} /
2M+iV-l-T)-fc-2pi
p i=
M-l-y-k]
E sup
|0?iViM^_T7_fc+r|
N+M-l-2r\t
. p i ! ( M + J V - l - J 7 - f c - 2 p i ) ! ( - r + p i +fc-J v + f ~ 1 +»?)!
fa2i-r-+Pl
N
+fc- +M-l
+v pk+1
°3N + M—n-2p1JN-t-M-r>-2]ii+l
»
Pr
" 3v-liv)
^ '
In the next section we will see what happens if we take these as independent variables. 3. T h e Galilean relativity principle in terms of the Lagrange multipliers Prom eqs. (16) and (17) we have that d\*' , - i i ^ av 3h+l
= (h
du*.'
, =(T?-1W;,.
n
-
Q
while
J n
x
+
'
-^— = 0
l)A;
x
,
' r JUl-3n-2
iovh
v
'
.
" T j l •••3n-l
5,
.
By defining ft = X^F**"*-
+A C ^ 1 " ^
6 1 6 1
H
ln)3
- -h
forrj>l ' —
454 we obtain t h a t (5) holds also if h and ipk are substituted by h and ^ fc respectively, i.e., h(lA) = h'[l'B(lA)} , ^ O i ) = f % ( k ) ] These become identities if calculated for v_ = 0 so t h a t they are equivalent t o their derivatives with respect t o Vj i.e., N-2
„j/ °Ah-3h
fc=0 M-l
E
QT,
^
r=l
& - 1) •
ritl...ir_a8ir_lir
+ (N + M + 1- 2rK 1 ... i r _ 1 5 i r J ] = 0,
"^n.-.ir
5> + 1 )^— A ^« + /l=0
jl---3h
M-l
E r=l
(19)
^ 7 ;
5^—K
r
" !) • ^ii...ir-2*v-iir + (AT + M + 1 - 2rK 1 ... i r _ 1 5 i r J -] = 0.
"^il-.-ir
B u t from eq. (15) we also have %1...lTe\e\...e N+M-\-2r
e w+ M-l-2r
dh
so t h a t eq. (9) and (12) follow as a consequence of (18). Similarly, if the entropy principle holds, we also have #fc = A:i.„i3rf^-'-fc+<„^dFHi-ireiei-ew+M2-i-2re"+^i-2r from which it follows rpi1...i3k
3
O^P
~ dK...i. '
i i . . . t r e i e i . . , e jtf + j^r-i-2r e ;y-[-ivf-i-2r
2
2
d(j>
=
*K...ir
so t h a t eqs (9) and (12) with s + 1 and r + 1 instead of s and r respectively, follow as consequence of eq (19). Consequently, only conditions (18) and (19) have t o be imposed. References 1. F. Borghero, F.Demontis, S. Pennisi, "The non-relativistic limit of Relativist s Extended Thermodynamics with many moments-The balance equations." presented to Wascom S005, 2. F . Demontis, S. Pennisi, F. Rundo, "Some further considerations on the galilean relativity principle in extended thermodynamics", presented to Wascom 2005, 3. Pennisi S., Ruggeri T., "A new method to exploit the Entropy Principle and Galilean invariance in the macroscopic approach of Extended Thermodynamics" , to be published. 4. I-Shin Liu, I. Muller, "Extended thermodynamics of classical and degenerate gases", Arch. Rational Mech. Anal 83 (1983).
EQUIVALENCE OF T W O K N O W N A P P R O A C H E S TO E X T E N D E D T H E R M O D Y N A M I C S W I T H 13 M O M E N T S
S. PENNISI, M.C. CARRISI, A. SCANU Universita e-mail:
Dipartimento di Matematica ed Informatica di Cagliari, Via Ospedale 12, 09124 Cagliari, Italy [email protected];[email protected];[email protected]
The recent paper "A new method to exploit the Entropy Principle and Galilean invariance in the macroscopic approach of Extended Thermodynamics", by Pennisi and Ruggeri, shows a very interesting way to overcome the difficult calculations involved in imposing these principles. As tests for the validity of their procedure, they indicate 2 models: the 5 moments model and the 13 moments one. The first of these is exactly that previously known in literature, while for the second one the comparison was not possible, because in literature this model hasn't until now been exploited, up to whatever order with respect to equilibrium. This gap is here filled because we prove that also the traditional approach leads to the same result found by Pennisi and Ruggeri. Obviously, this doesn't overcome their paper; in fact, a matter is proving that given functions are the unique and exact solution of the concerned conditions (which proof is the object of the present paper), and another matter is to find the expressions of these functions (which is the result of Pennisi and Ruggeri's paper). So we aim here only to furnish a further proof which confirms their procedure.
1. Introduction Let us consider the paper 1 , generalized in 2 ; here we call it "the first approach" to extended thermodynamics with 13 moments. In this paper the the Entropy and the Galilean Relativity principles are expressed by
*'
m«=
SA'
84>'k
^ < J m "
\^
t,
= ^r~s<-ssj>
,m*=*-
(1)
dXi
>
mr*
dX
dX
Ilk
d'Xij d<j>lk 8Xm
"
dX\j _fc 8Xr dXui
I (-$£•
mak
4k,i
- 8A °
) + \mmSii
=
m^k ,„.
455
456 0
= *xi T T + *«' 2 7 f - + -^T— *"*j 1,0 = 2Xiimik + XM (2miki + m fc „5«) + h'tf. J 9A y a\ij 8Xrs ) \ / (3) 3
Subsequently Pennisi and Ruggeri have found in a method capable to overcome the difficulties in calculations that arise from the first term at the right hand side of the equations (2); we refer to this as "the second approach". It consists in resolving the mathematical problem dh' dX
i
OX '
dh' d\i
4,
dXi '
dh' dXij
dh' dXui
iU
dX^ '
dX^
k
d$'
mikii
dX.ill - dh'
» dh'
-
(dh'
dh'
p
„.\
h'k . Qftk / Q^ik p,X,k 0 = A , - - ^ - + 2 A,jy - ^~- +\~Xm - +. -^—SrsS} "-ill\ I22 ^—-— dX dXi \ dX^ dXr
>*k + h'5 J
and then defining A, as inverse function of f&- = 0, inserting it in the expressions of h and h k finally finding in this way the functions h! and h k of the first approach. We compare here the results obtained by Pennisi and Ruggeri with what could be obtained directly from eqs. (1) — (3). In this way we give a further confirmation to the validity of their method. To this end we use the iterative procedure. 2. The iterative procedure. Firstly we notice t h a t t h e two results coincide at thermodynamical equilibrium. In fact eqs. (1) calculated in this state become: m
dhQ
dh0
= —T > m « = ~T ' m
because A and Xu are the only variables at equilibrium and there are no linear terms in h , for the representation theorems. Similarly eqs. (2) 1)2 amount to identities, eq. (2) 3 to m ^ t = 0, while eq. (2)4 gives rriikii at equilibrium, without conditions on (j> k. Finally eq. (3)! becomes an
457
identity and eq. (3) 2 yields ^Xumll5k+0+0+h'05k
= 0. Integrating this last
2
equation we find h0 = Xlt Ho(X) which coincides with the result of paper 1 . So we have proved that the two approaches coincide at thermodynamical equilibrium, because give the same result for h'0 and 0fe (which is zero). Let us suppose now to know h and (j)k up to order N, with respect to equilibrium. Equation (3)i at order n and for n = 0,1,..., JV gives Aj at order n (we know everything except Xi that we can obtain from the equation). The trace of eq. (3)2 yields 0 = 2Xu -j- + 2A< ii> + 3A»n — + 3h' Xu X
=
\ i 2 H ^X,X
,AJKAJJ 2 J .
Equation (3)2 is
\
6
dXu
^A<»j>-^
J dr ds
dXn +-AiH-^
OA
°
dA
h
CfXkll
^6<°sk>J^)5<^>
2Xm(_ȣ__?lJk \oX
3
dXn dX
dXc>
.
oX
Its symmetric trace-less part, at order N, needs the knowledge of h', 4> k and Xi up to order N (which we already have) except for the overbraced term (in which h! appears at order N + l ) ; then we use this equation to obtain N+l
N +
l
this term, i.e.,|A;;^§-^—S^S*-*, from which we can obtain h except for N+l
„
„
„_3
an arbitrary function h* (X,XuiX[[2). The integrability condition between (2)i and (2) 2a is dmrk dXr dmTk dXr 1 dmkU dXu dX
dX dXu
d2ti dXk d2h' —H ^ dXft dX dXudX
or
—r-
d2h'
dXk ,
dAdA// dAn
d2ti
+ dXdX -,:;
3 dX d
x
°a°b
,
*r dX
—~
=
, 1 d2h'
« r ^dXf +3 dXdXm a
458 The terms of this equation are known at order N except for the last one (remember that Ar = 0 at order 0) which we then obtain from this equation, /-t 1
„
2
iv+i
i-e. ~r^—• Calculating this expression in A< rs > = 0 we find - A — h* .
.
dXdXui „_2
c
°
^'°^ »
„
dXdXui „ „ „_3
JV + 1
(A,Aj H A H 2 ). Integrating with respect to Xm we find -^ h* (A, A iH A n 2 ) except for a function which doesn't depend on Xm; but this function is 0 because it doesn't depend on A < r s > and on Xm so that it is of order 0, while N+l
N+l
h* is of order N + l . So we have found ^
„ „
„_3
h* (X,XuiXa2):
N+l
N+l
„
Integrating „_3
with respect to A we find h* except for a function h** (Ajj;AH 2 ) ; because JV+l
h** is of order N + l we have N+i h
JO =
"
Vc1
if N + l is odd, (XuiXil%3)^
(6)
if N + l is even.
N+l
N+1
So h** has been found except for a constant of integration c which arises only when N + l is even. Now we already know the solutions of our equations obtained with the second approach which we call h[ and
c i, instead of _ c . So it suffices to choose c i= c
~>
N+l
to make coincide h and h[ also up to order N + l . On the other hand and
c
c i are both arbitrary constants and then it is not necessary to do N+l
N+l
any choice: h and h \ coincides at all! After that, eq. (3)i at order N + l gives A, at this order. Let's now turn our attention to ij> k and consider eq. (2)2fc and (2)3; they define m 1 ^ and mlku, but the symmetry conditions have to be satisfied:
d^ 9A
d\]il
Xa (^&a[k + 3A< r 3 > \d\11
\d^ll
9\
J^&^> 3
9\
d\k]u (7)
J dXi]u N+l
These, at order N, give — ^ — S r S 3 ^ and f|— as function of known quan&X
N+l
oXs^n
tities; because also <j)k satisfies these same conditions, substituting from
459 (7) the corresponding quantities with
^
= 0
(8)
JV+1
with tpk = 4>'k - 4>k . Now, because iph is of order N + l , it can be expressed in the form iV+l
Vfc = £
^ai6l-a^Cl-CN+1-rA
(9)
r=0
with iprai 1 " , a r r-ci...cN+1_r a ^ e n s o r depending only on A and A;;Eq. (9) doesn't change if we substitute ^a^-ar-brc1...cN+1-r ,kp1q1...prqT{c1...cN+i-T)p(aibi
parbr)
.
paibi
_
Aa-i M
with
]r
caibi
r vr piq\ •••rPrqr wneie r f t 9 . — uPi uqi 3 o PiQi u and the symmetrization is done treating a^bi as a single index. In other words we can still keep eq. (9) but with %l>rai 1 "' a r r 1---CN+1~r symmetric with respect to two generic indexes c^ and Cj, which remains the same exchanging any two couples of indexes afii and ajbj, and that gives 0 when we contract it with V 4 , ^ . After that eqs. (8) becomes ril){ki]ja2b2...arbrc1...cN+1^r k
1Lk1a1b1a,2b2...arbrc1...cN+i-rfil
fii]
TT
K\
_ g
fQr
_Q
for
r
_ ^ _ _]\f_|_J
r = 0 ...N
C/sT-(-l —r
'"'
In other words iprai ia2 2"'ar r C l - C N + 1 - r i s a tensor symmetric with respect to V couple of its indexes, so that jjkaibia.2b2-:arbrci...CN+i-T
_
_ Jo
ifN-risodd, ( kai bia2
~ \ipr(\,\ll)5 - 5
CN rCN r
...5 -
1
- + '>
if N-r is even.
Now, if r > 1 eq. (10) contracted with Saia2 gives _ Jo
ifN-risodd,
\Vv|^f^ a
(fca2 i 2a3
£'
..-<5
/kaibia,2b2--a r b r ci...CN + i- r
fl
CN rCjv r+l)
-
-
if N-r is even.
if 7- > 1
Thanks to this results, eq. (9) becomes k
fcci
civ+i"
"
JO [tpo{\ A//)A/£"(Ar.;Ar")2
if N is odd, if N is even.
460
M f>\ • Now, eq. (2)! is
^ dli d\k dK
Because A r = 0 a t order 0 =4> ^fe.
=
f~ is function of known quantities, so t h a t
0; consequently, we have t h a t tpo doesn't depend on A. Now we can
repeat all t h e considerations from t h e beginning of this section until eq. (7), but with N + l instead of N and noticing t h a t t h e results don't change if we N+l
^
JV+2
add at 4>'k a t e r m like 4>o(Xii)Xkll(XriiXrll)~2 Let's pass t o eq. (2) 2 a , i.e. (h^_dii_dXlL dXu
dXu dXu
dh'
. T h e n we find t h a t
JV+2
h = h± .
1 dh'
dX
dXu
3 dXku
N+l
which, a t t h e order N + l gives ^ — as function of known quantities => we JV+1
JV+1
have t h a t ^ - = 0; in other words ipo is a constant. So (f> k and ifc differ N+l
for t h e t e r m ipoXkll(XruXrl1)^ (present only when N is even); b u t in k was already present a t e r m like this with an arbitrary constant coefficient. JV+l N+l
So we can affirm, without loss of generality, t h a t cfi k =
t h e proof t h a t h = hx and <j> k = (f)k . We retain t h a t these results are very satisfactory and also give a further confirmation of t h e work 4 presented at Wascom 03 and applied in 5 t o t h e case of 20 moments. References 1. I-Shin Liu, I. Muller, "Extended thermodynamics of classical and degenerate gases", Arch. Rational Mech. Anal 83 (1983). 2. I. Muller, T. Ruggeri, "Rational Extended Thermodynamics, second Edition, Springer Verlag, New York, Berlin Heidelberg (1998). 3. Pennisi S., Ruggeri T., "A new method to exploit the Entropy Principle and Galilean invariance in the macroscopic approach of Extended Thermodynamics" , to be published. 4. S. Pennisi, A. Scanu, "Judicious interpretation of the conditions present in Extended Thermodynamics", Proceedings of WASCOM 2003, 12th Conference on Waves and Stability in Continuous Media, Villasimius (Cagliari), 393-399 (1-7 Giugno 2003), 5. M. C. Carrisi, S. Pennisi "On the exact macroscopic approach t o Extended Thermodynamics with 20 moments", Proceedings of WASCOM 2003, 12th Conference on Waves and Stability in Continuous Media, Villasimius (Cagliari), 386-392 (1-7 Giugno 2003).
FUNCTIONALS FOR THE COINCIDENCE BETWEEN LINEAR A N D NONLINEAR STABILITY WITH APPLICATIONS TO SPATIAL ECOLOGY A N D D O U B L E DIFFUSIVE CONVECTION
SALVATORE RIONERO University of Naples Federico II Department of Mathematics and Applications "R. Caccioppoli" Complesso Universitario Monte S. Angelo - Via Cinzia, 80126 Naples - ITALY E-mail: [email protected] In the framework of Liapunov Direct Method, coincidence between linear and nonlinear stability is studied . The advantage of determining and using functional depending (with their time derivative along the perturbations) directly on the eigenvalues of the involved linear operator is shown. Applications to spatial ecology end double diffusive convection in porous media are furnished.
1. I n t r o d u c t i o n Let i f b e a Hilbert space endowed with a scalar product < •, • > and associated norm || • ||. Denoting by L a linear operator (possibly unbounded) and N a nonlinear operator with N(Q) = 0 , consider in H the initial-value problem ( ut + Lu + Nu = 0
I «(0) = «o
, . [
'
assuming t h a t : i) L is densely defined, closed and sectorial such t h a t (L — A / ) - 1 is compact for some complex number A, I being the identity operator in H (i.e. L has compact resolvent); ii) the bilinear form associated with L is defined (and bounded) on a space H*, which is compactly embedded in H. Then the eigenvalues A n of L $ = A$ can be ordered in a sequence {Xn}nGJN+ i?e(Ai) < Re{\2)
sucn
(2) that
< ... < i?e(A n ) < ...
461
(3)
462
The zero solution to (1) is said to be linearly stable iff ite(Ai)>0.
a
(4)
As concerns the nonlinear stability of the zero solution to (1) with respect to ||u||, if L is symmetric and
VuGD(JV)
(5)
D(-) denoting the domain of the associated operator, then the eigenvalues A„ are real numbers and it can be shown that Ai > 0 implies the global nonlinear exponential stability with respect to ||u|| and hence there is coincidence between linear and nonlinear stability conditions. When L is not symmetric, if L = L\ + Z/2, with L\ symmetric and _L2 skew-symmetric, under (5) the global asymptotic exponential stability with respect to [|it|| can be obtained under the condition Ai = principal eigenvalue of L\ > 0 and generally Ai ^ i?e(Ai). In this case, in order to reach the coincidence - instead of the energy ||u|| - generalized energy (i.e. Liapunov functionals V(u) ^ ||w||) have been introduced 1 - 2 . In the present paper we assume that —A is a part of L, i.e. L = L\ — A, with L\ linear. The aim is to show the advantage of determining and using Liapunov functionals linked - with their time derivative along the perturbations - directly to the eigenvalues of the operator L*(u) = Li(u) — au where a is the principal eigenvalue of —A. The plan of the paper is as follows. Section 2 is dedicated to the application of the Direct Method with functionals depending on the eigenvalues of L. In particular we put in evidence the conditions guaranteeing the local and global nonlinear stability. In Section 3 the coincidence between the linear and nonlinear stability conditions for a general reactiondiffusion binary system of P.D.Es. of spatial ecology is shown. Motivated by the fact that the lowest I/2-energy dissipated by diffusion is linked to the principal eigenfunction of —A, we introduce the operator L* and reach the coincidence by furnishing a special Liapunov functional linked - with its time derivative along the perturbations - in a simple direct way to the L* eigenvalues. Sections 4-6 are dedicated to the stability of the rest state in double diffusive convection in porous media. a
(4) implies that the zero solution of ut + Lu = 0 is asymptotically exponentially stable and the bacin of attractivity is H (global stability).
463
2. Liapunov direct method with functionals depending on the L eigenvalues Theorem 2.1. Let V — V(Ai, A2,..., A„) be a positive definite functional, equivalent to ||u||, such that along (1) it follows that = /(A 1 ,A 2 ,...,A„)H| 2 + *(u)
^
(6)
with: i) {A„} sequence of eigenvalues of L; ii) f real function such that ite(Ai) > 0 =» / < 0; Hi) *(u) = o(||w|| 2 ). Then the zero solution of (1) is locally asymptotically exponentially stable with respect to \\u\\. Proof. By virtue of assumptions exist positive constants e, ki (i = 1, 2, 3,4) such that ki\\u\\2
f<-k3,
o(H|2)<£4|N2+e.
(7)
In view of (6)-(7) it turns out that
kzk1+e Then, by recursive argument, it follows that VJf = —r-\—m with m < 1 K2K4
implies —— < — 6V with S — — (1 — m) and hence y dt ~ kiK ' V
||U||2<||K||2e-«.
(9)
Theorem 2.2. Let (6) and the assumptions i)-ii) of Theorem 2.1 hold. If |$|
(10)
with k positive constant, then the zero solution of (1) is globally asymptotic exponentially stable with respect to ||u||. Proof. (6)-(10) imply
^<
( h
_*
) M
.<_(^*)v
(ii)
hence V
||n||2<^||Uo||2e-'5lt, Ki
S, = Uh
- k).
(12)
fci
Theorem 2.3. Let (6) and the assumptions i) and Hi) of Theorem 2.1 hold. If f > m — positive constant, then the zero solution of (1) is unstable with respect to ||u||.
464
Proof. In fact one obtains
^>m||«||-*4H|1+e
(13)
and hence ^>aiV-a2V1+s
(14)
with <2j (i = 1,2) positive constants. Integrating one obtains V
>
^
r
t
,
limV->^,
W&.
(15)
3. Spatial ecology Let Q C iR3 be a bounded smooth domain. The nonlinear stability analysis of an equilibrium state in 0. of two-species population dynamics with dispersal, very often can be traced back to the nonlinear stability analysis of the zero solution of a dimensionless nonlinear binary system of P.D.Es. like 3 - 8 = aiCi - b2C2 + 7 i A d + / ( C i , C2)
m
(16) dC —• = hd
+ a4C2 + 72A<72 + g{Cx,C2)
with / and g nonlinear and a*, bi > 0, ji > 0 constants and /(0,0) = g(0,0) = 0,
d:
{x, t) G n x JR+ -> d(x, t) G iR; % = 1,2 (17)
under Dirichlet boundary conditions Ci = C 2 = 0
o n f f l x JR+
(18)
or Neumann boundary conditions (n being the unit outward normal to dQ) ^ i = ^ = an dn with the additional conditions
0
ondnxM+
[CidQ= [(C2dfl = 0 JQ.
Denoting by
HQ(Q.)
(19) v '
V t G JR+ .
JQ. in
and .ff«(fi) the Sobolev spaces such that
^ an
= 0 on dQ
(20)
465
where < •, • > is the scalar product in L 2 (fi), [HQ(Q)]2 and [ ^ ( f i ) ] 2 are the natural spaces in which the problem is embedded according to the case (18) and the case (19)-(20) respectively. Denoting by a and f3 two rescaling constants and setting {C\ = au, C2 = 0v}, in view of (16), we obtain ut = hu
b2v + f* + f (21)
vt = -jb3u + b4v + g +g with 1 h = 01 - 71 a , 64 = 04 - 72a, / = - / ( c m , /3v) a 1 3 = -3d(au, (3v), /* = 71 (Au + a ) , g* = 72(At; + a)
(22)
a being the lowest eigenvalue A of A
b2v
(23)
Vt = -5O3U + 0 4 V
having the eigenvalues (A
/ ± y/P - 4A
I = bx + 64 = Ai + A2, A =
6164 + 62^3 = A1A2) and introduce the special functional
v = \ A(\\uf + \\vf)
p
(24)
a
Along the solutions of (21) it turns out that 7 2 — = AI(\\u\\ AHM2 2 +- \\v\\ H-II^ ) + ^* + ^ dt
(25)
with ' if>* =< a.\ — 0J3U, / * > + < OLIV — CX3U, g* >
$ = < a i — a^v, f > + < a2v — a^u, g > Oil
A
2h
2
^b b ,a22=A + -M Ui i + bi,a
2
b + ^bi,, ^ + bf az
(26)
Pr
a3 = -r&i&3 - -b2b4 p a
The following theorem holds. Theorem 3.1. The condition 7<0 A>0
(27)
with b\bi > 0 guarantees the stability of the zero solution while the instability is guaranteed by either I > 0 or A < 0.
466
We refer for the details of the proof to [7] in the case of the Dirichlet boundary data and to [8] in the case of Neumann boundary data. In [8] a dynamical system more general than (16) and conditions on 7, (i = 1,2) avoiding 6164 > 0 are considered. The spatial ecology is also studied in [9]. In particular, in the last section of [9], it is clarified the motivation for which we replace the linear operator ai+7iA
-b2
\ . ,
03
04 + 72 A )
, by
T* _ (o-i-otli L* = -1 "
~h "*
n
\ b3
.
(28)
04 - 0 7 2 )
4. Double diffusive convection: introduction The equation governing the motion of a binary fluid mixture bounded by the horizontal planes z = 0, z = d > 0, in the Darcy-Oberbeck-Boussinesq scheme, are 9 | V p = - ^ v + p0[l-7T(T-To)+7c(C-Co)]g, \ ATt + v • VT = kTAT,
eC t + v • V C =
V-v = 0
kcAC
with 7x thermal expansion coefficient,7c solute expansion coefficient, e porosity, v seepage velocity, C concentration, p pressure, T temperature, \x viscosity, kr thermal diffusivity, kc solute diffusivity, c specific heat of the solid, A =
^-, c p specific heat of fluid at constant pressure, po {P0Cp)f
fluid density at reference temperature and concentration To, Co, respectively. The subscripts m and / refer to the porous medium and the fluid respectively. To (29) we append the boundary conditions TL = T0 + ^(T1-T2)
,CL = C0 + i ( d - C 2 )
on z = 0 (30)
Tu = T0~ i ( T i - T2) ,Cu = C0-
| ( C i - C2)
onZ = ,
with Ti > T2 and C\ > C 2 . By introducing the scaling J * x = dx*, p*_k(p
+ fikTp0gz) ^'
, Ad2,* t= - — t * , kr T-TQ T i - T a '^ T . =
kT „ v = -^v* d Cx-C0 2 C*=C-C
the dimensionless version of (29) and (30) - omitting the stars - are respectively f V p = - v + (iir-<7C)k, V - v = 0 \ Tt + v VT = A T , eLeCtt + Lev • V C = AC
l
j
467
TL = \,CL
= i o n 2 = 0;
with R = vkT normalized porosity, C —
Tv = ~ , Cv = -l- on z = 1
(32)
the thermal Rayleigh number, e = — the A ; the solute Rayleigh number, VKT
v = — the kinematic viscosity and Le the Levis number. Equations (31)Po (32) admit the steady solution (motionless state v s = 0) Vps(z) = -{R + C)(z-
±\ k, Ta(z) = C.(z) = -(z-±\
.
(33)
The stability of (33) - also in the case that the layer rotates about the vertical axis z - has been considered by several authors [10], [11], [12]. Precisely, denoting by u = (u, v, to), 9, V respectively the velocity, temperature and concentration perturbations fields and assuming that they: i) are periodic in the x and y directions respectively of period — and — , ii) verify f
f
CLX
Cy
27T
2-7T
/ ud£l = / vd£l = 0 on the periodicity cell Q, = [0,a —] x [0,a—] x [0,1], Jo. Jo. x y iii)] belong to L2(Q.) Vt € JR + ; conditions guaranteeing the nonlinear conditional exponential asymptotic stability of (33), with respect to the L2(fl)norm, have been obtained. Now I reconsider the problem. Denoting by L\ (Q) the class of perturbations (u, 0, T) verifying I)—iii), the aim is to show that, with respect to the £ 2 (ft)-norm, if and only if there is linear stability, one obtains global nonlinear stability. 5. Double diffusive convection: preliminaries The equations governing the perturbations (u, 9, V) - as it is easily seen in view of (31) - are 'V7r = - u + ( i ? 0 - C r ) k , V - u = 0 eLeTtt + Leu • V r = AT + w t + u • V0 = AG + w ,
, (
under the boundary conditions w
= 0= r = O
o n z = 0,l
(35)
with w perturbation to the pressure field. Since the sequence {sinn7rz}, (n = 1,2,...), is a complete system for ^([0,1]), by virtue of the periodicity, it turns out that for Z e {w, 6, T} exists a sequence {Zn(x, y,t)} such that oo
Z = ^r,Zn{x,y,t) n=l
sinnirz
Vi>0
(36)
'
468
with AiZn = -a2Zn,
d2 9x 2
Ai
d2 dy 2 '
2
o
2
(37)
Setting ( = (V x u) • k in view of (34)2, one obtains d2w dxdz
dC dy'
d2w dydz
„
d( dy
d2w , Aiu dxdz
On the other hand (34)i implies C = 0, hence Aiu d2w , , , . dydz and therefore one obtains OO
,
OO
u = / 5 „ ( a ; , y , £ ) —(sinmrz); 2
Aiw„ = -a un,
7
i> = / ^ £ n (x, y,t) — (sin n7rz)
dZ
n=l
n=l
Ai^
tii-i
=
(39)
"*
1 3u>n-, U- „ a2 dx
2~ . ~ CI Vn,
(38)
1 9tDn a 2 <9y
Then tu, 5, T appear to be the effective perturbation fields. These fields are not independent. In fact from (34)i, by taking the third component of the double curl, it follows that [ Aw = AX{R9 - LeCT), 01 = A6 + w - u • V0
V-u = 0 (40)
r,t = 4-(Ar + t«)--u-vr. e
the
Setting Zn = Zn sin(n7T2:), Zn€{0n,Tn,wn}
in view of (36)-(39) one obtains
1 d2wn Awn = -(n 2 7r 2 + a2)wn , u n = (un, vn, wn), a 2 dxdz ' 1 d2wn Ai^n = —a #„ , AiT„ = —a r „ . a2 dydz ' Therefore, setting j
n
(41)
= —, £„ = a 2 + n27r2 it follows that #„, T„ and
w« = 1n{Rdn - LeCTn);
u„
1 92u;„ 1 d2wn a2 dxdz' a2 dydz', wr
(42)
verify (V a 2 > 0 and V n G N+): the boundary conditions w„ = 9n = Tn = 0
on z = 0,1;
(43)
469 the restrictions of ii) and the equations (40)i - (40)2. Then - by virtue of oo
their linearity - the general solutions of (40)i, (40)2 are w = y"\tfn, u = n=l
y~]u n with wn, u„ given by (42). n=l
6. Double diffusive convection: stability - instability Setting hn=lnR-£,n,
hn = -jnLeC, '
F=
°°
/ RQ
InR In ( in 6 3n = ~ — , 64„ = — ~ ( LeC+ — eLe ehe \ 7„
(44)
\
bli0
b2iTi + U W
Y.\-^~ * ~ • *j G = E ( V ~ hiTi ~ hiTt + eU ' Vr'
(45)
oo
with u = y j u i , (40) become i=l
F = 0 G= 0
(46)
and imply < < F, sin(n7rz) » = 0,
< < G, sin(nTrz) » = 0 Vn e IV+
(47)
where < < •, • > > denotes the scalar product in L 2 [0,1]. Lemma 6.1. Let p,q,n E IV + . Then max(p,q) < n implies « s i n ( p 7 r z ) , sin(g?rz) » =
«
\ J [0
• , \ . , x / x fO sin(G7rz)sm(n7r,z), cos(p7rz) > > = < „ ,, v 11/4
, for
P q
~
(48)
p^q
for p + qj^n . for p + q = n.
,.n. (49)
Proof. (48) i immediately follows from < < sm(p7r,z),sin(g7rz) > > = < < -,cos[(p — q)-nz\ — cos[(p + q)-Jrz} »
.
470
By virtue of sm(qirz) sm(mrz) = — {cos[(n — q)irz] — cos[(n + q)~nz]} cos[(n — q)ixz\ COS(PTTZ) = - {cos[(n — q — p)irz) — cos[(n + p — q)irz}} cos[(n + q)-!Tz] cos(pirz) = - {cos[(n + q + p)irz] — cos[(n + q— P)TTZ]} for n,p,q £ IV + it follows that «sm(qTcz)sm(nnz),cos(pTrz)
> > = < < -,cos[(n + p — q)irz) +
+ cos[(p + q — n)irz] — cos[(n + q — p)irz] »
, .
.
By virtue of (50), (49) immediately follows. oo
Lemma 6.2. Let Z = 2_,Zpi Zp ~ Zpsixi(pirz) with Zp £ {9P, Tp}, p £ JV + . Then it follows that oo
«
^
y^Zp, sin(n7rz) > > = -
(51)
2
P=I
< < Up • VZ g , sm(mrz) »— 0 = < * 7T ((pp ~ \ I 4 1—Vwp-VZq + qWpZq)
for
p + q^n
for
p + q = n.
(52)
Proof. Lemma 6.2 is immediately implied by Lemma 6.1. Theorem 6.1. / / and only if either f e Le < 1 \R
(53)
or
eLe>\,
R<-+(l £
C>C* \
RB
(sLe — l)Le
(54)
+ 4eLeJ - ) RB
with RB = 4-ff2, (33) is globally L2-stable. Moreover then (33) is asymptotically exponentially Instable.
471 Proof. We have to show that (33) is L 2 -stable with respect to any perturbation {S£\ S%]}, with (meN+)
5W = E«*. sg> = 5yt,
um = j >
»=i
(55)
i=l
The general proof will be given in [10]. Here for the sake of simplicity, we consider the case m = 2, i.e. — ^ - = 5 > u 0 i + hiTi) - U 2 • V 5 f } dt
i=i
95.
at
(56)
2
= X>3ifc + M\)--u 2 -vs:(H i=i
On multiplying (56) by sin(n7rz) with n = 1,2 and integrating on [0,1], by virtue of L e m m a 6.2, it easily follows t h a t BO
BY
^ = bu0i+b2iT1, dt
at
~=b3101+b41T1
1
BR
-~=b2iS2+b22T2--
«
BY 1 -~ =b2Z92 + b42Y2- — « - ot 2e
U! • V0i,sin(27rz) »
sin(27r.z)
(57)
m • Vr 1 ,sin(27rz) > > sin(27rz).
Introducing the peculiar functional (i = 1,2) Vi = -[MPiW2
+ \\Tif) + \\buTi - bzAf
+ \\b2iTi - 64i04|i
(58)
with M = biiba — b2ib3i, , Ii = bu + ba
(59)
it follows that ( dV^ = ^ 1 / 1 ( | | 5 1 | | 2 + ||r 1 || 2 ) dt dV2 = J 4 2 /2(||^|| 2 + I|r 2 || 2 ) + * I dt
(60)
472
with ( * = < ai<92 - a3T2,F* > + < a2F2 - a362, G* > ai= A2 + b\2 + 6| 2 , a2=A2
+ b\2 + b\2, a3 = 612632 + 622642 (61)
F* — - - «
U! • V0i, sin(27T2:) > > sin(27r.z)
« u x • V r i , sin(27rz) » sin(27rz) ~2e Either (53) or (54) imply {Ai > 0, h < 0, i = 1,2} therefore by virtue of the equivalence, for Ai > 0, between the norms Vi and (||#i||2 + ||rj|| 2 ), via well known imbedding inequalities, positive constants ki (i = 1,2,3) with k\ < k2, can be found such that G*
flrW
^.<-2k2V2
+
2k3e-^V21/2
and hence 'Vi(i) < Vi(0)e- felt (63) V2x/2(t) < V 2 1 / 2 ( 0 ) e - ^ + T-^T(e~k^ - e- fc2t ) k2 — ki By virtue of (63) the exponential attractivity withouth conditions on the initial data immediately follows. The exponential global asymptotic stabilty is immediately reached by taking into account that (56) imply the L2continuous dependence on any bounded time interval. As concerns the instability, by virtue of
£lNI2, II^HElir,
(64)
from
~
= bn01 + b21ru
—± = b3191 + b41r1
Mr, = b\mQm + b2mTm — UTO • VS*L/ dt o,
= b3m0m + bim^m
U m • VS^, >
(65)
473 it follows t h a t the instability is guaranteed by the instability of the zero solution of (ddl —
h a -u/, r = buOx + b 2 l i 1 (66)
BV ~ = v at
b3i91+b41T1
i.e., either if Ii > 0 or Ai < 0. It is easily verified t h a t one of these conditions is satisfied when at least one of (53), (54) is not verified.
Acknowledgments This work has been performed under the auspices of the G.N.F.M. of I.N.D.A. M. and M.I.U.R. (COFIN2005): "Propagazione non lineare e stabilita nei processi termodinamici del continuo".
References 1. B.Straughan: The energy method, stability and nonlinear convection. 2nd ed. Springer, Appl. Math. Sci. Ser., New York 2004 . 2. J.N. Flavin, S. Rionero: Qualitative estimates for partial differential equations. An introduction. CRC Press, Boca Raton, Florida 1996. 3. O.A. Ladyzenskaja, V. A. Solonikov: Linear and quasilinear equations of parabolic type. Vol. 23, Transl. Math. Monogr. A.M.S., Providence Rhode Island 1968. 4. A. Friedman: Partial differential equations of parabolic type. Prentice-Hall Inc. London. 5. J. Smoller: Shock waves and reaction - diffusion equations. "Series of Comprehensive Studies in Mathematics" n.258, Springer - Verlag 1983 6. H. Amann: Quasilinear parabolic systems under nonlinear boundary conditions, A.R.M.A. 92, 186; Dynamic theory of quasilinear parabolic equations II. Reaction - diffusion systems, Diff. & Integr. Equ., 3, 1, 190; Dynamic Theory of quasilinear parabolic systems III. Global existence, Math. Z. Ang. 202, 1989. 7. S. Rionero, A nonlinear stability analysis for two-species population dynamics with dispersal Math. Biosc. Eng. 3, n.l 189 (2006). 8. S. Rionero, "L -stability of the solutions to a nonlinear binary reactiondiffusion system of P.D.Es." Rend. Accademia dei Lincei. Fasc.4, 2005 (To appear). 9. S. Rionero, "A rigorous reduction of the L -stability of the solutions to a nonlinear binary reaction-diffusion system of P.D.Es. to the stability of the solutions to a linear binary system of O.D.Es." J. Math. Anal. Appl. (To appear).
474 10. S. Rionero, "Global nonlinear L -stability for double diffusive convection in porous media with and without rotation. (To appear).
SYMMETRY ANALYSIS FOR THE Q U A N T U M D R I F T - D I F F U S I O N M O D E L OF S E M I C O N D U C T O R S
V. ROMANO, M. TORRISI, R. TRACINA Dipartimento di Matematica e Informatica Universita di Catania Viale A. Doria, 6, 95125 Catania, Italy E-mail: [email protected], torrisiQdmi.unict.it and [email protected] The symmetry analysis of the one dimensional quantum drift-diffusion model for semiconductors, based on the Bohm potential, is performed and example of exact solutions are given.
1. T h e m o d e l In the last years continuum models for the description of charge carrier transport in semiconductors have interested applied mathematicians and engineers on account of their applications in the design of electron devices. Simple macroscopic models widely used in engineering applications are the drift-diffusion ones 1>2 ' 3 , t h a t are given by the balance equation for electron density a n d / o r hole density coupled t o the Poisson equation for t h e electric potential. T h e classical drift-diffusion models have been thoroughly investigated from an analytical point of view 3 ' 4 ' 5 and several suitable efficient numerical methods have been developed 6 . Recently a group analysis has been performed in 7 . However with shrinking dimensions of submicron semiconductor devices, the q u a n t u m effects are no longer negligible. One way to include them is based on the Bohm potential. T h e resulting q u a n t u m drif-diffusion model 4 is given in the unipolar case by the system
f + V.^O,
(!)
A2A$ = n - c ( x ) ,
(2)
where n is the electron density, J the electron m o m e n t u m density, A2 the dielectric constant divided by the elementary charge e, $ the electric po-
475
476
tential and c(x) the doping concentration that is a function of the position x. V is the divergence operator and A the Laplacian. The system (l)-(2) is supplemented by a constitutive relation for the momentum density J, which is expressed as the sum of a diffusion, a drift and a quantum term J = -KV{na)+iin{V<S>
+ Q).
(3)
K is the diffusion coefficient, fi the mobility and Q is the Bohm quantum correction given by Q=H
0
^
(4)
where h
H0 =
2m* e with h the reduced Planck constant andTO*the effective electron mass. a is a constant satisfying a > 1. The case a — 1 is the quantum analogous of the isothermal flow which is the basic assumption of the classical drift-diffusion models. As usual we assume that K and /J, are related by the Einstein relation K = -Uon,
(5)
where UQ = is the constant thermal potential with ks Boltzmann e constant and Tj, lattice temperature kept at equilibrium. The mobility \x is considered to be a function of the modulus \E\ of the electric field E = - V $ , that is l* = H(\E\).
(6)
In the next section we study the group properties of the quantum-drift diffusion model, classifying the functional forms of mobilities and doping profile for which Lie symmetry transformations are admitted. Some example of exact solution is presented in the last section. 2. The symmetry classification in the one-dimensional case In the one-dimensional case the Poisson equation can be rewritten in terms of the relevant component of the electric field. Because it is n > 0, we put w2 = n and in the following we use the new variable w.
477 So, the system becomes a w 2 +'
dt
^dx= 0 ,
(7)
dE -A^_,»!-c(*) dx
(8)
with
We will get the symmetry classification of the system (7)-(8) by the infinitesimal Lie method 8>9.i°.n.i2 Our goal will be to determine the functional forms of mobility n and doping profile c for which the system (7)-(8) does admit symmetries. The infinitesimal generator of the symmetry transformation has the general form
Y = Zl{x,t,w,E) +r]1{x,t,w,E)
~+£2{x,t,w,E)
~+r]\x,t,w,E)
A
—.
The analysis of the invarianee conditions of the solutions of system (7),(8) with respect to Y leads to the following classification. The principal Lie algebra Lp is spanned by
while in the cases below one has also the following extensions whose generator is indicated by X\. Case 1 c = Co, M = Mo, Y
m
d
dx
^ W 9 /Uo dE
with co, /io constitutive constants and 7(f) an arbitrary function of t.
478 Case 2 C = Co,
/i = /ii(|-E|) arbitrary, d
x with
CO
ax constitutive constant.
Case 3 CO
c =
(cix + c2)
3 2' ci x + c2 9 d n X1 = ow ci dx with /Lto, Ci and c2 constitutive constants. a =1
**w
Case 4 Cp
-
(Cl X + C2)
a =
v 1
3 2 ' d ~~ at
with fi0, k ^
Ci x + c2 d ci(3H4)&;
—43|' ,
o
W
^ 3 H 4 9W
3
E
d
3fc + 4 3 E '
ci and c2 constitutive constants
In the last two cases we require E > 0. We remark that the case 1 and 2 occur also in the classical case, that is for H0 = 0. 3. An example of reduced system and exact solutions The reduce systems are obtained by introducing in the original system of PDEs suitable similarity variables, determined as invariant functions with respect to the infinitesimal generator of the symmetry transformation. So it is possible to find solutions of the PDEs by solving the ODEs. As example we consider the case 3 and show how to reduce the original 4
system of PDEs to ODEs. We set fi = noE~ 5 and c = ^ and recall that
a=§.
479
We use the generator £\
X!=x—
r\
-2w—
ox
r\
-3E—-. oE
dw
The invariance conditions lead to dx
dw 2w~
X
dE ZE
and the similarity solution is
w = -V xd where / and g are solutions for g ^ 0 of the reduced system / ( l 8 / V o % - 36fgH0^o - 3 / 5 V o - 2 / V / 3 ) = 0, 2
3gX - f + co = 0.
(11) (12)
In the case f(t) — constant ^ 0, one obtains the following stationary solutions: , v
iyi(x) =
_ , .
9t/0A2 + ^81U$\* *
c U o y /81U*\
£q(X) = b
y~2
4
- 36F 0 A 2 + c0
(13)
- 36ff0A2 + co + 9c/02A2 - 2 g 0 3
(14)
and , , 9C/0A2 - V81t/02A4 - 36iJ0A2 + c0 u>2 (a:) = -2
(15)
2 4 2 2 „ , . a -Uo y/81J70 A - 36g 0 A2 + c 0 + 9L^0 A - 2ff0 Y £ 2 (x) = 6 o •
(16)
Acknowledgments The authors acknowledge the financial support from P.R.A (ex 60 %) University of Catania and from INdAM through G.N.F.M.. V.R acknowledges also the support from by M.I.U.R. (PRIN 2004 Mathematical Problems in Kinetic Theories). R.T and M.T acknowledge also the support from M.I.U.R. (PRIN 2003/05 Nonlinear Mathematical Problems of Wave Propagation and Stability in Models of Continuous Media).
480
References 1. S. Selberherr, Analysis and Simulation of Semiconductor Devices, Springer, Wien, New York, 1984. 2. W. Hansch, The Drift-Diffusion Equation and its Applications in MOSFET modeling, Springer, Wien, 1991. 3. P. Markowich, C. A. Ringhofer, C. Schmeiser, Semiconductor Equations, Springer, Wien, 1990. 4. A. Jiingel, Quasi-Hydrodynamics Semiconductor Equations, Birkhauser, Basel, 2001. 5. J. W. Jerome, Analysis of Charge Transport, Springer, Berlin, 1996. 6. F. Brezzi, L.D. Marini, S. Micheletti, P. Pietra, R. sacco, S. Wang, Discretization of semiconductor device problems (I), chapter 5 in Handbook of Numerical Analysis Vol. XIII, Elsevier North-Holland, 2005. 7. V. Romano, J. M. Sellier, M. Torrisi, J. Phys. A 341, 62 (2004). 8. L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982. 9. P. J. Olver, Applications of Lie Groups to Differential Equations, SpringerVerlag, New York, 1986. 10. G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, New-York, 1989. 11. N. H. Ibragimov, CRC Hanbook of Lie Group Analysis of Differential Equations, CRC Press, Boca Raton FL, 1994. 12. W. I. Fushchych and W. M. Shtelen, Symmetry Analysis and Exact Solutions of Nonlinear Equations of Mathematical Physics, Kluwer, Dordrecht , 1993.
SYMMETRIES A N D R E D U C T I O N T E C H N I Q U E S FOR A DISSIPATIVE MODEL
M. RUGGIERI, A. VALENTI Dipartimento di Matematica e Informatica, Universita di Catania, viale A. Doria, 6, 95125 Catania, Italy E-mail: [email protected], [email protected]
Symmetries and reduction techniques are applied to a mathematical model describing one-dimensional motion in nonlinear dissipative media. Reduced equations through the optimal systems of subalgebras are performed and an application is shown.
1. Introduction We take in consideration the third order PDE wtt = / K ) wxx + A0 wxxt,
/ , A0 > 0,
/ ' ^ 0,
(1)
where / is an arbitrary function of its argument, Ao is a positive real parameter, w(t,x) is the dependent variable and subscripts denote partial derivative with respect to the independent variables t and x. Hereafter, prime denotes derivative of a function with respect to the only variable upon which it depends. Some mathematical questions related to (1), as the existence, uniqueness and stability of weak solutions can be found in ref. 1, while a study related to a generalized "shock structure" is shown in ref. 2. Recently, a symmetry analysis of the equation (1) was performed in ref. 3. Moreover, when Ao is considered as a small parameter, the approximate symmetries of the equation (1) were studied in ref. 4. The equation (1) can describe wave propagation in a nonlinear dissipative medium. In fact, it arises when one considers purely longitudinal motions of a homogeneous viscoelastic bar of uniform cross-section and assumes that the material is a nonlinear Kelvin solid. This model is described by the classical equation of motion (the constant density is normalized to 481
482
1 and the mass forces are neglected) Wtt = rx
(2)
and by assuming a stress-strain relation of the following form: T = a(wx) + X0wxt,
(3)
where r is the stress, x the position of a cross-section in the homogeneous rest configuration of the bar, w(t,x) the displacement at time t of the section from its rest position, <J(WX) the elastic tension (wx is the strain), Ao the viscosity positive coefficient. Taking (3) into account and setting / = a'(wx) the equation (2) assumes the form (1). Moreover, making use of the following change of variables: u = wx,
v = wt,
(4)
equation (1) can be written as an equivalent 2 x 2 system of the form ut = vx, vt = f{u)ux
(5) + X0vxx.
(6)
This system occurs in the more well-known setting of one-dimensional motion of a viscous isentropic gas, treated from the lagrangian point of view when, u corresponds to the specific volume, p(u) = Ju f(s)ds is the pressure and v is the velocity. Still, the system (5)-(6) can be viewed as the potential system 5 associated to the following equation: u
tt = [/(«) ux + Ao uxt]x .
(7)
In ref. 3 the complete group classifications of the equation (1), the system (5)-(6) and the equation (7) were presented and a theorem affirming that those classifications are identical in the sense that, for any / , a point symmetry admitted by any one of them induces a point symmetry admitted by the remaining two ones, was stated. In this paper, on the basis of that theorem, in order to find exact invariant solutions, we reduce through the optimal system of subalgebras, only one of (1), (5)-(6) and (7). Obviously, it is convenient to reduce the equation (1) because of, once obtained its solutions, simply deriving we can obtain the corresponding solutions of (5)(6) and (7). The plan of the paper is the following. In Sec. 2 we recall the main results of the symmetry classifications obtained in ref. 3. The optimal systems of one-dimensional subalgebras are constructed in Sec. 3. The reductions of (1) to an ODE through the optimal systems are performed in Sec. 4. Finally in Sec. 5 an application is shown.
483
2. On the symmetry classifications In this section we recall the main results of the symmetry classifications of the equation (1), the system (5)-(6) and the equation (7) obtained in ref. 3. The infinitesimal operator of (1) is chosen in the form ~+£2(t,x,w)
X = S\t,x,w)
JL-+ri(t,x,w)
-^
(8)
and the following result was obtained: for arbitrary / the Principal Lie Algebra L-p of equation (1) is four-dimensional and it is spanned by the operators X
-
9
X
-
9
X
d
Y
t
at ox aw otherwise the classification is summarized in Table 1.
d
0)
aw
Table 1. The group classification of the equation (1). fo, p and q are constitutive constants with / o > 0 , p ^ O . Case
/
Forms of
f(wx)
/ K ) = foe
Extensions of L-p
X5 = 2t-§-t+x£
P I
II
f&x) = fo (vix + q)p
X5 = 2t-§-t+x£
+
+
(w-2px)£
[(l~2p)w-2pqx]£
The infinitesimal operator of (5)-(6) is chosen as X = ^(t,x,
u, rf-fo+^it, x->u: v)-Q-+rl1(t,x,
u
> •v)^+??2^'x>M'u) av
_(io)
and the classification gives: for / arbitrary the Principal Lie Algebra Lp of the system (5)-(6) is three-dimensional and it is spanned by the operators
x Xl
-
d
~¥t'
x2 X
-
d
~a?
x x-
d
"-^
(11)
otherwise we obtain the results summarized in Table 2. The infinitesimal operator of (7) is taken in the form X = I 1 (t, x,u) — + i2(t, x,u) — + fj(t, x, u) —
(12)
484 Table 2. The group classification of the system (5)-(6). jo, p and q are constitutive constants with /o > 0, p ^ 0.
Case
Forms of f(u)
Extensions of L-p
/
f(u) = foev
*4 = 2 t & + * & - 2 p & - t ; &
II
/(u) = /o (u + q)i ,
X4 = 2t£ + x£-2p(u
+ q)£-(l
+ 2p)v-§;
and the result of the classification is: for arbitrary / the Principal Lie Algebra L-p of the equation (7) is two-dimensional and it is spanned by the operators
* ' - ! •
(13)
* - ! •
otherwise the group classification is summarized in Table 3. Table 3. The group classification of the equation (7). /o, p and q are constitutive constants with /o > 0, p ^ 0.
Case
Forms of f(u)
Extensions of L-p
J
/(u) = / 0 e ?
*4 = 2 t £ + x £ - 2 P £
II
/(«) = /o (u + q)p ,
I
X4 = 2t-§-t+x£-2p(u
+ q)£
Finally, the Theorem 2, which has been proven in ref. 3, reads: The classifications of (1), (5)-(6) and (7) are identical in the sense that, for any f, a point symmetry admitted by any one of them induces a point symmetry admitted by the remaining two ones. Then, on the basis of this theorem, in order to search for invariant solutions of (1), (5)-(6) and (7), it is not necessary to reduce all them, but it is more convenient to reduce only one of them. Obviously, it is convenient to reduce the equation (1) because of, once obtained its solutions, simply deriving we can obtain the corresponding solutions of (5)-(6) and (7).
485
3. Optimal systems of subalgebras To obtain all the essentially different invariant solutions (i.e. the invariant solutions that cannot be carried over into each other by the admissible transformations) we recourse to the concept of optimal system of subalgebras. This concept follows from the fact that, given a Lie algebra L of order r > 1 with G the corresponding group of transformations, if two subalgebras of L are similar, i.e. they are connected with each other by a transformation of G, then their corresponding invariant solutions are connected with each other by the same transformation. Therefore, in order to construct all the non similar s-dimensional subalgebras of L, it is sufficient to put into one class all similar subalgebras of a given dimension, say s < r, and select a representative from each class. The set of all representatives of these classes is called optimal system of s-dimensional subalgebras of L 6 . It is well known that for the one-dimensional subalgebras, the problem of finding the optimal system is the same as the problem of classifying the orbits of the adjoint transformations. In ref. 6 in constructing the one-dimensional optimal system the global matrix of the adjoint transformations is used. In ref. 7 a slightly different technique is employed. It consists in constructing a table (adjoint table) showing as the separate adjoint action of each element of the Lie algebra acts on all the other elements. In this paper, in order to construct the one-dimensional optimal system of subalgebras, we utilize both the adjoint table and the global matrix. We show the details of the analysis only for the Principal Lie Algebra L-p of (1). So, as a basis of the adjoint algebra Lp, we take the infinitesimal adjoint operators, namely Ai = -[Xi,Xj}^-,
( M = 1,2,..,4).
(14)
Taking into account of commutators reported in Table 4 Table 4. Commutator table of Ljy. The (ij)-th entry indicates [Aj, Xj\ = Aj y(.j — Xj AiXi Xi
xXi2
0 0 0
x4 -x3
x2 0 0 0 0
x3 0 0 0 0
x4
X3 0 0 0
each infinitesimal operator Ai generates an one-parameter group of linear transformations, i. e.
486
Ai = ~ X s ^
(15)
generates the linear transformations X1=Xi,
X2 = X2,
X3 — X3,
X4=X4—£iXs,
(16)
represented by the matrix
Mi On) = The linear transformations generate by each Ai can be deduced simply by inspecting the first, second, ecc. row of the adjoint table of L-p (Table 5). Table 5. Adjoint table of L-p. The (ij)-th entry indicates Ad(exp(siXi))Xj = Xj — £j [Xi,X,].
Ad X1
x2 xXi3
Xi Xi Xi
Xx X\ 4- £4X3
x2 x2 x2 x2 x2
x3 x3 x3 x3 x3
Xi
Xi — £lX3 Xi
xXi4
The global matrix M of the adjoint transformations is the product of matrices Mj(ej) associated to each infinitesimal operator Ai. For L-p we have
(11! S l 0 e4
Q\
0 0 - £ l 1,
In order to obtain the global action of operators Ai, (i = 1, ...,4) ply the matrix MT, transposed matrix of M, to an element of e. X = Xw=i al Xi. Actually, it is preferable to work with the a = (a1, a2, a3, a4). The coordinates of the transformed vector of a al=a},
a2 = a2,
a3 = a3 + e 4 a 1 - £ 1 a 4 ,
a* = a4
we apL-p, i. vector are (17)
and firstly we remark that these transformations leave invariant the components o 1 , a2 and a4 of the vector under consideration. These latter transformations give rise to the adjoint group GA of Lp. The construction of the optimal system of one-dimensional subalgebras of L-p can be carried out using a very simple natural approach. Namely, we simplify any given vector a = (a\, 02,^3, o-i) by means of transformations (17) and divide the
487
obtained vectors into nonequivalent classes; in any class we select a representative having as simple form as possible, this can be made through the choices as + £4 ai (a 4 / 0), or e 4 = — (ai ^ 0). (18) £\ a4
Oi
Hence, the non trivial operator of the optimal system of Lp is v v v v ^ 9 d (19) X0i = Ai + c 2 A2 + C4A4 = — + c 2 — + c 4 t - — , ot ox aw where c2 and c 4 are real parameters with c2 ^ 0. The other Cases I and II, taking the Tables reported in Appendix A into account, can be investigated in a straightforward way. Because of the optimal systems of their Lie algebras are an extension of the optimal system of L-p, we show in the Table 6, only the extensions with respect to (19). Table 6. Cases I and 77: extensions of the optimal system of L-p Case
Extensions
I
Xo2 = X5 = 21 & + x £ + (w - 2px) £
Ha
Ih
He
Xo2 = C3 X3 + X5 = 2i §-t + X £ + (C3 - QX) £;
Xo2 = c4 X4 + X5 = 2t §-t + x £ + (c4 t + 2w + qx)
Xo2 = X5 = 2t^+x£
+
£.
[(l-2p)w-2pqx]£
4. Reductions to ODEs One of the advantages of the symmetry analysis is the possibility to find solutions of the original PDEs by solving ODEs. These ODEs, called reduced
488
equations, are obtained by introducing suitable new variables, determined as invariant functions with respect to the infinitesimal generators. On the basis of the infinitesimal generator (19) of the optimal system of L-p and those reported in Table 6, we can construct the reduced ODEs of (1). We show the details of the analysis only for L-p . By applying the invariant surface condition, through (19), we obtain dt
dx
dw
. .
which gives the following similarity variable and solution, respectively w = ip(z) + -c^t2,
z = x-c2t,
(21)
where tfj must satisfy the ODE to which (1) is reduced by means of (19) A 0 c 2 V'" + ( c i - / ) V " + C4 = 0,
(22)
with / an arbitrary function of ip'(z). The other Cases in Table 6 can be investigated in a straightforward way and the result is summarized in Table 7. 5. An application The optimal system of subalgebras gives the possibility to construct the optimal system of solutions 6 . In fact, two solutions wi and w-i are said essentially different 6 with respect to a Lie group of transformations G, if «2 does not belong to the orbit (wi, G) (that is the set of the solutions generated by transforming u\ through G) or, of course, if w\ <£ (ui2,G). Starting from this definition it is possible to separate the solutions, invariant with respect to G, in classes of equivalence. The list of these classes is called optimal system of solutions. Moreover, we recall that a solution of a PDE, non invariant with respect to its Lie group of transformations G, is mapped by a transformation of G, in a non invariant solution. Hence, the knowledge of the optimal system of subalgebras permit to generate new solutions starting from invariant or non invariant solutions. As an application, we consider the reduction presented in the Case IIa of the Table 7. By setting c 3 = ^Ao,, an exact solution of the corresponding reduced equation of (1) is
m
z2
= UK-
(23)
489 Table 7. Reductions of the equation (1) to an ODE. Case
Similarity variables and solutions
Reduced ODEs
>~i
z — -2-
w = \/tij){z) — p x log t
- 4 / o V " e P +ztp'
z 2 V" - 4 /o i/>'2 ip"
Z = -2= Ha
- V + 4pz = 0
Vt
+ 2 A 0 ( z V ' " + 2V") + 3 z V ' - 2 c 3 = 0
w = ip(z) + C3 logVt — gx z — -2-
Z2V/2l/>"-4/oV"
w = t if)(z) + | C41 log t — q x
+ 2 A o z V ' 2 V ' " ' - ^ V ; ' 3 + 2c4i/'' 2 = 0
Hb
z 2 V"-4/o(V'')^V'" z— — + 2 A 0 [zi/ ) '" + ( 2 p + l ) V " ]
He w
t(l-2p)/2^z-)_qx
=
+ (4p + 1) z V' + (4p 2 - 1) V = 0
Coming back to the original variables and taking (23) into account, the solution can be written as w
eVfot
H—-= log t - q x, vTo
(24)
So that, the solution of the system (5)-(6) is 2x u =
T=eVfot
x-r-2
e/fot
(25)
- q, A0 2
vTtot'
(26)
moreover (25) is also the solution of the equation (7). Other invariant solutions can be obtained by integrating the ODE (22) in which we must introduce the specific functional form that / assumes in the Case IIa, namely / = /o (wx + q) • Therefore (22) assumes the
490
following form: Ao c2 V/" + \c\ - / 0 ( ^ ' + q)2} r
= 0,
(27)
where we have set C4 = 0. So that, another family of invariant solution in the case under consideration is
V3A0 .
J,
vh
i^*
V k e x°
w = —j=r- log
Idl
4l\
+ e x°
— e xo
V
—
/
/ V3c2
\
— - = - + q ) x.
V
Vfo
(28)
J
with k a positive arbitrary constant. Hence, the solution of the system (5)-(6) is \/3c2 u = —==-
e*o ,
===== - q,
V k e x°
V3 v = — -'
fo
(29)
+ e *»
2
(30) /
ke
2c2 1 x
2c|f x
o + e °
These solutions are an example of exact solutions in the field of nonlinear dissipative media. Apart their own theoretical value, they can be used as benchmark test for numerical schemes and codes. Moreover, because of known exact solutions in viscoelasticity are very few (see ref. 8 and bibliography therein), these ones found in the present paper give a contribution to the literature in this field. Acknowledgements The authors acknowledge the financial support by MIUR-COFIN 2003/05 through the project Nonlinear Mathematical Problems of Wave Propagation and Stability in Models of Continuous Media and by GNFM of INdAM through the project Simmetrie e Tecniche di Riduzione per Equazioni Differenziali di Interesse Fisico-Matematico.
491
Appendix A In this Appendix are reported the commutator and adjoint tables of the Cases / and II. Table 8.
Commutator table of Case I. x-,.
*1 0 0 0
Xl
x2
*3 X4 Xb
-x3
-2X]
-X2
+ 2pX3
Table 9. Ad
x2 x2 xX 2 2 x2
*1 X-i Xl *1
*1 X2
xX3
Xl +SiX3 e2e5 Xj
4
Xb
ee5 X 2 - 2 p e 5
Table 10. x-, x2 x3 x4 x$
-x3
~X2
x2 x3 x4 X5
X-, Xl X! Xl
Xl+£4X3 e2e5Xi
e
*3 -Xi 0
x4
x3
x3 x3 x3 x3 e x3
xs
X4 x
i —
e\X3
Xi X4 X4
-"5
X5 - 2 £ l X , - e2 * 2 + 2 p e 2 X 3 X5 — £3X3
X5 + e 4 X 4
e~ e 5 x 4
e 5 x3
*5
Commutator table of Case II. x3
Xi
0 0 0 0
+ 2 p q X3
Table 11. Ad Xl
X2 — 2 p X3
Adjoint table of Case / .
0 0 0 0
-2Xj
*K 2X1
X3 0 0 0
-x3
x2
0 0 0
Xi
x4
X3 0 0 0 0
0 0 0 0
x2 x2 x2 x2 x2
-(l-2p) *3
0 0
(l + 2 p X 4
0
Xi
X4 -
X3
, ^3 e(l-2p)e5
£!X3
Xi Xi Xi
x3 x3
3
«!
X2 - 2 p g X 3 (1 - 2 p ) X 3 -(l + 2p)X4
Adjoint table of Case II. x3
e 5 x 2 - 2 p g e 5x
Xa
x03
x3
e
-(l+2p)e5x
x
Xr, ^5-2£lXl 5 -«2X2 + 2pge2X3 X 5 - (1 - 2 p ) £ 3 X 3 X 5 + (1 + 2 p ) e 4 X 4 X5
492
References 1. J. M. Greenberg, R. C. Mac Camy, V. J. Mizel, J. Math. Mech. 17, 707 (1968). 2. A. Donato, G. Vermiglio, J. de Mecanique Theor. Appl. 1, 359 (1982). 3. M. Ruggieri and A. Valenti, Proceedings of MOGRAN X, N. H. Ibragimov et al. Eds., 175 (2005). 4. A. Valenti, Proceedings of MOGRAN X, N. H. Ibragimov et al. Eds., 236 (2005). 5. G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, New-York (1989). 6. L. V. Ovsiannikov, Group Analysis of Differential Equations, New York, Academic Press (1982). 7. P. J. Olver, Applications of Lie Groups to Differential Equations, SpringerVerlag, New York (1986). 8. K. R. Rajagopal and G. Saccomandi, Q.Jl. Appl. Math. 56, 311 (2002).
S Y M M E T R Y ANALYSIS A N D LINEARIZATION OP T H E ( 2 + 1 ) D I M E N S I O N A L B U R G E R S EQUATION
M. SENTHILVELAN Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirapalli 620 024, India Email: [email protected] M. T O R R I S I Dipartimento
di Matematica Viale A. Doria, Email:
e Informatica, Universita di 6, 95125 Catania, Italy [email protected]
Catania,
We investigate group invariance properties of a (2 + 1) dimensional Burgers equation. We show that it is one of the higher dimensional nonlinear partial differential equations which does not admit Kac-Moody-Virasoro type sub-algebras. Through Lie symmetry analysis we derive a wide class of interesting solutions. Further, we deduce a transformation which maps the (2 + 1) dimensional Burgers equation to a (1 + 1) dimensional linear partial differential equation.
1. I n t r o d u c t i o n During the past three decades considerable progress has been made in understanding the mathematical properties of higher dimensional nonlinear partial differential equations (PDEs) [1]. In particular several (2 + 1) dimensional nonlinear evolution equations have been shown to be integrable and possess a rich variety of solutions. To explore these solutions several approaches have been adopted. The most widely used techniques are Inverse Scattering Transform, Hirota method, Painleve analysis, separation of variables, Lie group analysis and direct method [1-7]. Among these the Lie group analysis is playing a vital role. The method provides symmetries, Lie algebras, similarity reduced PDEs and their underlying geometry [8-10]. As far as (2 + 1) dimensional nonlinear PDEs are concerned the method has been used not only to explore the solutions but also to point out the existence of infinite dimensional Lie algebras in particular Kac-Moody-Virasoro type algebras. In this direction it has shown 493
494
that many integrable (2 + l)-dimensional nonlinear PDEs admit infinitedimensional Lie algebras, often of the Kac-Moody-Virasoro type. Typical examples include Kadomtsev-Petviashvili equation, Davey-Stewartson equation, Nizhnik-Novikov-Veselov equation, nonlinear Schrodinger equation introduced by Fokas and the (2+1) dimensional sine-Gordon equation. However, there are certain (2+1) dimensional integrable PDEs though admit infinite- dimensional Lie algebras, do not admit Virasoro-type algebras. Examples of this category are breaking soliton equation and the nonlinear Schrodinger equation introduced by Strachn [11]. In this paper we consider the following (2+1) dimensional generalized Burgers equation [12]
Uf + UXy + UUy + UX /
Uydx'
= 0.
(l)
Eq. (1) reduces to the Burgers equation when y = x. It describes wave propagation in (2 + 1) dimensional space. Eq. (1) passes Painleve test and admits several soliton type solutions [12,13]. In this paper we perform Lie symmetry analysis to (1) and point out that it does not admit Virasoro type sub-algebras eventhough it is integrable. Further, we show that it is transformable to one dimensional linear partial differential equation. The plan of the paper is as follows. In the following section we study Lie symmetries and Lie algebra of Eq. (1). In Sec. 3, we construct general similarity reduced (1 + 1) dimensional PDEs and some special solutions. We investigate the group invariance properties of the similarity reduced PDEs and deduce general similarity solutions in Sec. 4. We perform the linearization in Sec. 5. Finally, we present our conclusions in Sec. 6.
2. Lie algebra To study classical Lie symmetries of Eq. (1) we introduce the transformation uy = vx so that (1) can be rewritten as ut + uxy + uvx + vux = 0,
uy~vx.
(2)
The invariance of Eqs. (2) under the one parameter Lie group of infinitesimal point transformations leads to the following expressions for the in-
495 finitesimal components, 6 = —citx - c2x + a(t),
& — -city - c3y + c4t + c 5 ,
6 = ~(c2 + c3)t - cit2 + c 6 , cj>i = (at + c2)u - ay + c4, 4>2 — {cit + c3)v-cix
+ a(t),
(3)
where ^ s and <j>'jS, i = 1,2,3 and j = 1,2, are the infinitesimal components associated with the variables x,y,t and u,v respectively. In the above, c^s, i = 1 , . . . ,6, are arbitrary constants, a(t) is an arbitrary function of t and dot denotes differentiation with respect to t. The presence of an arbitrary function a(t) in the infinitesimal symmetries leads to an infinite dimensional Lie algebra. The general element of this Lie algebra can be written as a linear combination of the following infinitesimal generators, namely, d
Vi = —, ay
T,
d
V2 = —, at
,r d 8 T, d V3=t— + -—, Vi = ~xay au ox
v
° = ~xth - ytH - t2li+
d
d t—+u—, at au
iut y ) + {vt x)
- l
(4)
- i-
The non-zero commutation relations between the vector fields are given by [V1,V5] = -V1,
[VUVQ\ = -V3,
[V2,V5} = -V5,
[V2,V6}=V4 + V5,
[V3, V5] = -V3, [V5,V6] = -V6,
^2,^3] = - ^ ,
[V4, V6] = -V6, [V5, V7\ = V7(-ta(t)),
[V2,V4] = -V2,
[V2,V7} = V7(a),
\V3,V4]
[V4, V7] = V7(-ta(t)
=-V3,
+ a(t)),
[V6,V7] = V7(ta(t) - t2a(t)). (5)
The commutation relation between V7{a\) and V7(a2) turns out to be [V7(a1),V7(a2))=0
(6)
which is not of Virasoro type. 3. General similarity reductions While solving the characteristic equations associated with the infinitesimal symmetries we observed that they can be integrated only under the following three different parametric restrictions, namely, (i) (c2 + c3)2 — 4C\CQ =
496
0, (ii) (C2 + C3)2 — 4cic 6 > 0, (iii) (C2 + C3)2 — 4cic 6 < 0. We consider each case separately and derive t h e similarity variables. As t h e calculations are straightforward, in t h e following, we omit some details. Case 1 (02 + c 3 ) 2 = 4ciC6
fHt)
J
fHt')
g(t)fHt)
J
g(t')fHt')
where 9{t)
= 6 X P [(c^+^t)
'
'<*> = (ce + (c2 + c3)t +
cj
and
^ C ) = - C 1 (C-/ 1 /
J^^tfAtf)_/*('>*
- / ' ( /
g(t")fHt")
J
9{t)
dt/_ r* df ~ J 9(f) g(t')fi(t')' f Hi')' C4
v(v,0 = -ciV + vgWfHQ+Cif a(t)g(t)
fHt)
(f
^ ^ d A
r (c2+Clt')a(t')g(t')
+ /
V,.,
J
dt'
dt^
-
fHf)
(7)
Case 2 (c 2 + C3)2 - 4cic 6 > 0
77 = F(i)x + y c =
a(t')F(t>) 2
(dt' + ht' + c6)
dt,
V , / " ( ^ + c 5 )F(tQ ^ ( d ^ + fcjt + c e ) ^ ) J ( C l i' 2 + Ait' + c 6 ) 2
where (C2 - c 3 )
^
_
1 2
/ 2cit + fci + fc2 \
(cii + fcit + c 6 )5 ^2cit +
fci-*2>/
&1 = (C2 + C 3 ), fc2 = (c 2 + C 3 ) 2 - 4C!C6) 2 ,
2fc2
'
497 and t I
II
Kv, 0 - -ciC* + -^rrr + c i F(t) J
rt"
\J
(c4t" + c5)dt" F(t")(at"2
+ kit" + ce)2,
dt'
1
f* +C4 / — J F(t')(at'2
at, + h? + c6) t / rt" a(t")F(t") -dt" v{r],0 = -ci?7i + ci / I / {dt"2 + kit" + ce) +(cif 2'"••"~)F{t)v-f. + kit + ce) Case 3 77 =
dt'
.(8) ^ ' ^ (cii' 2 + M ' + C6)F(t')
(c2 + c 3 ) 2 - 4cic 6 < 0 F(t)a: + I
a(t')F(t') -dt', (at' + ht' + c6) 2
y , f {cjt' + c6) 2 2 F(i)(cit + fcii + c6) 7 F(i)(cii' + M ' + c 6 ) 2
, '
where exp(fe£f)tan-^(2^±^)) F(t) =
(cii 2 + fcit + c 6 )5 fci = (C2 + C3), fc2 = (4cic 6 - A;2)2,
and
M=
*
+Cl
W) f{f
-Ci(t
+ C4
f
v(V, 0 = -C\T}t + CX
(c4" + c'l) -dt" F(t")(at"2 + kit" + ce)2
F{t'){cit'2 + kit'+
c6)
dt,
[f f a(t")F(t") J J (Clt"2 + ht" + ce)
+{cit2 + ht + ce )F(t)v
+J
dt'
dt
d(t')F(t') -dt'. (at' + kit' + c6 2
(9)
498 Substituting the similarity transformations (7) into (2) we obtain the following (1 + 1) dimensional PDE system uvc + (u- c3C)u^ + (v - c277)'U7) - c2u + c\Ce( — 0,
u^ - vv = 0. (10)
One may note that the similarity variables (8) and (9) also provide exactly the same equation (10). The results of the above derivations are quite general. However, for certain special values of the parameters the similarity variables could loss their present forms. These special cases are to be treated separately. We consider some of these cases in Sees. 4 and 5.
3.1. Special
solutions
In this subsection we show that one can generate some nontrivial solutions for the system (2) from a trivial solution of (10) through the similarity transformations (7)-(9). For example, let us consider c& = 0 in (10). In this case the latter possesses the trivial solution u = v = 0. One may note that the condition CQ = 0 imposes a further restriction C2 = —C3 in the Casel. Substituting u = v = 0 and Ci = —C3 in (7) we obtain the following solution for the PDE, (2), namely,
ciy /(*)
cl9{t) r* {at' - c5) ^ _ ci9(t) tl dt' fHt) J 9(f) fHt') /*(«)•/ 9(f) /§(*')
( f f' ( ^ ' - ^ / U ci9(t) l9(t)f*fU
•i(t)J fi(t)J
[J \J
g(t")fHt")
„ = _£!£ + . Cl /"* «(*')»(*') ^ /(*) 9(t)fi(t)J fHf)
wJ^'
Cl
•Ht)J 9(t)fHt)J c ^
g(t)fHt)J
{* / /•*' a(t")g(t")
[J \J
fHt")
fHc2+ct')a(t')9(t')
fHf)
_ dt"
dt' ^
)
yK
'
= c
(^)>
f{t)
=
cit,_
n )
In the Case 2 we observed that there is no restriction on the parameters C2 and C3 when c§ = 0. As a result we obtain another class of solution of the form
499
CitFjt) ( Cl t + M + C6)(V2)y 2
-c4F(t)f cAtF{tXx F(t)
fl (c4t' + c5) J FWicit12 + kit'+ <%)&*) df 2
F(t'){Clt>
56if)F{t)^_
F{t)
+ M ' + c 6 )' f
F{t)J
F(t')
/ f \J
apHn ,„) 2 ( t o " + to" + ce)
M
'
Since the parameters C2 and cs are real it is not possible to extract any solution for the Case 3. 4. Group invariant solutions The invariance of (10J under the one parameter Lie group of infinitesimal transformations leads to the following symmetries £i = -{car] + cb), £2 = ca(, fa = cau, fa = -(c2cb + cav),
(13)
where ca and cb are arbitrary constants. In the above, &, fa, i = 1, 2, are the infinitesimals associated with the independent and dependent variables respectively. The corresponding Lie vector fields are
Solving the characteristic equations (by assuming ca / 0) associated with the symmetries (13) we obtain the following similarity transformation Z = C{CaV + cb),
U = Cwi(z),
V=
.
(15)
which transforms Eq. (10) to the following system of ODEs a2 cazw'( + zwiw[ + 2caw'1 + w1 + W2w'1 — a.w\ — azw[ + — = 0, w'2-zw'i-wi
=0,
a = (c2 + c3), (16)
where prime denotes differentiation with respect to the new independent variable z. Integrating the second equation in (16) we get W2 — ZW1+I1,
(17)
500
where I\ is an integration constant. Substituting (17) into the first equation and integrating the resultant equation we obtain zw[ + zw\ + ((1 + 7 0 - (c2 + c3)z)
+ ( ( C 2 ^ ° 3 ) z - I2) = 0,
Wl
(18)
where I2 is the second integration constant and we have taken c\ = 1 for convenience. Now, introducing the transformation w\ — ^TJ) , eq. (18) can be brought to a linear second order ODE of the form
<17 + W )+ +(( _ | + j>"M -((<* + « ) - ^ ^ )^ »'(*>
fci±2!>!WH 0.(19)
The general solution of (19) can be written in terms of confluent hypergeometric function. However, certain interesting special solutions can be extracted by appropriately choosing the arbitrary constants. For example, let us choose, h — 0, h — 1 and (C2 + c3) = 2. Then the resultant ODE can be shown to admit the following general solution p(z) = (I3+I4logz)ez,
*
(20)
where .Z3 and I4 are integration constants, which in turn provides an explicit solution for w\ and w^ of the form, z(I3 + J 4 log z)
\
z(I3 +
hlogz)J
respectively. Substituting (21) into (15) we get a solution for the similarity reduced PDE (10) of the form
h u = C 1 + C(ri + cb) (h + h log(C(c8r? + eg))) v = (rj + cb)
C(T? + Cb) (I 3 + h log(C(c8r) + co))).
C2Cb-
(22)
Going back to the original variables we arrive at the following solution for the (2 + 1) dimensional Burgers equation
=
_ ciy_ ciff(t) r faf - c5) _ C4g(t) r /(*)
fHt) J 9(f)fHf)
c\g(t)
/•' / /•*'
(c4t"-c5
'fHt) J \J 9(t")fHt'9(t)
c+ fHt)
fHt) J g(t' dt"
dt'
h C(r) + cb) (h + hlog{((r] + cb)))
dt'
501
v=
_c,faifWldt, 'Ht)J g(t)fi(t)J
+
1
/iff) fi{t')
g(t)fi(t)J g(t)fi(t)J
fWliffltf /*(*')
h &(*)/* (<)fo + C6)(/3+/4l0g[Cfo + C6)) g(t)fHt)J
\J
/§(<")
(?? + ct) _ aft) <#)/*(*) /(*)
y
/(*)
^
;
We mention here that the solution (23) exist in all three cases of Sec. 3. 5. Some special similarity reductions Restricting some of the parameters, c\s in the infinitesimal symmetries one can derive a large class of particular solutions including the travelling wave solution. In the following we discuss one such interesting case. 5.1. Classical self-similar
solutions
We note that this case is a sub-case of Case 2 of the general similarity reduction discussed in Sec.2. Choosing c2 = 1, a(t) = c\ = C3 = C4 = C5 = CQ = 0 in the similarity variables we get uvc, + w£>£ + vv,£ — (u^ — u = 0, un — t>£.
(24)
where '7 = 2/. £ = 7 >
u = wi
> u = u.
(25)
Eq. (24) is invariant under the following infinitesimal symmetries £l = ~Ca, & = Cfe, 01 = 0 , 02 = Cfc.
(26)
Solving the characteristic equation associated with the infinitesimal symmetries (26) we obtain the following similarity variables, that is, z = T)+—C,
u = wi{z),
v = t + w2(z),
(27)
Cb
which transforms (24) to the following ODE w" + —wiw[ + w2w[ + —wi = 0, w[ - —w'2 - 1 = 0. c
a
Ca
(28)
Ca
Integrating the second equation in (28) and substituting it into the first equation we arrive at // , 2cfc / w{ -\ wiw1 Ca
c
b
1 hcb , cb zwx w1 Wi = 0,
Ca
Ca
'
Ca
. (29)
502
which can be again transformed to the Bessel equation. However, the subcase I\ = 0 provides us an explicit solution exp{I2 + Wi =
W2
^z)z
I
*+(t)SexP^ + £;z)zdz
\ca)
h + fjexp(h
{M)
+ %z)zdz•
Rewriting (30) in terms of original variables we obtain an exponential solution for the (2 + 1) dimensional Burgers equation. 6. Linearization In this section we show that the system (2) can be transformed to a (1 + 1) dimensional linear partial differential equation. To illustrate this we consider the following infinitesimal symmetries
£i = x, £ 2 = y , 6 = 0, i = - u , cj>2 = v.
(31)
Using (31) we can transform Eqs. (2) to the following form: u^ + t]um + rjuvrj + rjvuv = 0,
un — v — r}vn = 0,
(32)
1
where ri = xy, (; = t, u =
, v = xv(ri,Q.
(33)
X
Integrating the second equation in (32) we get (34)
u = rjv + 7 ( 0 , where 7(C) is an arbitrary function of £. equation we obtain
Substituting it into the first
V( + r]vvv + 2r/vvv + 2vv + v + ^vv -\ 7' = 0 , where prime denotes differentiation with respect to (. rewritten in the form
(35)
Eq. (35) can be
V{ + (v + r)vv + rjv2 + "f'ln(r)) + ~fv)n = 0.
(36)
By introducing an auxiliary variable w(rj, Q eq. (36) can be split into wv=v,
W£ = -(v + r]vv + rjv2 +y'ln(rf) +jv).
(37)
503
Seeking Lie symmetries to this system we find (37) is invariant under the infinitesimal transformation characterized by 6=0,
6 = 0 ,
fo
=-cd
+F(r,,()e-W, (38)
where F(r}, £) is a solution of the following linear PDE r]Fm + (1 + 7 ) ^ +Fi+7'
ln(r?)F = 0.
(39)
The vector field associated with the infinitesimal components (38) is V = [-vF(V, C) + Fv(r,, C)] e " * ^ + F(T?, C K ^ .
(40)
Taking into account that (40) generates an infinite dimensional algebras, in agreement with the algorithm given in [8], we construct the following overdetermined system of PDEs for the unknown function $(r?,C, v,w), that is, - « $ 5 + $c = 0, $ c = 0.
(41)
Let us choose two particular solutions of eqs. (41) as new independent variables z\ and z2 zi=V,
Z2=(-
(42)
The new dependent variables can be obtained from the solution, tpi(v,w) and ip2(v, w), of the following system of equations (-v^+iPl)e-w
= l, $ = 0 , - i J ^ + ^ = 0 ,
e - ^ ? = l.
(43)
Solving (43) we get w = logil)i,
v=—,
(44)
where ipi and ip2 are new dependent variables. Substituting (44) and (42) into (37) we can transform the latter into the following linearized system ipiz, = 1P2, ipiz2 = -(V>2 + ziip2zi + i>\.l'{z2) ln(zi) + 7(22)^2)- (45) Choosing 7 = — 1 in (45), we get fe2 +zi^2zlZl
=0.
(46)
504 7.
Conclusions
In this paper we have studied the group invariance properties of the (2+1) dimensional generalized Burgers equation. We have pointed out t h a t Eq. (1) does not admit Virasoro type algebras eventhough it is integrable. As we mentioned in the introduction the (2+1) dimensional breaking soliton equation and nonlinear Schrodinger equation introduced by Strachn are the only equations which do not admit Virasoro type algebras in higher dimensions. Equation (1) is another example in this category. Further, we have explored several interesting solutions for this equation which are all new t o this problem. In addition to t h e above, we have shown t h a t (2+1) dimensional Burgers equation can be written as a (1 + 1 ) dimensional linear partial differential equation. Acknowledgments T h e work of MS forms part of a Department of Science and Technology, Government of India, sponsored research project. T h e work of M T is supported by P.R.A (ex 60 %) of University of Catania, by INdAM through G.N.F.M. and by M.I.U.R. ( P R I N 2003/05 Nonlinear Mathematical Problems of Wave Propagation and Stability in Models of Continuous Media). References 1. M. J. Ablowitz and P. A. Clarkson, Solitons: Nonlinear Evolution Equations and Inverse Scattering, (Cambridge University Press, Cambridge, 1991). 2. V. B. Matveev and M. A. Salle, Darboux transformations and solitons, (Springer, New York, 1991). 3. J. Weiss, M. Tabor and G. Carnevale, J. Math. Phys., 24, 522 (1983). 4. S. Y. Lou and L. L. Chen, J. Math. Phys., 40, 6491 (1989). 5. S. Y. Lou, Phys. Lett, 277A, 94 (2000). 6. M. L. Wang, Phys. Lett, 199A, 169 (1995). 7. E. G. Fan and H. Q. Zhang, Phys. Lett, 246A, 403 (1998). 8. G. W. Bluman and S. Kumei, Symmetries and Differential Equations, (Springer-Verlag, New York, 1989). 9. P. J. Olver, Applications of Lie Groups to Differential Equations, (Spring, New York 1986). 10. N. H. Ibragimov, CRC Hanbook of Lie Group analysis of Differential Equations, (Boca, Raton, 1996). 11. M. Senthil Velan and M. Lakshmanan, J. Nonlinear Math. Phys. 5, 190 (1998). 12. Z. Yan, Chinese J. Phys., 40, 203 (2002). 13. Z. Yan, J. Phys. A. Math. Gen., 35, 9923 (2002).
T R A N S F E R P R O P E R T I E S OF ELASTIC MATERIALS W I T H T H I N LAYERS
M. P. SPECIALE1, M. BROCATO2 Department
of Mathematics, University of Messina, [email protected], Ecole Nationale Superieure a"Architecture de Versailles, maurizio. brocato@mac. com
We study the transfer properties of a material body made of thin and thick layers of two different elastic constituents. The body is modelled under the assumption that layers in the first family have almost vanishing thickness when compared with those in the second family, as proposed in Ref. . Elastic waves are considered following the linearized dynamic model presented in Refs. and in two dimensions, and the study of the transfer properties of the system is made through the transfer matrix technique given in Refs. ^' .
1. Introduction We study the transfer properties of layered periodic materials made of thin and thick layers (denoted by B and A respectively) of two different elastic and isotropic constituents, the softer filling the thin beds, the thickness (denoted by 26) of which is much smaller than that of the next layers. For this purpose we call upon the model proposed in Refs. x ' , and use the transfer matrix technique of Refs. "*' , to compare results obtained for thin beds of different stiffness. In this case the overall transfer matrix is obtained by the composition of the matrices pertaining to the different layers. Thus the obtained expression can be simplified in the present circumstance thanks to the assumption of vanishing thickness of the inner bed. Attenuation and amplification of the anisotropy due to the layered structure are thus shown to depend on the ratio of the elastic parameters relative to the two families of layers. The result can be applied to the study of wave propagation in materials such as masonry or layered rocks, especially for the inverse identification of softening of the materials filling the thin beds, which can be induced, for instance, by damage. In section 2, we present the model and the solution of the wave propagation problem; then, in section 3, we build the transfer matrices and make
505
506
the dispersion equation explicit for our system. In the last section we show some numerical results obtained when the material parameters are chosen to represent masonry. 2. Position of the problem The model we call upon starts from a microscopic description of a system compound of one thin layer and the two half thick layers next to it. This is a microscopic point of view if one considers that the details of the solution within the thin layers can be disregarded when bodies including hundreds or even thousands of such layers have to be studied. The model is given by the Cauchy equations of motion — with boundary conditions assigned at the boundary of the system (not at the interfaces) —, the linear elasticity constitutive equations, and the continuity conditions at the interfaces between the layers. Cauchy field equations and the constitutive assumptions are: divTA = pAuA
in VA
divT B = pBuB
TA = 2fiAEA + \A(trEA)I,
in VB,
TB = 2p>BEB + \B(tiEB)I,
(1) (2)
TA, TB being the Cauchy stresses, PA, PB the mass densities, UA and uB the displacements, fiA, AA, P>B, and XB Lame's coefficients, EA := symgradw^ and EB := symgradw^ the (small) strains. The following contact conditions are imposed on the surfaces T+ and r ~ delimiting the thin layer (A+ and A" indicate the thick layer on the side of r + and r ~ respectively — a mark that was unnecessary in the previous equations — and n r the common normal to these surfaces): TBnr = TA~nr
on T~ ,
TBnr = TA+nr
on F + .
When the displacement field uA is extended to the whole body V — VA+UBUA- J IUA} and UA\T indicate its jump and average across T, we may write uA = uA + luA]x,
X-= {-^mVA-,0'mVB,
iinV^+},
with uA continuous on V and such that uA\r =uA\r. The contact between layers is assumed to be perfect uB\r+=uA\r+
uB\r-=uA\r-.
(3)
This model is made coarser to get a macroscopic description of the system, and an approximate solution is sought, through the assumption that the stress TB is constant along the direction normal to the bed.
507
As a consequence TBnT = TA\r± nr
on r + U T~ , = > !T A ]n r = 0,
and the compatibility conditions on the components of EB entail ^(T)X2-^X3+1I(T) uB —W(XI,X2,T)
+
-a3{T)Xl-^x3+l2{T) ai(r)xi + a2{T)x2 + e(r)
(4)
x3,
where w(xi, X2, T) is the displacement of T, cti (i — 1, 2, 3) are rotations of planes || T, jj (j = 1,2) are shears, and e elongation of fibers _L T, (xi, x2) being coordinates on Y, and r time. As a further consequence, Q:3(T)X 2 + 7 I ( T )
- a 3 ( r ) x i +72(r) a i C ^ i + a2(T)Z2 + e(r)
MA =u^(xi,X2,a;3,7-) + 2<J
X-
(5)
The total energy of the system is fv ipA dv + 28 J r (ipB - VAIF)
d a
- lay ( " A + luA\x)
• f + o{52).
By neglecting terms of o(5) and applying Euler's principle to UA and w we get: ( HA^UA + ((iA + Ax) grad( divwA) = PA^A \ (2(iA symgradwA + AA diviiA-On = /
in V, on dV,
' (i*Aw + ((i* + A*)grad(div«/) + AB s = ,9*u>+ /^Adiv (wA,3| r ) " r + AAgrad(uA3,3| r ) in T, 2(i*m • (symgradw)ra + A* divw + \B(O.\XI + a2x2 + e) = AAWA3,3|r in9r, 2(i*nr • (symgia,dw)m + JJ,B9 • m = (IA UA,3\T • m on dV, where A* = AB - AA, P* = P-B - PA, P* = PB - PA (being m • n r = 0) and 9 = [7i +oizx2, 72 -Oi3xi,
0]
s = [ a i , a 2 , 0] .
Taking the derivative of the total energy with respect to a, (i = 1 , . . . 3), 7j (j = 1,2) and e and defined J8J- := Jr%iXj da, Si := J r x, da and T :—
508
Jr da, we get the linear equations Ju J12 S\ Ju J22 S2
"ai"
+
Q2
Si 52 r J _ e
A B + 2/XB
Jr(wi,i +w 2 ; 2) Jr
da
£2
1 ^2
AB+2/XB J a v - ' 3
(6)
Xda,
1
Ju + J22 5*2 — Si
s2 -5i
r 0 0
r
0:3
"W3,la
+ ir
7i .72.
da
™3,1 ^3,2
(7)
7lZ2 - / 2 X 1 1 2MB
X^a.
A
Jav
/2
3. Transfer M a t r i x To study the propagation of plane elastic waves, let us assume the velocity field to be vA = grad<E> + r o t * $ = (j)iei{ocx3)
+(j)2e~i(ax3)ei(ax1-^t)
^
^
=
(8)
^iei([3x3)
+1p2e-i(Px3)ei(<7X1-u:t)
where a = KLcos6L,
p = KTcos6T,
&(
3(cr)
where KL = f-= w.L PJ, and KT = ± w, / ^ ^ are the wave numMA bers relative to the longitudinal CL and transverse CT velocities, respectively, K an unknown wave number, and c € (c x ,c z ) (according to the direction of propagation). Moreover, the Snell's Law is applied, so that K sin 8 =
KL
sin 6L — KT sin
6T-
By introducing as in Ref. 4 the vectors PA =
[VA1,VA3,TA13,TA33}T,
f = [01 +4>2,4>1 - ^ 2 , ^ 1 - ^ 2 , ^ 1 + V^F ,
we can write p ^ at the upper boundary of ^4+ and at the lower boundary of A" as follows (h the thickness of A): PA+\X3=±+5 PA-
»=-*-*
= ( C ^ ^
+<5Die^)) f e - ^ \
(c 2 e i ( C T a ; i ) - £D 2 e i ( < T °) f e - i( " T) .
509 As a consequence, we get PA+\Xa=h+S
(9)
= A p A - | I S = _• i-s _1
where A = ( C i + <$Di)(C2 — <5D 2 ) is t h e overall transfer matrix, or t h e matrix transferring the mechanical signal across layer A and through the structured line T. T h e other matrices entering the definition of A are
Ci
icr cos p —a s n i p -i^ cosp 2/^Off7
—i/3cosg —a sing
-asinp —IQ; a; cosp skip
/Jsing ia cos g sing
HAi^-P)
sinp
/x,4(^-,3 2 )
smg
cosg
where C,A = i^A + 2jJiA)a2 + XAcr2, p = a ( f + 5), q = /?(§ + 5), and 87+
87+
07+
a/i
a/2
a/3
97+
a/4
a ( a + x i + £ + ) a ( a + x i + e + ) 3(o:+xi+e+)
D, =
a(g+xi+e+)
9/i
a/ 2
a/3
a/ 4
0
0
0
0
3a+
3a+
3a+
l"A s/a
^ A a/2
VA df3
3a+
VA -gj^
where 7 + , d\ and e + are the solutions of the equations of motion (6)-(7), with X=\ (explicit form are omitted for shortness). The matrix C2 is defined as C i , but replacingp with p = — a ( | +S) and q with q = — / 3 ( | + 5); D 2 is defined as D 1 ; taking 7J" instead of 7 ^ , a^~ instead of a^" and e~ instead of e + , all of t h e m being determined by solving the equations of motion (6)-(7) as already mentioned for their analogue at the + side, b u t with \ — ~\ (again, the explicit forms are omitted). T h e dispersion equation is obtained by imposing periodic b o u n d a r y conditions. Since we deal with an approximate solution, these periodic boundary conditions cannot be satisfied pointwise, whence the need of approximations. We consider PA+
x 3 = f+<5
PA-
\x3 =
-§-S
<5Q(zi),
with Q(a;i) the list of misfits and residuals occurring in the approximation. T h e n we assume rl/2
Q(xi)(p(xi)dxi
=0
\fip(xi) test function
J-i-1/2
and drive consequences depending on the choice of t h e class of test functions. Here we take
+ 5ei(-al^ / l , ( D i + B2)dx1)
= 0.
510 playing the role of the dispersion equation. The approximate dispersion equation, when neglecting o(S2), admits two solutions:
cxT = KT + i5K2T,
(10)
where
j, K2L K2T
3(2\B - \A)fMAKL sin(KLl) sm(KLl) 4pA{XB + 2fiB)l2htj2 v v " ' KLl ' KLl (2XB - XA)XAKL ,„, fv. n sm(KLl) 2 . , „ n2\ = - J ^ - T T — , o M2h —-— Y + sm(KLl) ), i 3(3(cos(KLl) 4/OA(AB +2nB)lhu) KLl ((HA - 2/jB)KT sin(KTl)2 4hfiBu> KljX2
In the above equations the contribution of K\L to
the damping coefficients.
4. Numerical solution of the dispersion equation The numerical values of the coefficients Ku,, K2L and K2T have been obtained by considering the material parameters listed in the table Layers A Layers B
Young modulus 2.0 GPa 0.1 GPa
Poisson ratio 0.3 0.2
Mass density 1.8-10 3 k g m " 3 1.6 • 103 kg m " 3
(i.e., nA = 10 9 /L3, XA = 1-5 • 10 9 /1.3, \iB = 10 8 /2.6, \ B = 0.4 • 10 8 /0.72) and assuming I = 12 cm and h = 6 cm. The results are plotted in Fig. 1 and 2. In Fig. 3, we compare the numerical value of 3J(cr) = KL + SKIL with that of 3?(<J) = KL, obtained when 6 = 0 (i.e., when there is no layer B), in order to stress the influence on the velocity of propagation of the presence of a thin layer. In conclusion, it is worth of noticing that if we consider the material of layer B to be possibly subject to damage v 6 [0,1], i.e., if we replace \iB
— • MB^,
XB —>
XBv,
we get results that depend on the attained damage level as displayed in Fig. 4. These results show how the characteristics of the wave depend on damage and thus open to experimental applications in the field of inverse identification.
511 References 1. M. P. Speciale, M. B r o c a t o . Proc. "Wascom 2001", World Scientific Publishing, 548, Singapore (2002). 2. M. P. Speciale, M. B r o c a t o , Math. Comp. Model., 3 7 , 477 (2003). 3. W . T . T h o m s o n , J. Appl. Phys., 2 1 , 89 (1950). 4. L. M. Brekhovskikh, Waves in layered media, Acad. P r e s s , N e w York (1960).
20000
•40000
Tegs 60000
3QOQO
"
TOOE
—" '
-1E- i o
Figure 1.
20000
10-
-3E
1D
-4E
io-
Numerical value of ifix, (left) and K2L (right) functions of w.
40000
eoooo
Figure 2. Figure 3.
-2E
eoooo
100000
SOOO
1OOOO
15000 omega
20000
25000
30000
(left) Numerical value of Kyr function of UJ.
(right) 5ft(
5000 10000 15000 20000 25000 30000 omega nu=i Without layer B nu=f/2
5000 10000 15000 20000 25000 300OO omega
nu=1/4
nu=1/4
Figure 4.
, , r S v , , ••>
nu=1/2
2ft(cr) for longitudinal waves and 5 ( c ) for transversal waves.
WAVE PROPAGATION P H E N O M E N A IN SOLIDS N E A R T H E MELTING P O I N T
M. SUGIYAMA Graduate
School of Engineering, Nagoya Institute Nagoya 4-66-8555, Japan E-mail: [email protected]
of
Technology,
The analyses of linear and nonlinear wave propagation phenomena in solids at finite temperatures made so far on the basis of a model, which was proposed recently by the present author, are reviewed. The model is a finite-deformation continuum model for solids at finite temperatures, which is valid in a wide temperature range including the melting point as a limiting case. Some peculiar temperaturedependences of the phenomena near the melting point found in the analyses are summarized. Future problems to be solved are also pointed out and discussed.
1. I n t r o d u c t i o n When the temperature of a solid increases and then reaches a critical temperature, the solid becomes unstable and melts. The study of wave propagation phenomena in a solid near such an instability point seems to be interesting theoretically. Moreover, the study is quite important practically. In fact, nowadays, solid materials are utilized in many practical situations at high temperatures even near the melting temperature. However such studies, both theoretical and experimental, have not been fully made until now. Recently a new finite-deformation continuum model for solids at finite temperatures incorporating explicitly microscopic thermal vibrations of the constituent atoms was proposed by the present author 1,2 . The model was derived from the statistical-mechanical study of an anharmonic crystal lattice. And it was shown that the model is valid in a wide temperature range including the melting point as a limiting case. On the basis of the model, linear and nonlinear wave propagation phenomena in solids at finite temperatures have been analyzed. Some peculiar temperature-dependences of the phenomena near the melting point were found and their physical implications have been discussed.
512
513
In the present paper, the characteristic features of the waves obtained so far by analyzing the model, especially those near the melting point, are summarized. The problems remained to be solved are also pointed out. 2. Wave Propagation Phenomena in One-Dimensional Solids In this section, wave propagation phenomena in one-dimensional (ID) solids at finite temperatures are discussed. The analysis here can be viewed as preliminaries to the analysis of the realistic three-dimensional (3D) continuum model mentioned later, and it may also be interesting when we study the effect of dimensionality in the wave propagation phenomena. An advantage of ID model is that we can obtain many interesting and rigorous analytical results explicitly from the model. 2.1. ID
Model
The physical system we want to study here is a ID anharmonic crystal lattice (crystal chain). For simplicity, we assume identical constituent atoms each with mass M and with nearest-neighbor interactions. The potential energy v of two atoms separated by a distance x is assumed, for definiteness, to be given by the Morse function: v(x) = £>[exp{-2a(x - R0)} - 2 exp{-a(ar - R0)}} , 3 4
(1)
where D , a and RQ are the material constants ' . The present author proposed a nonequilibrium statistical-mechanical model 5 for describing nonequilibrium phenomena in anharmonic crystal lattices at finite temperatures by extending the concept of the self-consistent Einstein model (SCEM) 6 ' 7 ' 8 , where a self-consistent Gaussian one-body approximation is made. The basic equations for the model are, therefore, expressed by the self-consistent evolution equations for the parameters in the Gaussian distribution functions, i.e., means and variances of the displacements of the constituent atoms from their equilibrium points and their conjugate momenta. The basic equations can be seen as a simplified version of the Liouville equation for a distribution function of the crystal lattice. The model contains all dynamical modes from microscopic to macroscopic scale, which interact with each other in a complicated manner. By adopting the continuum apprximation where only macroscopic modes are explicitly taken into consideration, a system of basic equations for the continuum model 1 can be derived from the statistical-mechanical model. In
514
the continuum model explained here, we neglect, for simplicity, irreversible processes, and adopt the thermodynamical local-equilibrium-assumption. Below, we summarize such a system of basic equations. For more details of them, see the reference1. The basic equations are expressed by the mechanical dimensionless quantities d(X, t) and q(X,t), and the thermal dimensionless quantities g(X, t) and r{x,t) defined below. Here X is the position of a material point in a reference configuration and t is the time. Hereafter, the thermal equilibrium state at absolute temperature T with no external force and no translational motion is adopted as the reference configuration. d {w{X,t)) = dX
{aae)-"d{X,t)
d_ (w(X,t)) dt
q(X,t)
, (2)
2
2
iw(X,t)-(W(X,t))) )=a- l*
Ft(w(X,t)-(w(X,t)))
+ 9(X,t)} D M
kBT + r(X,t) D
where ( ) stands for a statistical average over the nonequilibrium distribution function, w{X,t) is the displacement of a constituent atom from its thermal equilibrium point X and k-Q is the Boltzmann constant. The quantity ae is the lattice constant and A is the reduced mean square displacement due to the thermal vibration in the reference thermal equilibrium state: A = a2{w(X,t)2)equiiibrium
These are related to the temperature T such that ae = RQ + 3a" A , kBT Ae -2A AD
•
(3)
5
(4) (5)
The melting temperature T m of the ID crystal lattice is given by fcBTm/(4Z>) = l/(2e) ^ 0.1839. Mechanical quantities d(X,t) and q{X,t) are the scaled strain and the dimensionless velocity of a material point, respectively. Thermal quantities g(X,t) and r(X,t) indicate the deviations, respectively, of the amplitude
515 of atomic thermal vibration and of a temperature induced by a wave from the reference thermal equilibrium state. The field equations, that is, the conservation laws of mass, momentum and energy, are summarized as follows1: / __i
^
dd
T
da
da
BIT
where Q, is a microscopic characteristic frequency of atomic vibration defined by fi = ay/D/M. The dimensionless pressure IT and the dimensionless energy density e are given by ix = 2e" 2A (e4°-2d - ev-d) , (7)
=\(c§-+q2+r)+e~2X {e49~2d'2e9~d)=^2 with u being the dimensionless internal energy density. In addition to the field equations, there exists an equation of state 1 such that r = 4e- 2A (A + g) (2ei9~2d - eg~d) - 4Xe~2X .
(8)
By using this equation we may eliminate the quantity r in the quantities e and u. The characteristic feature of the continuum model expressed by these equations resides in the formulation with the explicit use of the field variable g indicating the change of the amplitude of atomic thermal vibration. The quantity g is specifically apt to describe thermal properties, including those near the melting point, in crystals, and the quantity can be observed directly, for example, by X-ray diffraction experiments. It is well-known that the variance of the momenta of constituent atoms, which is proportional to the temperature, plays an important role in the analysis of wave propagation phenomena. It seems to be natural, therefore, that its dual quantity g also plays an essential role in the analysis. The system (6) is a particular case of a system of conservation laws9: dt
+
dX
U
{
'
516 by identifying
F(u) = lq J , G(u) = aeQ I TT J ,
(10)
and choosing, for example, as field variables
(11) We can determine globally not only the strict hyperbolicity and convexity regions but also the elliptic and parabolic regions in the space of the state of the system 9 . Physical states are within the convexity region and, therefore, the melting point is on the boundary of the region. 2.2. Waves in ID
Solids
Linear and nonlinear waves analyzed by using the ID model are briefly summarized. 2.2.1. Linear waves On the basis of the linearized basic equations with respect to the independent variables, linear harmonic waves were studied in detail 10 ' 11 : (a) Temperature dependences of the propagation speeds were obtained explicitly. Near the melting point, it is found that the propagation speed decrases rapidly with the increase of the temperature of the solid, and its decreasing rate becomes to be singular at the melting point. (b) Temperature dependences of the amplitude ratios were also obtained. Similar singularities at the melting point in the amplitude ratios were also found. (c) Importance of the quantity g in the model was clearly shown. By using this quantity, we can analyze the thermal properties of the waves deeply, especially near the melting point. (d) Two sets of the results obtained from microscopic and macroscopic models were compared with each other. And the applicability range of the macroscopic continuum model was elucidated. In connection with this, elastic waves with ultrasonic frequencies propagating in a crystal lattice were studied 12 .
517 2.2.2. Acceleration waves Acceleration waves were studied by applying the method of singular surfaces to the model 13 : (a) Temperature dependences of the propagation speeds and their singularities at the melting point were analyzed. (b) Temperature dependences of the ratios of the mechanical and thermal amplitudes, and their singularities at the melting point were also analyzed. (c) Amplitude equations, i.e., the differential equations which govern the variation of the amplitudes with time were derived explicitly. On the basis of these equations, the temperature dependence of the critical time, at which the amplitudes of progressive acceleration waves diverge, was found. And a criterion about the formation of shock waves was formulated. (d) It was pointed out that by monitoring experimentally the peculiar behavior of the wave near the melting point, which was predicted in the analysis, we can obtain useful information about instability phenomena in solids accompanied by melting. 2.2.3. Shock waves The following points about shock waves in ID solids were analyzed 1 ' 14 : (a) Temperature dependences of the propagation speeds of shock waves were obtained explicitly. (b) Temperature dependences of the Rankine-Hugoniot relations for both mechanical and thermal quantities were studied. Physical quantities involved in the Rankine-Hugoniot relations are the propagation speed of a shock wave, relative displacement, shock Mach number, particle velocity, the change of the mean square amplitude of atomic thermal vibration, temperature change, entropy change, pressure, and the change of internal energy density. (c) Stability of shock waves was discussed on the basis of the second law of thermodynamics. (d) Comparison between two sets of the results from microscopic and macroscopic analyses was made. And the applicability range of the macroscopic continuum model was studied.
2.3. Problems
Remained
Problems remained to be solved may be summarized as follows:
518 (a) Study of the reflection and transmission of a wave at a boundary 15 . (b) Study of rarefaction waves. (c) Study of the Riemann problem. (d) Study of the collision between an acceleration wave and a shock wave. These problems are now being under study, and their details will soon be reported elsewhere. 3. Wave P r o p a g a t i o n P h e n o m e n a in 3D Solids In this section, after explaining the model for realistic 3D solids at finite temperatures, characteristic features of the wave propagation phenomena analyzed so far by using the 3D model are summarized. 3.1. 3D
Model
A 3D continuum model for solids at finite temperatures was also proposed 2 on the basis of a nonequilibrium statistical-mechanical model of 3D anharmonic crystal lattices 16 by using the continuum approximation. As the model is a natural extension of the ID continuum model to the 3D case, we summarize briefly the set of field equations for the 3D continuum model. Here, for simplicity, we restrict ourselves within studying isotropic solids. The field variables in the set of the field equations are the deformation gradient F(X,t), the dimensionless velocity q(X,t), and the thermal dimensionless quantities g(X,t) and r(X,t). Here X is the position of a material point in the reference configuration. As in the ID model, the thermal equilibrium state at absolute temperature T with no external force and no translational motion is adopted as the reference configuration. And a single Cartesian coordinate system for both reference and current configurations is adopted. Then, the above-mentioned field variables are defined in a component form as follows: (wi(X,t))
=
^qi(X,t),
{(wi{X,t)
- ((Wi(X,t)))(Wj(X,t)
-
(Wj(X,t))))
2
= a- [A%+5ii(X,i)] , ((wi{X,t)
- ((wi(X,t)))(tij(X,t) _ D_ k -jf+r(X,t) = ~M
~ dij ,
(12) (wj(X,t))))
519 where w(X, t) is the displacement of a constituent atom from its thermal equilibrium point corresponding to a material point X at a time t, ({X, t) = d((X,t)/dt for a generic quantity £, and 6ij is the Kronecker's delta. The quantity A is the reduced mean square displacement due to the atomic thermal vibration in the reference equilibrium state: A<%
Q-2(Wi(X,t)Wj{X
,i))equilibr
(13)
Here we have taken into account the assumption that solids under study are isotropic. This quantity A can be determined explicitly as the function of the temperature T16. In the third relation in the definitions (12), we have adopted the local equilibrium assumption by introducing the scalor quantity r. A possible generalization of this assumption will be discussed in the last section. In terms of the dimensionless quantities defined above, the basic equations for the 3D model are given as follows2 (the summation convention is adopted): pdet{Fik] = p 0 , da 0Fik da dFik o-i.
(A4
da dFjk
?jk
Z
-i
(14)
9 ( da\ dXk \C'dFik) '
da """'dgmj
kBT D 1+ 2D V kBT
where p and po are mass densities in the current and reference configurations, respectively, and the quantity po is the known function of the temperature 2 . The quantity a is the dimensionless potential energy density, and it is the function of the quantities F and g. Explicit functional form of a can be derived from the microscopic theory 16 ' 17 . The quantity e is the dimensionless energy density defined by 1 +
3 fkv,T +r D
2
(15)
The basic equations (14)i_4 represent the conservation laws of mass, momentum, angular momentum and energy, respectively, while the relation (14)5 means the equation of state.
520 The basic equations (14)i_4 can also be expressed in a conventional form of the conservation laws (9) 17 . And the thermodynamic stability of solids was also studied in detail, but only in a local region near an equilibrium state 18 .
3.2. Waves in 3D Solids Linear and nonlinear waves analyzed by using the 3D model are briefly discussed.
3.2.1. Linear waves By using the linearized basic equations, linear harmonic waves were studied in detail 19 : (a) Temperature dependences of the propagation speeds of longitudinal and transverse waves were derived. For example, in Fig.l, the propagation speeds of the longitudinal and transverse waves VL and Vp in several fee metals are shown. From Fig.l, we notice the singularities of the speeds at the melting points. Propagation speeds predicted by the model agree fairly well with experimental data at high temperatures 20 .
0
1000
T(K)
2000
0
1000
j
( K )
2000
Figure 1. Temperature dependences of the propagation speeds of longitudinal and transverse waves VL and VT in Ag, Al, Cu, Ni and Pb derived from the 3D model.
521 (b) Temperature dependences of the amplitude ratios of longitudinal and transverse waves were also derived, and similar singularities at the melting point were found. (c) Anisotropy in the thermal vibrations of constituent atoms induced by the wave was studied in detail by analyzing the quantity g. And their temperature dependences were also elucidated. (d) Applicability range of the continuum model with the local equilibrium assumption was studied through analyzing the temperature dependences of the propagation speeds and the quantity r. It was found that, as far as the linear waves are concerned, the assumption is good except for the temperature range near the melting point. 3.2.2. Acceleration waves Acceleration waves were studied 17 , and the following points were found: (a) Temperature dependences of the propagation speeds of longitudinal and transverse waves were obtained and their singularities at the melting point were found. (b) Temperature dependences of the ratios of the mechanical and thermal amplitudes for longitudinal and transverse waves were obtained and their singularities at the melting point were found. (c) Amplitude ratios of standing waves were derived. (d) Amplitude equations were derived analytically. And the temperature dependence of the critical time when the amplitudes of progressive longitudinal acceleration waves diverge was derived. The critical time for plane acceleration waves, for example, becomes much smaller as the temperature of a solid tends to the melting temperature. (e) A criterion about the formation of shock waves was formulated and discussed. 3.2.3. Shock waves Recently weak shock waves in solids were analyzed in detail 21 : (a) Entropy jump across a shock wave was analyzed and the stability condition for the wave was discussed. (b) Temperature dependences of the propagation speeds of shock waves were analyzed. (c) Temperature dependences of the Rankine-Hugoniot relations were derived. Physical quantities involved in the Rankine-Hugoniot relations are the propagation speed of a shock wave, particle velocity, deformation gra-
522
dient, mass density, Piola-Kirchhoff stress, the change of the mean square amplitude of atomic thermal vibration, temperature change, entropy change and the change of internal energy density. (d) Anisotropy in the thermal vibrations of constituent atoms induced by a shock wave was also studied. (e) Singular properties in the Rankine-Hugoniot relations near the melting point that may be observed by experiments were pointed out and discussed. 3.3. Problems
Remained
Problems remained to be solved may be summarized as follows: (a) Study of linear and nonlinear waves propagating in anisotropic solids such as single crystals. (b) Study of reflection, refraction and transmission of a wave at a boundary. A preliminary study was made recently 22 . (c) Study of ultrasonic waves taking their dispersion effect into account. A preliminary study was made for the waves in ID solids 12 . 4. Concluding Remarks As concluding remarks, let us comment on possible improvements on the model explained above. (l)In the preceding sections, only the model with the local equilibrium assumption has been explained. However, some preliminary analyses reveal that, in order to obtain quantitatively better results particularly near the melting point, we should go beyond the local equilibrium assumption. Therefore extended-thermodynamical approach 23 , where we take more thermodynamic quantities into account, is promising. (2)The above model is the so-called thermo-elastic solids. However, it is well-known that at a high temperature near the melting point, effects from viscosity and/or plasticity become to be important in the wave propagation phenomena. Therefore the model explained in the present paper should be regarded as that at the first step of our approach. The above-mentioned effects may be taken into the model step by step. Certainly the quantities analyzed here, that is, propagation speeds of the waves, amplitude ratios and so on are not so strongly dependent on the dissipation effects. Furthermore the shock waves in 3D solids studied here were limited only within weak shock waves. However, when we try to study, for example, attenuation of a wave, we
523 should go on t o the next step. T h e next study in this direction seems to be interesting and important b o t h theoretically and practically.
Acknowledgments T h e author would like t o express his thanks t o Profs. G. Valenti, C. Curro and T. Ruggeri for their valuable discussions and comments. This work was supported by a Grant-in-Aid from the J a p a n Society for the Promotion of Science (No. 15560042).
References 1. M. Sugiyama and T. Isogai, Jpn. J. Appl. Phys. 35, 3505 (1996). 2. M. Sugiyama, J. Phys. Soc. Jpn. 72, 1989 (2003). 3. I. M. Torrens, Interatomic Potentials (Academic Press, New York, 1972). 4. L. A. Girifalco and V. G. Weizer, Phys. Rev. 114, 687 (1959). 5. M. Sugiyama, J. Phys. Soc. Jpn. 61, 2260 (1992). 6. T. Matsubara and K. Kamiya, Prog. Theor. Phys. 58, 767 (1977). 7. T. Hama and T. Matsubara, Prog. Theor Phys. 59, 1407 (1978) 8. M. Sugiyama and H. Okamoto, Prog. Theor. Phys. 64, 1945 (1980). 9. T. Ruggeri and M. Sugiyama, J. Phys. A: Math. Gen. 38, 4337 (2005). 10. M. Sugiyama and T. Fukuta, J. Phys. Soc. Jpn. 61, 2269 (1992). 11. T. Fukuta and M. Sugiyama, J. Phys. Soc. Jpn. 61, 4367 (1992). 12. M. Sugiyama, H. Kikkawa and K. Mizuno Jpn. J. Appl. Phys. 32, N o . 5 B , 2202 (1993). 13. N. Kameyama and M. Sugiyama, Continuum. Mech. Thermodyn. 8, 351 (1996). 14. M. Sugiyama, H. Kikkawa and K. Mizuno, Jpn. J. Appl. Phys. 33, 1450 (1994). 15. M. Sugiyama, N. Kameyama and K. Mizuno, J. Phys. Soc. Jpn. 64, 1 (1995). 16. M. Sugiyama and K. Goto, J. Phys. Soc. Jpn. 72, 545 (2003). 17. G. Valenti, C. Curro and M. Sugiyama, Continuum. Mech. Thermodyn. 16, 185 (2004). 18. M. Sugiyama and H. Suzumura, J. Phys. Soc. Jpn. 74, 631 (2005). 19. M. Sugiyama, K. Goto, K. Takada, G. Valenti and C. Curro, J. Phys. Soc. Jpn. 72, 3132 (2003). 20. I. Ihara, D. Burhan and Y. Seda, Proc. Symp. Ultrasonic Electronics 25, 147 (2004). 21. C. Curro, M. Sugiyama, H. Suzumura and G. Valenti, Continuum. Mech. Thermodyn. (submitted). 22. M. Sugiyama and M. Chaki, Jpn. J. Appl. Phys. 44, N 0 . 6 B , 4309 (2005). 23. I. Miiller and T. Ruggeri, Extended Thermodynamics (Springer, New York, 1993), Rational Extended Thermodynamics (Springer, New York, 1998).
P L A N E WAVES A N D V I B R A T I O N S IN T H E ELASTIC MIXTURES
M E R A B SVANADZE Department of Elasticity Theory, I. Vekua Institute Applied Mathematics, Tbilisi State University, 2, University St., Tbilisi, 0186, Georgia e-mail: [email protected]
of
In this paper the diffusion and shift models of the linear theory of elasticity of binary mixtures are considered. The basic properties of wave numbers of the plane harmonic waves are established. The existence theorems of eigenoscillation frequencies (eigenfrequencies) of interior boundary value problem (BVP) of steady vibrations are proved. T h e connection between plane waves and existence of eigenfrequencies is established. Lorentz's postulate on the asymptotic distribution of eigenfrequencies for binary mixtures is proved.
1. Introduction The modern formulation of continuum theories of mixtures goes back to papers by Truesdell and Toupin, Eringen and Ingram, Green and Naghdi, 1. Muller, Bowen and Wiese. The construction and intensive investigation of the mathematical models of the elastic mixtures arise by the wide use of composites into engineering. The diffusion and shift models of the linear theory of elasticity and thermoelasticity of binary mixtures are presented by several authors (Green and Naghdi, Steel, Bowen, Bedford and Stern, Rushchitskii, McNiven and Mengi, Tiersten and Jahanmir, Iesan, Khoroshun and Soltanov). An extensive review of the results of the theory of binary mixtures can be found in [1-6]. 2. Basic Equations The system of equations of linear dynamical theory of binary mixtures with two isotropic elastic materials can be written as follows a\ A M + b\ graddivu + cAw + dgraddivw — a(u — w) — v{u — w)
524
525 = pnu -
Puw,
c A u + d grad div u + a2 A w + b2 grad div w + a(u — w) + v(u - w) = -pl2U
+ P22W,
where the partial displacements u(x, t) and vector functions, a > 0, v > 0; a\,02,61,62,c, and p\2 are the partial densities, p\\ = pi + denotes differentiation with respect to time. For different values of the parameters a models: a) the diffusion model (a = 0, v > 0) [7]; b) the shift model ( a > 0, v = 0) [3, 8].
w(x, t) are three-component d are elastic constants, pi, P2 P12, P22 = P2 + P12, and dot and v we have the different
3. Plane Waves Suppose that plane harmonic waves corresponding to the wave number k and angular frequency 10 are propagated in the x\ direction through the binary mixtures, that is u(x,t) = A'exp{i(kxi
— ut)},
w(x,t) = A"exp{i(kxi
- cJt)},
where A' = (A[,A'2,A'3) and A" = {A'^A'^A'D are constant vectors, x — {x\,X2,x-i) is a point of the Euclidean three-dimensional space E3. The wave number k is the solution of the dispersion equations of longitudinal and transverse plane waves dik4 — [u)2pi +
— a)qi]k2 + co4po + {iwv — a)uj2p = 0
(1)
d2kA — [u2p2 + [iijjv - a)q2]k2 + uj4po + (iu>v — a)uj2p — 0
(2)
(ILOV
and
respectively, where di — ab — c 0 ,
d2 = a\a2 — c ,
Pi — ap22 + bpn+2c0pi2, q2 = a1+a2
+ 2c,
a = a\ + b\,
b = a2 + b2,
P2 = aip22+a2pn+2cpi2, p = p\ + p2,
qi=a
Co — c + d, + b + 2c0,
Po = P11P22 - p\2-
Obviously, if k is real, then the corresponding plane wave has the constant amplitude, and if k is complex with Im k > 0, then the plane wave is damped as x\ —> 00.
526 Let kf,k2 and k\,k\ be roots of Eqs. (1) and (2), respectively. Six plane harmonic waves (2 longitudinal and 4 transverse) propagate in t h e binary mixture, fci, k2 and k3, k^ are t h e wave numbers of t h e longitudinal and transverse plane waves, respectively (two pairs of transverse waves having the equal wave numbers) [9]. Let Di = (a + co)p2 - (b + co)pi,
D2 = (ai + c)p2 - (a 2 +
c)p\.
T h e numbers k\, k2, k\ and k\ have t h e following properties [9]: T h e o r e m 3 . 1 . In the diffusion model(a = 0, v > 0): a)lmk]^0 0 = 1,2), for DX ± 0; b) kf > 0, Im k\ ^ 0, /or £>i = 0. T h e o r e m 3 . 2 . 7n the diffusion model: a)lmk?jk0 (j = 3 , 4 ) , /or D 2 ^ 0; b) kj > 0, Im k\ ^ 0, /or £>2 = 0. C o r o l l a r y 3 . 1 . If DiD2 4 0 , then in the diffusion plane waves propagate through the binary mixture.
model only
damped
T h e o r e m 3 . 3 . In the shift model (v — 0, a > 0 ) : a) k) > 0 (j = 1, 2 , 3 , 4 ) , for LJ > co0; b) k\ > 0, fcf > 0, k\ = k\= 0, for w = OJ0; c) kj > 0, k\< 0, fcf > 0, k\ < 0, / o r 0 < w < w0, 1 w/iere WQ = app^ . 4. E x i s t e n c e o f E i g e n f r e q u e n c i e s Let S be a closed surface surrounding a finite domain Q in E3. T h e basic interior homogeneous B V P of steady oscillations (vibrations) is formulated in the following way. Find a regular solution U — (u',w') to the system a\ Au' cAu'
+ b\ grad div u ' + cAw'
+ dgvaddivw'
— j3\u' + (32w' = 0,
+ dgrad div u' + a2 A w' + b2 grad div w' + f32u' - /33w' = 0,
(3)
in f2, satisfying the b o u n d a r y condition U(z) = 0,
for
zeS,
(4)
where /3\ = a — iuiv — to pn,
(32 = a — iuiv — ui pi2,
03 = a — iuiv — u> p22,
527
and u(x,t) = Re{u'(x)e-iut),
w{x,t) =
Re{w'(x)e~iuJt}.
Theorem 4.1. If D\D^ ^ 0 then in the diffusion model the interior homogeneous BVP (3), (4) has only the trivial solution in class of regular vectors, that is, there exists no vibrations (no eigenfrequencies). Theorem 4.2. If D\Di — 0 then in the diffusion model the interior homogeneous BVP (3), (4) has a non-trivial solution V = (u',u') in the class of regular vectors. In addition, a) if D\ ^0,D<2 = 0, then the vector u' is a solution to the BVP: (A+u2pq21)u'{x)=0,
divu'(x)=0,
u\z) = 0,
xeil,
z€S;
(5)
the problems (3), (4) and (5) have the same eigenfrequencies, b) if Di = 0,D2 ^ 0, then the vector u' is a solution to the BVP: (A + uj2pq^1)ul(x)
= 0,
u'(z) = 0,
curlu'(x) = 0 ,
i6fl,
z G S;
(6)
the problems (3), (4) and (6) have the same eigenfrequencies, c) if D\ = Z>2 = 0 , then the vector u' is a solution to the BVP: q2Au'(x) + (qi — q2) grad div u'(x) + pu2 u'(x) = 0 , u'(z) = 0,
z G S;
iffi,
(7)
the problems (3), (4) and (7) have the same eigenfrequencies, Theorem 4.3. In the shift model the homogeneous BVP (3), (4) has a discrete spectrum of eigenfrequencies {ion}'^L1 and lim tun — +oo. n—>oo
Theorems 4.1 to 4.3 can be proved similarly to the corresponding theorems in [9].
528 5.
Connection Between Plane Waves and Existence of Eigenfrequencies
By virtue of theorems 3.1 - 3.3, 4.1 - 4.3 and corollary 3.1 we have the following results: 1. If all plane waves propagating through a binary mixture are damped, then the interior homogeneous BVP (3), (4) has only the trivial solution, that is, the exist no vibration (no eigenfrequencies). 2. If all longitudinal (transverse) plane waves propagating through a binary mixture are damped, then in the interior homogeneous BVP (3), (4) divergence (curl) of partial displacements vanishes. 3. If through a binary mixture there propagates as least one plane wave with constant amplitude, then the existence of eigenfrequencies is possible in the interior homogeneous BVP (3), (4).
6. Asymptotic Distribution of Eigenfrequencies Lorentz's well-known postulate that "asymptotic distribution of eigenfrequencies of elastic solids does not depend on the shape of the body but depends on its volume" was proved by Weyl [10] for an isotropic threedimensional elastic body and developed the law of asymptotic distribution of eigenfrequencies. The basic results on this subject are obtained by T. Carleman, R. Courant, H. Weyl, A. Plejel, H. Niemeyer, T. Burchuladze, R. Dikhamindzhia, M. Svanadze. Let ci,C2 and 03,04 are the roots of the equations (with respect £) Po£2 — P\£, + o!i = 0 and p0£2 — p 2 £ + d2 — 0 , respectively. Obviously, if a = v = 0, then ci,C2 and 03,04 are the velocities of longitudinal and transverse plane harmonic waves [9], respectively. We have the following result (for details see [11]). Theorem 6.1. The asymptotic distribution of the eigenfrequencies {ujn}'^L1, of the BVP (3), (4) in the shift model is expressed by formula N(t)~-^\n\Mt\ where N(t) is the number of eigenfrequencies u>n not greater that t, |Q| is the volume ofT2, t —> +00 and M = c^3 + c^3 + 2(c^3 + c j 3 ) . Hence, Lorentz's postulate on the asymptotic distribution of eigenfrequencies for binary mixtures is proved.
529 7. C o n c l u d i n g R e m a r k s In the binary mixture t h e elastic constants and t h e partial densities of which satisfy the condition D\Di ^ 0, exists no vibrations. This material will be interesting in seismology and engineering. T h e above-mentioned connections 2 and 3 between plane waves and eigenfrequencies hold for isotropic elastic solids in t h e classical theory of elasticity, thermoelasticity, generalized thermoelasticity, micropolar theory of elasticity and thermoelasticity, theory of mixtures. Q u e s t i o n 7 . 1 . Is it possible for engineers condition which will be without vibration? Q u e s t i o n 7 . 2 . Are connections mechanics?
1-3
to make a composite
valid in all theories
of
with the
continuum
T h e answers for these questions do not have nowadays.
References 1. R. M. Bowen,Continuum Physics (A. C. Eringen, ed.), Ill, Academic Press, New York, 1 (1976). 2. K.R. Rajagopai and L. Tao, Mechanics of Mixtures, World Sci. Publ., Teaneck, NJ (1995). 3. Ya. Ya. Rushchitskii, Elements of Mixture Theory, Naukova Dumka, Kiev (1991). 4. L. P. Khoroshun and N. S. Soltanov, Thermoelasticity of Two Component Mixtures, Naukova Dumka, Kiev (1984). 5. R. J. Atkin and R. E. Craine, Quart. J. Mech. Appl. Math. 29, 209 (1976). 6. A. Bedford and D. S. Drumheller, Int. J. Engng. Sci. 2 1 , 863 (1983). 7. T. R. Steel, Quart. J. Mech. Appl. Math. 20, 57 (1967). 8. H. D. McNiven and Y. A. Mengi, Inter. J. Solids Struct. 15, 541 (1979). 9. M. Svanadze, Quart. J. Mech. Appl. Math. 5 1 , 427 (1998). 10. H. Weyl, Rendiconti del Circolo Matem. di Palermo 39, 1 (1915). 11. M. Svanadze, Georgian Math. J. 3, 177 (1996).
M A X I M U M E N T R O P Y P R I N C I P L E FOR H Y D R O D Y N A M I C ANALYSIS OF T H E FLUCTUATIONS OF M O M E N T S FOR T H E H O T C A R R I E R S IN SEMICONDUCTORS
M. T R O V A T O Dipartimento di Matematica e Informatica Citta Universitaria, Viale Andrea Doria 6, 95125 Catania, ITALY E-mail: [email protected] Within the maximum entropy principle (MEP) we present a general theory to obtain, under spatially homogeneous conditions, a closed set of balance HD equations for the ac small signal (dynamic) response using an arbitrary number of moments of the distribution function. The theoretical approach is applied to n-Si at 300 K and is validated by comparing numerical calculations with ensemble Monte Carlo simulations and with experimental data.
1. General theory The aim of this work is to develop and apply a theoretical study of MEP 1 to the case of high field transport in semiconductor materials 2>3>4>5>6>7>8. Within the framework of the moment theory, for a general many-valley band model, we must consider the following set of generalized kinetic fields 4'7-8-9 ipA(k) = {e m , emuil, ..., emuil •••uis} and the corresponding macroscopic quantities FA = {F(m), F{m)lh, . . . , F{m)lh...is} where e(fe) is a general single particle band energy dispersion of arbitrary form, being m = 0,1, ...iV and s — 1,2, ...M. This approach makes it possible to include the full-band effects within a total energy scheme to describe the full complexity of the band in terms of a single particle with an effective mass function of its average total energy W 4 ' 7 ' 8 ' 9 . In this way the unique independent mean quantities are the traceless part •F1(p)|(i1...is) of the tensor i^pjiij.-i,- In particular, for problems with axial symmetry only the independent components F(p)\{s) = F(P)\{i-i) are of concern, thus by considering the single carrier quantities Fa = {F(q), F(P)\i, • • • ,F(P)\(s)} the corresponding balance equations under homogeneous conditions read °£. + lRaE
530
+ Pa = 0
(1)
531 where e is the unit charge, E is the electric field, Ra, Pa indicate respectively the external field productions and the collisional productions explicitly reported, in the linear approximation, in Ref. [7]. By assuming that the carrier ensemble is perturbed by an electric field 5E£(t) along the direction of E, we calculate the deviations from the stationary values of the moments denoted, respectively, with 5Fa. After linearizing Eq.(l) around the stationary state, we obtain a system of equations for the perturbations ^jf^
= Ta06F0(t)
- e5Et(t)rlE)
(2)
where the relaxation of the carrier ensemble to the stationary state is described by the response matrix Vap, and the — e5E£,(t) Ta are the perturbing forces. Since SFa(0) = 0, the (2) has the formal solution 10 5F(t) = -e5E
f K(s)Z(t-s)ds Jo
with
K(s) = exp(Ts) T^
(3)
being exp(rt) = <& diag{exp(\it), ••• , exp(Aw-i£)} * _ 1 with Aa the eigenvalues of r a / 3 and $ the matrix of its eigenvectors. The small-signal analysis is described by the explicit form of the function £(t). In the particular case of a step-like switching perturbation we have £(t) = 1 for all t > 0 and the differential response 5Fa(t)/SE is the solution of the differential equation
This means that to the extreme position of SFa at time t it corresponds Ka(t) = 0. Analogously, by a further differentiation of Eq. (4) we obtain that one flex point of the perturbation 5FQ at time t' can be associated with an extreme position of the corresponding response function Ka(t'). In the case of a small harmonic perturbation £(£) = exp (itot) we obtain a perturbation of the single-carrier moments which is also harmonic SF(t) = SF(u) exp (iuit) being 6Fa(w) = fi'a(cj) 5E 8 ' 9 with /•oo
fi'a(uj) =—e
/ ./o
Ka(s) exp (—iojs) ds
and
A4(0) =
dFa dE
(5)
where the right side of (5)2 represent the slope (i.e. the dc differential mobility) of curves Fa (see, for example, the slope of curves in Fig.l). With this approach, the vector Ta , the asymmetric response matrix Tap and the vectorial response function Ka(t) can be explicitly evaluated 9 starting from the knowledge of the stationary values Fa of the system.
532
2. The numerical results for n-Silicon In this section we apply the theoretical results to the relevant case of n-Si.
Figure 1. HD stationary values for {.F(o)|(s), ^(1)1(3)}. MC stationary values for velocity v = f(o)|ii energy W = F^ and flux energy S = F(i)\i and experimental data for v.
Figure 1 reports, as example, the HD values for the moments {-F(o)|{s)} and {f(i)|(s>} as a function of the electric field, calculated for N = 2 and M = 5, both in the parabolic and full-band case. For the velocity, energy, and energy flux we report the MC values of full-band simulations n (open circles) and analytic nonparabolic-band simulations 12 (crosses); for the velocity we report also the experimental data 13 . All moments F^o)i(s) (starting from v = .F(o)|i and for increasing values of s) exhibit an initial increase and then tend to saturate at the highest electric fields where the dc differentials mobility vanishes. By contrast, for p > 1, all moments f(p)|(s> exhibit an initial increase, but they do not saturate at high field values.
0
Figure 2.
50
100
150
200
0
50
100
150
200
E (KV/cm) E (KV/cm) Eigenvalues of the response matrix Tap evaluated for N = 1 and M — 5.
In particular, the behavior of moments F(p)\{s) (for different values of indices p and s) is due essentially to the combined action of the acceleration impressed by the external field in the direction of its application and of the scattering mechanisms which dissipate energy and distribute momentum in different directions. The applied field acts mainly on the isotropic part p of
533
the moments while the mechanisms of scattering act mainly on the deviatoric part s of the moments. Indeed, at high fields and for increasing values of the index p, all moments increase faster because of the smaller efficiency of scattering to dissipate the excess energy gained by the field. By contrast, for increasing values of the deviatoric part s, all moments increase slower with increasing fields because of the increasing efficiency of scattering to dissipate through randomization the momentum 9 gained by the field. Fig. 2 reports the eigenvalue spectrum (continuous lines the real part and dashed lines the imaginary part) of Ta/3 for N = 1, M = 5, both in the parabolic (P) and nonparabolic (NP) case. All moments -F(o)|<s} show different coupling regions. Thus, velocity and energy rates are coupled by the electric field and, in the low-field region, they are characterized by complex values. Also the generalized rates (A(0)|<2), A(o)|(3>) and (A(0)|(4), A(o)|<5)) are found to be strongly coupled by exhibiting complex values that, in the nonparabolic case, may extend to the whole range of electric fields. These numerical results confirm the interpretation given above of the saturation regions shown in Fig. 1, that strong dissipative phenomena affect 9 all moments .F(o)|(s) which have increasing values of the deviatoric part s.
Time (ps) Time (ps) Figure 3. Response functions and of the differential responses evaluated for s = 1,2. Fig. 3 shows, as example, the time evolution of the response functions and of the differential responses {K^\^, <5F(0)|(s)} in the nonparabolic case (NP) for s = 1,2. For high electric fields, all the response functions -Fi"(o)|(s> show an nonexponential decay by exhibiting .a negative part 8>9'10 which
534
corresponds to an overshoot in the differential response to the perturbation. Its possible to prove that initially the response functions if( 0 )|( s )(0) are always strictly positive 8 ' 9 , besides from Eq. (5)2 we obtain that, in the regions of saturation for the moments F(o)|(s), the integral in Eq. (5)i vanishes for w = 0. Thus, if the initial part of the response functions is positive then, the contribution at long times of K(fi)\(s)(t) should be negative to compensate. In this transition, the response functions fall through a zero and the corresponding derivative is negative. Thus from Eq. (4), to a zero value of lf(o)|(s) it corresponds a maximum of &F(o)|(«> (open squares in Fig. 3), and analogously, to one flex point of the perturbation <5F(0)|(S) (full squares) can be associated a minimum of the response function.
Figure 4.
f(Hz) f(Hz) Real part and imaginary part of the ac differential mobility p!v and M(OM/2V
Fig. 4 reports, as example, the spectra of the ac differential mobility n'(0\\is\ for s = 1,2 in the nonparabolic (NP) case. Its possible to prove 9 that the peaks of real parts Re[/z',0s|. J, and the maxima (with the consequent fall through a zero) of the imaginary parts Im[/i^,, ,] are always imputable to the negative parts of if (o)|<s) • We can thus demonstrate 9 that, at high fields, the vanishing dc differential mobility of different moments, the presence of complex eigenvalues, the negative values taken by the response functions, the positive overshoot of differentials responses, the maximum of the real and imaginary parts of the ac differentials mobility, are all related to the presence of dissipative scattering processes. Accordingly, Fig. 5 reports the low frequency (f « 108 Hz) and high fre-
535 quency (f = 123.3 GHz) differential mobility 8 Re[/z^] for t h e velocity v as a function of E evaluated respectively at T0 = 300 K and T 0 = 293 K. 0.14
Re[ | i ' v ] N=5, M=1 (NP) ^
°'
12
>\
>
c5~ E r> 0.06
Re[(l'v] N=5_ M=1 (P)
~"~^X
+
MC (NP Analytic-Band)
+
B
MC (FULL-BAND)
»
Exp. Data
\ *
\N
.
"~^NB > 0.10 c5^ E - ^
+*£> ™Sk
Low Frequency (10 8 Hz) T 0 = 300 K
10°
10'
10 2
^ v .
0.06
'35' 0.02
™W«|«llii»U
0.00
High Frequency (123.3 GHz) T0 = 293K
a
N=5, M=1 (P) N=5, M=1 (NP) Exp. Data
^ ^ \ - - - . ^""""^^r
EIDQO 10 3
0.0
2.5
5.0
7.5
10.0
E (KV/cm) E (KV/cm) Figure 5. Real part of the ac differential mobility for velocity vs electric field E at low and high frequency with To = 300 K and To = 293 K respectively.
In particular we compare t h e H D results with M C (full-band n , analytic 1 2 ) d a t a and with experimental d a t a 1 3 ' 1 4 . We remark t h a t t h e nonparabolic HD results agree very well b o t h with full-band M C and experimental data. Acknowledgments Drs. M.V. Fischetti, T . Gonzales, and M.J. Martin are thanked for providing t h e M C data. Partial support from GNFM-INDAM and MIUR Research Project is gratefully acknowledged. References 1 I. Miiller and T. Ruggeri, Rational Extended Thermodynamics. Springer Tracts in Natural Philosophy, Vol. 37, Springer-Verlag New York (1998). M. Trovato and P. Falsaperla, Phys. Rev. B, 57, 4456 (1998). M. Trovato and L. Reggiani, J. Appl. Phys. 85, 4050 (1999). M. Trovato, P. Falsaperla and L. Reggiani, J. Appl. Phys. 86, 5906 (1999). H. Struchtrup, Physica A, 275, 229 (2000). S.F. Liotta ,H. Struchtrup, Solid-state Elect, 44, 95 (2000). M. Trovato and L. Reggiani, Phys. Rev. B, 6 1 , 16667 (2000). M. Trovato, Proceedings of the XII Symposium STAMM-02 Eds S. Rionero and G. Romano, (Springer-Verlag, Italy 2005), p. 269-285. M. Trovato and L. Reggiani, Maximum entropy principle for static and dynamic high field transport in semiconductors .(submitted), (2005). 10 V. Gruzhinskis, E. Starikov, P. Shiktorov, L. Reggiani, M. Saranniti and L. Varani, Semicond. Sci. Technol. 8, 1283, (1993). 11 M. Fischetti, IEEE, Trans. Electron. Devices, 38, 634 (1991). 12 M.J. Martin, T. Gonzalez, J.E. Velasquez, D. Pardo, Semicond. Sci. Technol. 8, 1291, (1993). 13. P. M. Smith, M. Inoue and Jeffrey Frey, Appl. Phys. Lett. 37 (9), 797, (1980). 14. J. Zimmermann, Y. Leroy and E. Constant, J. Appl. Phys. 49, 3378 (1978).
SOLITON EXCITATIONS IN A N I N H O M O G E N E O U S D N A MOLECULAR C H A I N
V. V A S U M A T H I A N D M. D A N I E L Centre for Nonlinear Dynamics, Department of Physics, Bharathidasan University, Tiruchirappalli - 620 024, India. E-mail:[email protected] and [email protected] We study nonlinear dynamics of a periodic inhomogeneous DNA double helical chain under dynamic plane-base rotator model by considering angular rotation of bases in a plane normal to the helical axis. The dynamics is governed by a perturbed sine-Gordon equation. The perturbed soliton solution is obtained using a multiple scale soliton perturbation theory. The perturbed kink-antikink solitons represent formation of open state configuration with fluctuation in DNA.
1. Introduction Deoxyribonucleic acid (DNA) plays an important role in the conservation and transformation of genetic information in biological systems l. Recently, solitons have been found to govern the fluctuation of double helix between an open state and its equilibrium states 2 ~ 5 . Among the different possible motions, rotational motion of bases in DNA is found to contribute more towards the opening of base pairs 6 _ 9 . In this context, a number of theoretical models have been proposed to study the nonlinear dynamics of DNA. In this paper, we study nonlinear dynamics of DNA double helix with periodic inhomogeneous (site-dependent) strands by considering a plane base rotator model 6 ' 7 . The Hamiltonian for such an inhomogeneous DNA model with the helical axis along z-direction is written as 8 ' 9
* = £ -(<}>l + i>l) +
Jfn{2-COs(cj>n+1
-COs(lpn+1 - Ipn)] ~ V [1 - COs((j)n - 1pn)]] .
(1)
Here
537
sent a measure of stacking and hydrogen bonding energies respectively, / „ indicate inhomogeneities in the stacking engery. Using the above Hamiltonian we derive the dynamical equation in section 2 and solve the same using multiscale soliton perturbation theory in section 3. The results are concluded in section 4. 2.
Dynamical equation
The Hamilton's equations of motion for DNA bases corresponding to Hamiltonian (1) is written as 14>n = J [fn sin{(pn+i ~
Iiptt = Ja [f{z)i>zz + fzipz] + r] sin(i/j - <j>).
(4) (5)
Here the suffices t and z represent partial derivatives with respect to t and z respectively. After choosing rj = — ^- and rescaling the time variable as i = \f^j-t in Eqs. (4) and (5), the resultant equations after adding and subtracting and upon choosing <j> = —ip, become * H - $zz + sinV = e \g(z)9z]z
.
(6)
While writing Eq. (6), we have further chosen ^ = 2<j> and expanded the inhomogeneity in stacking energy in powers of a small parameter e as f{z) = 1 +eg(z). When e = 0, Eq. (6) reduces to the completely integrable sine-Gordon equation which admits kink-antikink solitons 6 and hence we call Eq. (6) as perturbed sine-Gordon equation. Perturbed sine-Gordon equations with different perturbations have been studied by several authors in the past by using inverse scattering transform and related mathods 1 0 - 1 2 . Mann 13,14 has made a detailed study on the derivation of soliton solutions to perturbed sine-Gordon equation using Greens function formulisom dispensing with the inverse scattering method. The perturbed sine-Gordon equation (6) is derived from an entirely different context here and in the next section we carry out a soliton perturbation analysis.
538
3. Soliton perturbation theory 3.1. Linearization
of the perturbed
sine-Gordon
equation
When the perturbation is absent (e = 0), the integrable sine-Gordon equation admits the following kink-antikink soliton solution. ty(z,i) = 4arctanexp[±m(z
. (7) V1 — v2 Here v and m _ 1 are real parameters that represent the velocity and width of the soliton respectively. In order to study the effect of perturbation 15 the time derivative and * in Eq.(6) are replaced by the expansion J* =
-sk + e 5?r + e2m^ + ••••
and
— vi)],
m =
* = * (0) + e * (1) + e2 * (2) + •••• a n d
equate the coefficients of different powers of e. Thus, at O(e^) )
*i o °i o -*£ +«n*(°)=0,
then we
we obtain
(8)
for which the one soliton solution takes the form (7) with t replaced by toDue to perturbation, the soliton parameters namely m and £(£ = vt) are now treated as functions of the slow time variables to,ti,t2, •••• However, m is treated as independent of to. The equation at 0(e^) takes the form
~ *& + co S *(°)*« = gV?) + g2^
- 2*^.
(9)
We rewrite the above equation by using the result costy^ — (1 — 2sech2() where C — ±m(z — £)>£to = wo> obtained from the solution of Eq.(8) and make the transformations z — -*- + vto and to —• to + C to represent everything in a co-ordinate system moving with the soliton. Further, upon using a new transformation, r = ^ — " + ^°K for later convenience, the resultant equation of (9) becomes # r C « - * W + (1 - secft 2 C)* (1) = F™,
(10)
where F ( 1 ) = 2 [fii(C)sec/iC]c + 4w0sec/iC [m tl + (m 2 £ tl - Cm tl ) ton/iC] . (11) The solution of Eq. (10) is searched by assuming * ( 1 ) (C,r) = and FM = X((()H(T) which leads to X c c + (2sec/i 2 C-l)X = A0Xc(C),
TT - \0T = H{T),
X(()T(T)
(12)
where Ao is a constant. Eqs.(12) can be solved without using explicit forms for Xf (£) and H{Q but by using F^> after simple manipulations. Also, on using the first order correction oo
/
X(C,k)T(r,k)dk+ J2 XjiQTjir), "°°
j=o,i
(13)
539 we get the continuous and discrete eigen states as =
(1
1.2
k Xktanht)etkC V27r(l + k2)
= secK:Xi{0
T
^=;wWI
To(T) =
fl-|^
3.2. Variation
j°°
dKFVXM),
of soliton
= CsecK;
-1]
(14)
<15)
T,(T)=0.
(16)
parameters
In order to find the first order correction to soliton we need to evaluate the eigen states (15) and (16) explicitly for which we need the values of mtl and £tj which can be found from the nonsecularity conditions by substituting the values of F^((,T),XO(() and Xi(() respectively from Eqs. (11) and (14). The results give the time evolution of the inverse of the width (m) and the velocity (£ tl ) of the soliton as 1
f°°
mtl = - — / \g(()sech(]c sech(d(, *Vo 7-oo i r°° 6 l = ~ 2 ^ j _ l9(0sech(](tsech<;dC.
(17) (18)
We consider a situation where the sequence of bases of DNA appear in a periodic form. Hence, we substitute g(£) = cosC, in Eqs. (17) and (18) and evaluate the integrals in the right hand side. Finally, we obtain mjj = 0, £tj = 16 ^ which can be written in terms of the original time variable t as m = mo, £t = v — vo + j^i—, where 1/mo and VQ are the initial width and initial velocity of the soliton respectively. The above results show that while the width ( m _ 1 ) of the soliton remains constant the velocity of the soliton increases.
3.3. First order perturbed
soliton
Now, we explicitly construct the first order perturbation correction to the one soliton by substituting the values of the basis functions {X} —
{*(C,fc),x 0 (C),Xi(0}
and { T }
= { T ( T > fc )' T o( r ). T i( T )}
after u s i n
e
the
1
values of F^ \rnt1 and ^ in Eq. (13). The integrals are then evaluated using standard residue theorem and after very lengthy calculations, we finally obtain \P(z,io) ~ 4 arctanexp[±mo(z - «o
540
where b = ^(v2 - 1) and d = g ^ L . The rotation of bases 0(z,t o ) can be immediately written down by using the relation 0 = f. In Figs. (la,b) we plot 0(2;, t 0 ) for the parametric choice v0 = 0.4. From the figure, we observe that there appears fluctuation in the form of a train of periodic pulses closely resembling the shape of the inhomogeneity profile in the width of the soliton as time progresses. Also, as time passes, the amplitude of the
to 5
\
i \
10
(a)
20
30
(b)
Figure 1. The perturbed (a) kink and (b) antikink solitons (Eq.(l9)) for v0 = 0.4.
pulses generating this fluctuation increases. However, in the asymptotic region of the soliton there is no change in the topological character and no fluctuation exists in that region. It shows that the periodic inhomogeneity in stacking energy does not affect very much the opening of bases except fluctuations in the form of a train of periodic pulses in the localized region of the kink and antikink-soliton. Recently, base flipping in DNA was supported experimentally through flourescence 16 and electron microscopy measurements 17 . 4. Conclusion In this paper, we studied the nonlinear dynamics of DNA double helix with oscillatory stacking inhomogeneity by using a multiple scale soliton perturbation theory. The results show that, the speed of the soliton increases wrth a correction that is proportional to the square of the initial width of the soliton and inversely proportional to its initial velocity. We find
541 that fluctuation in the form of periodic pulse trains resembling the shape of the inhomogeneity is generated in the width of the soliton representing open state configuration of DNA, the amplitude of which increases as time passes on. The results indicate that inhomogeneity in stacking energy in DNA double helix can (i) introduce small fluctuations during the process of opening and closing of base pairs and (ii) change the speed with which the open state configuration can travel along the double helical chain. As nature selects generally inhomogeneous DNA, the functions such as replication and transcription can be explained more viably through formation of open states through our inhomogeneous model. Acknowledgements M. D and V. V acknowledge the support of the Abdus Salam International Centre for Theoretical Physics , Trieste, Italy under Senior Associateship and Young Collaborator Programme respectively. References 1. L. Stryer, Biochemistry. 4 ed. (W. H. Freeman and Company, New York, 1995). 2. L. V. Yakushevich, Nonlinear Physics of DNA (Wiley-VCH, Berlin, 2004). 3. L. V. Yakushevich, J. Biosci. 26, 305 (2001). 4. M. Peyrard, Nonlinearity 17, Rl (2004). 5. A. Campa, Phys. Rev. E 63, 021901 (2001). 6. S. Yomosa, Phys. Rev. A 27, 2120 (1983). 7. S. Yomosa, Phys. Rev. A 30, 474 (1984). 8. S. Takeno and S. Homma, Prog. Theor. Phys. 70, 308 (1983). 9. S. Takeno and S. Homma, Prog. Theor. Phys. 72, 679 (1984). 10. Y. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989). 11. D. J. Kaup and A. C. Newell, Proc. R. Soc. London A 361, 413 (1978). 12. W. Hai, Z. Zhang and J. Fang, Eur. Phys. J. B 21, 103 (2001). 13. E. Mann, J. Phys. A: Math. Gen. 30, 1227 (1997). 14. E. Mann, Nonlinear Coherent Structures in Physics and Biology (NATO ASI Series B 329) ed. K.H. Spatschek and F. G. Mertens ( Plenum, New York, 1994). 15. J. Yan, Y. Tang, G. Zhou and Z. Chen, Phys. Rev. E 58, 1064 (1998). 16. K. Liebert, A. Hermann, M. Schlickenrieder and A. Jeltsch, J. Mol. Biol. 341, 443 (2004). 17. A. S. Borovik, Y.A. Kalambet, Y. L. Lyubchenko, T.V. Shitov and E. Golovanov, Nuc. Acids. Res. 8, 4065 (1980)
STABILITY OF S T R U C T U R E D P R E Y - P R E D A T O R MODEL
W E N D I WANG* Department
of Mathematics, Southwest Normal Chongqing, 400715, P. R. China E-mail: [email protected]
University,
A stage structure is incorporated into a prey-predator model in which predators are split into immature predators and mature predators. The local stability of equilibria are analyzed and the Hopf bifurcation is found. Sufficient conditions for the global stability of the positive equilibrium is studied by compound matrix.
1. Introduction Classical population models such as the Volterra prey-predator model and the Holling prey-predator models assume that all individuals within each population are functionally identical. As a result, the dynamical behavior of the models is simple in many cases. However, it is known that the heterogeneity of individuals could induce rich dynamical behaviors. In Wang and Chen 3 , a stage structure for predators is incorporated into the Volterra prey-predator model where predators are split into immature predators and mature predators. It is found that the model admits large amplitude periodic oscillations, which disappear when the stage structure is neglected. In Wang and Mulone 4 , the ratio-dependent functional response is introduced into the predator-prey model with the structure for predators and thresholds for the permanence and extinction of populations are obtained. The common point of these models is that the mortality rate of juvenile predators is a constant. However, from biological perspective, juvenile predators assimilate nutrients for maintaining life and increasing body size. Motivated by these observations, we improve the model in paper 3 by assuming the death rate of juvenile predators depends upon the availability of nutrient. Let x be the density of the prey, yi denote the density of immature predators and y2 denote the density of mature predators. We assume that *Work partially supported by the Scientific Fund of Southwest Normal University.
542
543
only adult predators attack at the prey and provide nutrients to their juveniles. Following general modelling methodology, we suppose that adult predators assimilate nutrients for reproduction. We also assume that the death rate of juvenile predators is regulated by the availability of nutrient, which is of the form of Leslie 1. we obtain the following structured model: (3V2 y\ TVl y[ = kf3xy2 - d x = x (r
• ax
1)
(y'2 = Tyi - d2y2 where r is the intrinsic growth rate of the prey, a is the density dependent coefficient, (3 is the attacking coefficient of predators, k is the conversion coefficient of nutrient to juvenile predators, d\ is the mortality rate of juvenile predators and d2 is the death rate of adult predators, T is the maturation rate. By introducing scaling variables x = uL,y\ = V\A\,y2 = v2A2,t = e9 with e = l/r,u = r/a,v\ = r2/(adi),v2 = r/(32 and then representing dimensionless variables L, Ai and A2 by x, y\ and y2, respectively, we obtain x' = x(l -x
-y2),
yx = axy2
byi,
(2)
X
j ' 2 =PVi ~qV2, where kd\
, T o= - , r
a=
Tj3 p= —, ad\
_rf2 q= r
2. Analysis Let us look for a positive equilibrium of (2). Setting the right-hand sides of (2) to zero, we obtain (pyi- QV2 = o, I 1-1-2/2=0, y2 axy2 <
It follows that
--byi=0 X
x (apx — qb) Vi =
q p x (apx — qb)
< V2=
2
,
ap 2x2 + q(q - qb)x - q2
(3)
544
Then it is easy to see that there exists a unique positive equilibrium E — (ar*,|/J,i/5)in(2)if ap > bq,
(4)
and there is no positive equilibrium in (2) if the inequality (4) is reversed. Note that (4) implies that the parameter d\ does not affect the existence of the equilibrium. By similar arguments to those in paper 3 , we obtain Theorem 2.1. / / (4) holds, the populations of the prey and predators are permanent. Further, if ap < bq, adult predators and juvenile predators go to extinction as time tends to infinity. We are now in a stage to discuss the stability of the positive equilibrium. The characteristic equation of E is A0X3 + AXX2 + A2X + A3 = 0, where Ao = x* > 0,
Ax = x*2 + 2y\ + (b + q)x* > 0,
A2 = 2x*y\ + (b + q)x*2 + 2qy\* + bqx* - ax*2p > 0, A3 =2qx*yl+bqx*2
-apx*3
+ apy2x*2
+pyf
> 0,
where (3) and x* > qb/(ap) are used. Then by the Routh-Hurwitz criteria, we see that the positive equilibrium E is asymptotically stable if and only if AiA2 > A0A3.
(5)
Numerical calculations show that (5) holds for a large range of parameter values, and there exists a bifurcation set such that stability switches occur. For illustration purpose, we consider the following example. Example 2.1. We fix a = k = r = 1,T = 0.1,/? = 20 and d2 = 0.2. Then a = di, b = 0.1,p = 2/d\ and q = 0.2. By (5), we see that the equilibrium is stable if d\ > 0.2578 and is unstable if d\ < 0.2578. Numerical calculations show that a Hopf bifurcation occurs at di — 0.2578 and there are periodic solutions when d\ decreases from 0.2578 (see the bifurcation graph of Figure 1). This shows that the coefficient d\ has the tendency to stabilize the equilibrium.
545
Figure 1. The graph of the bifurcation of 3/2 component versus di which indicates the emergence of the Hopf bifurcation when d\ decreases from 0.2578 where a = di,b = 0.1,p = 2/di and q = 0.2.
Next, let us assume that (4) holds and look for sufficient conditions that ensure the global stability of E. To this end, we invoke the technique of the compound matrix in paper 2 . By Theorem 2.1, there is a compact region D in the interior of R\ such that positive solutions of (2) eventually enter and stay in D. Let <^(t,T]) = (x(t),yi(t),y2(t)) be a solution of (2) in D. Then the second compound matrix of the Jacobian matrix of (2) is \-2x-
j[2]mv))
2^-b
"2/2
P
0 Let z — (X,Y,Z). pound system
l-2x-y2-q
ay2 + (^)
2
0
-q-2f-bj
2
By paper , E is globally attractive if the second com-
z\t) = jM(at,v))4t)
(6)
is uniformly asymptotically stable, and the exponential decay rate of all solutions to (6) is uniform for rj in each compact subset of D. Theorem 2.2. Let (4) and (5) hold. If a < 1 andp < q, then E is globally stable.
546
Proof. We consider the following Liapunov function V =
x
maX{^\X\,^\Y\,\Z\}. x
Calculating the right derivative along solutions of (6), we have
DJ*\X\)<(*-*)*\X\ \ x
'
\V\ x J x + ^ ( ( l - 2 x - y
(7) 2
- 2 ^ - 6)1X1 + oa:|y| + x\Z\)
X \
X
J
and ^x
>
\y\
x) x
+ ^(p\X\
+
(8)
(l-2x-y2-q)\Y\),
Note also that D+lzl<{^yi
+
yi)yilY\-(q
+ 2^ + b)\z\
2/i x x = {2^-+b+^-)^-\Y\-(q x yi x
x + 2^-+b)\Z\. x
(9)
If V(t) = *$f-\X(t)\ with V(t) > m a x { ^ | Y ( £ ) | , \Z{t)\}, it follows from (7) and the first equation of (2) that D+V<(^-((l-a)x + 2^ + b-yi))v. (10) \yi x j By the first equation of (2), it is easy to see that l i m s u p t ^ ^ x(t) < 1. For notational simplicity, we assume that x(t) < 1 for all t. It follows from (10) that the first equation of (2) that
D+v<(^--by. Similarly, if V(t) = ^§\Y(t)\ we have
(ii)
with V(t) > max{»$\X(t)\,
\Z(t)\}, by (8)
D+V<(y±-x\v, if V(t) = \Z(t)\ with V(t) > msx{*$\X{t)\,
D+V
-{lt~9)V-
*$\Y(t)\},
(12) ™ have
(13)
547
In other cases, it is easy to verify that one of (11), (12) and (13) must be satisfied. T h u s , if x(t) is confined t o D and e = inf
{b,q,x(t)},
then e > 0. Therefore, we obtain t h a t D+V < (^— - e ) V. yi
(14)
Note t h a t
tJo
| - e ) d S = iln(yi(t)/yi(0))-e<-|
when t is large enough. We conclude t h a t (6) is uniformly asymptotically stable, and t h e exponential decay r a t e of all solutions t o (6) is uniform for r\ in each compact subset of D. u
References 1. N.F. Britton, Essential mathematical biology. Springer Undergraduate Mathematics Series. Springer-Verlag London, Ltd., London, 2003. 2. M.Y. Li; Hal L. Smith and L. Wang, Global dynamics an SEIR epidemic model with vertical transmission. SIAM J. Appl. Math. 62 (2001), 58-69. 3. Wendi Wang and Lansun Chen, A predator-prey system with stage-structure for predator. Corn-put. Math. Appl. 33 (1997), 83-91. 4. Wendi Wang; G. Mulone; F. Salemi and V. Salone, Permanence and stability of a stage-structured predator-prey model. J. Math. Anal. Appl. 262 (2001), 499-528.
D E C A Y ESTIMATES IN CHEMOTAXIS: AGGREGATION OF GLIA A N D A POSSIBLE APPLICATION TO ALZHEIMER'S DISEASE SENILE PLAQUES
M. WEBBER AND B. STRAUGHAN Department of Mathematical Sciences, Durham University, Science Laboratories, South Road, Durham, DH1 3LE, UK E-mail: [email protected]; [email protected]
Some models of chemotaxis are reviewed, particularly those involving three coupled nonlinear partial differential equations. It is shown how decay bounds may be formulated in these cases. Applications are considered, in particular to a model for glia aggregation, and the possible connection with Alzheimer's disease.
1. Introduction The subject of chemotactically driven movement of cells has been a major topic since models were developed by Keller and Segel [1, 2, 3]. In particular, mathematical analysis has concentrated on questions of pattern formation, existence, control, blow-up and collapse, see e.g. Nagai [4], Hiller and Painter [5], Biler [6, 7], Myerscough et al. [8], Nagai et al. [9, 10], Owen and Sherratt [11], Ryu and Yagi [12], Senba and Suzuki [13, 14], Straughan [15], Jager and Luckhaus [16], Herrero and Velaquez [17, 18, 19], Othmer and Stevens [20]. These and other mathematical topics in chemotaxis are reviewed in detail by Horstmann [21]. More recently, studies involving the question of deriving bounds which guarantee blow-up (or cell aggregation) will not occur have been gaining impetus. We mention work of Gajewski and Zacharias [22], Horstmann [23], Kowalczyk [24], Osaki et al. [25], Payne and Straughan [26] and Quinlan and Straughan [27]. 548
549
2. Chemotaxis Models Payne and Straughan [26] adressed the system da = V • (p(p)Va) - V • (ax(p)Vp), (1) dt Op = -k(p)p + af(p) + DAp, (2) dt which is one of the original models of Keller and Segel [1, 2, 3] for describing the behaviour of the life cycle of the cell dictyostelium discoideum. Their method hinges on constructing a suitable "energy", or Lyapunov function for demonstrating decay for a solution to the full nonlinear system. The energy method, in general, is described in the book by Straughan [28], and recent developments of the energy method including some in a biological context are those of Webber [29], Mulone [30], Mulone at al. [31], and Rionero [32, 33]. More complicated models involving chemotaxis leading to three coupled nonlinear partial differential equations have been developed for describing cell aggregation in Escherichia coli (E. coli) and Salmonella typhimurium, see Murray [34], p.253, and for describing aggregation of glia cells in the human brain, by Luca et al. [35]. The systems of Murray [34], pp.264, 265, have form kin " 9n ^ . _ Us2 \ ~ = DnAn-V. 2 (3) 0"
l(k2+c)
\
sn2 ic, OT k6 + nz ds ^ . , ns2 — = DsAs k8-——3, dt kg + s2 when the growth medium is semi-solid, or dc
_ .
,
dn
^
dt
Uc A C + f c
.
-=DnAn-VA r- 1 h
ds
=
-As,
„
L(fc2+c)2VCJ )
^6+n2,
_
n
(4) (5)
(6) (7) (8)
when the growth medium is a liquid. In these systems the chemotaxis effect is manifest through the term involving Vc in the n equation. In particular, we note that the chemotaxis function x(n, c) has form k\n X(n,c) = (9) (*2 + c)2
550
see Murray [34], p.263. By contrast, the system effectively employed by Luca et al [35] is -^
= fxAm -
XV
• (mV0) + X2V - (mVV>),
(10)
-£ = DxAcf) + a i m - brf,
(11)
-^
(12)
= D2Aip + a2m - b2tp.
Here, m is the cell density of microglia, <j> is the concentration of attractant, interleukin —1 (IL—1/3), and ip is the concentration of chemorepellent, tumour-necrosis factor—a (TNF—a). In contrast to the above systems of Murray [34] the chemoattractant factor xi(m,c) is chosen as x i = Xim> for xi constant. The chemorepulsion term involving X2 is new. Luca et al [35] give a detailed linear instability analysis of the steady solution (m,, V>) = {ino,bi(f>o/ai,b2ipo/a2) and produce a numerical simulation in one space dimension. They link this with senile plaque formation in the human brain and destruction of neurons and connection with Alzheimer's disease. 3. Nonlinear Analysis Quinlan and Straughan [27] deal with the non-dimensional form of equations (10)-(12), namely -^
= Am - AiV • (mV
e1^=A(j>
+ a2(m-cj)),
dib t2-^- = Atp + m-ip
(13) (14) (15)
on some domain Q. bounded by the surface T, with the boundary conditions dn dn dn at T. The analysis of [27] perturbs nonlinearly about the non-dimensional steady state (m, , ip). If m, , "ip denote nonlinear perturbations to m, 4>, •^ then Quinlan and Straughan [27] employ a solution measure ^ ) = ^IH|2 + ^ | | V ^ | | 2 + . ^ | | V ^ | | 2 ,
(17)
551 where A, A2 are positive constants to be selected judiciously, and ||.|| is the norm on L2(Cl), when 0, is a bounded domain in R 2 . When Q. is a bounded domain in R 3 then the threshold of [27] is essentially the same, but the analysis has to proceed via the more involved functional
E{t) = \\\mf + ^\m\\2
+ ^ I I W H 2 + §||A
f° r £1 >&2 > 0 to be chosen. Quinlan and Straughan [27] show that if Ax < 1 + ^
(19)
for a suitable e € (0,1), where X\ = \\(£l) > 0 and E(0) is suitably restricted, then E(t) decays to zero at least exponentially fast. In this case glia aggregation cannot occur. They interpret this by showing that in a one-dimensional spatial domain aggregation cannot occur if fh < O(10" 5 - 10" 3 ) cells /jm - 3 , where the specific value of the right hand side is calculated. This represents a threshold glia density below which cell aggregation will not occur. We mention that a possible extension is to analyse the model with the possibly more realistic chemotaxis function xi given by - ^ Xim Xl
(h + W
This reflects the fact that increasing chemoattractant concentration may saturate, and attraction will not increase indefinitely. Acknowledgments The work of M. Webber is funded by a research studentship of the Engineering and Physical Sciences Research Council. References 1. 2. 3. 4. 5.
E. E. E. T. T.
F. Keller and L. A. Segel, J. Theor. Biol. 26, 399 (1970) F. Keller and L. A. Segel, J. Theor. Biol. 30, 225 (1971) F. Keller and L. A. Segel, J. Theor. Biol. 30, 235 (1971) Nagai, Adv. Math. Sci. Appl. 5, 581 (1995) Hiller and K. Painter, Advances in Mathematics 26, 280 (2001)
552 6. P. Biler, Adv. Math. Sci. Appl. 8, 715 (1998) 7. P. Biler, Adv. Math. Sci. Appl. 9, 347 (1999) 8. M. Myerscough, P. Maini and K. Painter, Bull. Math. Biol. 60, 1 (1998) 9. T, Nagai, T. Senba and T. Suzuki, Hiroshima Math. J. 30, 463 (2000) 10. T. Nagai, T. Senba and K. Yoshida, Funkcialaj Ekvacioj. 40, 411 (1997) 11. M. R. Owen and J. A. Sherratt, J. Theor. Biol. 189, 63 (1993) 12. S. U. Ryu and A. Yagi, J. Math. Anal. Appl. 256, 45 (2001) 13. T. Senba and T. Suzuki, Advances in Mathematical Sciences and Applications. 10, 191 (2000) 14. T. Senba and T. Suzuki, Advances in Differential Equations. 6, 21 (2001) 15. B. Straughan, Explosive instabilities in mechanics. Springer, Heidelberg (1998) 16. W. Jager and S. Luckhaus, Trans. Amer. Math. Soc. 329, 819 (1992) 17. M. A. Herrero and J. J. L. Velaquez, J. Math. Biol. 35, 177 (1996) 18. M. A. Herrero and J. J. L. Velaquez, Math. Ann. 306, 583 (1996) 19. M. A. Herrero and J. J. L. Velaquez, Ann. Savola Normale Sup. Pisa. 26, 633 (1997) 20. H. G. Othmer and A. Stevens, SIAM J. Appl. Math. 57, 1044 (1997) 21. D. Horstmann, Jahresbericht der DMV. 105, 103 (2003) 22. H. Gajewski and K. Zacharias, Math. Nachr. 159, 77 (1998) 23. D. Horstmann, Colloquium Mathematicum. 87, 113 (2001) 24. R. Kowalczyk, J. Math. Anal. Appl. 305, 566 (2005) 25. K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Nonlinear Anal. 51, 119 (2002) 26. L. E. Payne and B. Straughan. Manuscript, 2001. 27. R. A. Quinlan and B. Straughan, Proc. Roy. Soc. London A. 461, 2887 (2005) 28. B. Straughan, The energy method, stability, and nonlinear convection. Second edition, Springer, New York (2004) 29. M. Webber, Math. Meth. Appl. Sci. In the press (2005) 30. G. Mulone, Far East J. Appl. Math. 15, 117 (2004) 31. G. Mulone, B. Straughan and W. Wang. Stability of epidemic models with evolution. Manuscript. 32. S. Rionero, II Nuovo Cimento. 119, 773 (2004) 33. S. Rioner, Math. Biosci. Eng. To appear (2005) 34. J. D. Murray, Mathematical Biology. II. Spatial models and biomedical applications. Springer, New York (2003) 35. M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Bull. Math. Biol. 65, 693 (2003)
E X T E N D E D T H E R M O D Y N A M I C S W I T H CONSISTENT ORDER
WOLF WEISS Weierstrass Institute for Applied Analysis and Stochastics Mohrenstr. 39 10117 Berlin, Germany E-mail: weiss@wias-berlin. de
The choice of variables in extended thermodynamics is arbitrary and this leads to some poblems. In the version with consistend order, every variable has a certain order of magnitude and the set of variables taken into account contains only variables up to a choosen order. In this paper the determination of the order of magnitude is calculated with a method similar to the Chapman- Enskog method.
1. Kinetic Theory We consider a monatomic ideal gas of particles with mass m. The basic quantity in the Kinetic Theory is the distribution function / ( x , c, t) which depends on the space coordinate x, the absolute velocity of the particles c and the time t. All quantities of interest may be obtained by the so called moments of / , namely 0th moment
g = m J fdc
mass density
1th moment
QVi— m J Cifdc
momentum density
moment
ge = ff P
C2fdC =mfC
internal energy pressure deviator (1)
2
moment
4th moment
fjC CifdC qi = 6
553
/M)<^C
heat flux full traceless part double trace
554
Here v denotes the macroscopic velocity, C the peculiar velocity and / M the Maxwell distribution which is given by
The quantity 9 is an abbreviation for kT/m where k denotes the Boltzmann constant and T the absolute temperature. Angular brackets denote the symmetric and traceless part. 2. Extended Thermodynamics 2.1. Some choices of
variables
In equilibrium the state of a gas is complete described by the five moments density g, velocity v and temperature 9. The concerning field equations for these variables are given by the well known Euler equations. In order to describe a nonequilibrium process, Grad 1 derived the so called 13 moment theory, which extends the Euler case by the additional moments pressure deviator p
g
Vi
9
13 moments
g
Vi
9
P
1i
20 moments
g
v^
9
P
qi
14 moments
g
Vi
9
P
qi
g
Table 1, Set of variables for different theories. Especially in the high moment cases the choice of the variables is quite arbitrary. One may ask for instance: Which quantity is more important, the full traceless part Q
555 are additional higher moments of the same magnitude like g
of the order of
magnitude
For the distribution function / the Boltzmann equation holds. For simplicity we consider the BGK model 7 in one-dimensional case
W+C^
=
7(/-/M)-
(3)
T is a relaxation time and may be interpreted as a Knudsen number. We multiply (3) with r and consider equilibrium. Setting / equal to the Maxwell distribution the right hand side vanishes and for the left hand side follows, by using the Euler equations in order to replace the time derivatives
We conclude that in equilibrium the derivatives of vx and 9 vanishes. However vx and 9 are equilibrium quantities, they may have an arbitrary value in equilibrium. Therefore we say vx and 9 are of order zero. If we consider a process which is close to equilibrium, the derivatives of vx and 9 have to be small. We conclude that the derivative with respect to space increases the smallness of the expression. The same holds for the time derivative and also Q is of order zero. We generalize and assume that for every quantity the derivatives da da T-— and (5) dx dt increases the order of magnitude by one. Now we expand / in a series of Hermite polynomials / = / M ^
—2^Gr,li>r,l-
r,l Q\/9
(6)
556 Some of the first Hermite polynomials Vv,i are given by ^o,o = l , ,/,
^0,1 = 7f>
_ V3C<XCX>
^1,0 = J
_
- ^ ^ ^ ,
(7)
C2-56r
1
The expansion coefficients Gr^ are the moments of the Hermite polynomials Grti = V62r+lm
fipr,ifdC.
(8)
The Gr?; moments are connected with the classical moments Go,o — Q, Go,i = 0,
Gito = 0,
\/3 Go,2 — —p<xx>,
1 Gi,i — —ITTA*' (9)
Now we determine the order of magnitude. First step: We insert / M into the left hand side and the expansion (6) into the right hand side of (3). After replacing the polynomials in C by Hermite polynomials and divided by / M we get O(l)
2 ,
dvx
/~5 ,
09 (10)
O(l)
0(>1)
From the comparison of the Hermite polynomials we conclude that the moments Go,2 and Gi,i have the order one, all other moments have an order grater than one. Second step: Now we insert into the left hand side of (3) the expansion (6) in which the sum runs up to the first order terms, namely Go,2 and Gi,i, we get 0(2)
2^ '
]
dG0<2
'
OX • • • + G0,3lp0,3 + G2filp2,0 H ^ ^ > v °(1) 0(2)
OGZ~
~
~~
OX
de OX
\- G2,2^2,2 '
+ G0,5^0,5 H ^ v 0(>2)
'
/-,-,-,
557
The coefficients ar,/ depend only on g, v, 6 and are not of interest here. From the comparison of the coefficients of the Hermite polynomials we conclude that the moments Go,3, G2,o, Gi,2, Go,4, G2,i, ^i,3» ^3,o and <J2,2, are of order two. The further steps are following in the same manner. The results are listed in Tab. 2. Prom this table we may read off the sets of variables for the zeroth order (Euler case), the first order (Grad's 13 moments) and the second order (a selection of 51 moments). The classical 14 and 20 moment cases do not occur! Order of magnitude 0 0 1 2
1
3
Q
Vi
9
3
Go,2
Gi,i
4
Rank of moment
2
5 6
Go,3 G2,0 Gj.,2 Go,4 G2,l Gi,3 G3,o G2,2
7
Go,5 Gi,4 Go,6 G3,l
Table 2, Order of magnitude of the G-moments. 2.3. Field
equations
The derivation of the field equations for the variables listed in Tab. 2 is quite simple. From the BGK model follows an infinite set of transfer equations for the Gr>i moments by multiplying (3) with ipr,i and integrating over the whole velocity space. As an example the transfer equation for Go,2 is shown 0(2)
,dG 0 , 2 dt
0(2)
<9G0,2 ,
2
dv a
0(2)
0(2)
O(l)
x
7 r,
dvx
dGhl 15 dx T
3
T9G0'3 = y/5 dx 0(3)
C -—^s
O(l)
(12)
558 By using the results of Tab. 2 we are able t o indicate the order of magnitude of every t e r m in the whole hierarchy. We choose the order of magnitude e.g. 2 and neglect all terms of higher order. In (12) this is the term with the derivative of Go,3. An explicit set of field equations automatically results. There are no additional moments which have to be considered as constitutive functions. This is t h e case for every order of magnitude. 3.
Conclusions
Extended Thermodynamics with consistent order has some important advantages. First we have t o decide only the order of magnitude. All variables follow automatically and the concerning field equations are also given explicitly. However, the order of magnitude which is needed to describe an process satisfactorily has t o b e chosen carefully. T h e problems of classical extended thermodynamics, mentioned above, are solved 6 ' 8 . T h e solutions of systems with consistent order compared with classical systems with the same number of variables converge faster 9 . T h e systems are stable, even though they are derived by a method which is similar to the Chapman-Enskog method. References 1. H. Grad, Principles of the kinetic theory of gases, Hanbuch der Physik XII, Springer (1958). 2. C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys. 83, 1021-1065 (1996) 3. I. Miiller and T. Ruggeri, Rational extended thermodynamics, Springer Tracts in Natural Philosophy Springer 37, New York (1998). 4. F. Brini and T. Ruggeri, Entropy principle for the moment systems of degree associated to the Boitzmann equation. Critical derivatives and non controllable boundary data, Cont. Mech. Thermodyn. 14, 2, 165-189 (2002). 5. M. Junk, Domain of definition of Levermore's five-moment system, J. Stat. Phys. 93, 1143-1167 (1998). 6. I. Miiller, D. Reitebuch and W. Weiss, Extended thermodynamics - consistent in order of magnitude, Cont. Mech. Thermodyn. 15, 2, 113-146 (2003). 7. P. L. Bhatnagar, E. P. Gross and M. Krook, A model for collision processes in gases I., Phys. Rev. 94, 511-525 (1954). 8. D. Reitebuch, Konsistent geordnete Erweiterte Thermodynamik, Dissertation, TU Berlin (2005). 9. I. Miiller and D. Reitebuch, Consistent Order Extended Thermodynamics and its Application to Light Scattering, to appear in the Proceedings of the Symposium on Trends in Applications of Mathematics to Mechanics (STAMM) Sept 30 - Oct 4, 2002 in Maiori, Italy,
ONE-DIMENSIONAL STATIONARY HEAT C O N D U C T I O N IN A R A R E F I E D GAS AT R E S T ANALYZED B Y CONSISTENT-ORDER EXTENDED THERMODYNAMICS
NANRONG ZHAO1'2 AND MASARU SUGIYAMA1 Graduate
School of Engineering, Nagoya Institute of Technology, Nagoya 466-8555, Japan Department of Physics, Sichuan University, Chengdu 610064, China E-mail: [email protected], [email protected]
One-dimensional stationary heat conduction in a rarefied gas at rest is studied consistently by the 3rd-order theory of consistent-order extended thermodynamics. The temperature field is obtained in a series expansion form, which agrees quantitatively with numerical calculations. The consistency of such a series solution is specially checked. As illustrations, temperature jump and entropy production at the boundary are explicitly analyzed.
1. Introduction Recently consistent-order extended thermodynamics (COET) 1 was proposed as a revised version of extended thermodynamics (ET) 2 , which makes use of combinations of ordinary moments in ET as its field variables, called G-moments. Each G-moment can be assigned an order of magnitude as a measure of its importance in an irreversible process under consideration. In the context of COET, there still remains the so-called uncontrollable boundary value problem. A reliable resolution to such a problem should be based on a consistent general solution of the field equations at the very beginning of the analysis. Otherwise, it would be entangled with the inconsistency of the solution and then becomes more complicated. Main purpose of the present paper is to explore a consistent analysis for one-dimensional stationary heat conduction in a rarefied gas at rest on the basis of the 3rd-order COET. Temperature field is obtained in a series expansion form. The full-consistency of such a solution is emphasized and checked. Finally, we explicitly analyze the temperature jump and the entropy production at the boundary. 559
560
2. A Consistent Solution for the 3rd-Order COET The basic equations for the 3rd-order COET can be reduced to a linear differential equation of 2nd-order as follows1: 42
GltlK;
dx2
+
5
2*ndx
M
Gi,
(1)
where dimensionless quantities have been introduced according to the definitions adopted in ref. 3. 0 is the temperature field which depends only on the x-coordinate (0 = kT/m, where T is the kinetic temperature of the gas, k is the Boltzmann constant and m is the mass of an atom). Kn is the f2 Knudsen number and Gn = q where q is the heat flux. We remind that G14 is a constant due to the energy balance. The mathematically "exact" solution for Eq. (1) can be easily obtained. However, it is not physically acceptable. Indeed, the 2nd and 3rd-order moments in the 3rd-order theory 1 evaluated by such an "exact" solution have evident inconsistency in magnitude at one boundary as seen in Fig. 1.
2nd-order moments 3rd-order moments
Figure 1. The "exact" solution for the 3rd-order COET under a condition: 9L = 0.886, OR = 1.109 and G i l = 0.0252.
Therefore, it is natural from the very concept of "order" in COET that a consistent solution of the temperature field in the 3rd-order theory should be in the form of a power series with respect to x up to cubic such that 6(x) = 90 + 9xx + 02x2 + 63x3
(2)
561 with expansion coefficients 60, 6\, 02 and #3, which can be specified by an iteration method applied to Eq.(l). Then we have
" < x ) = ( , ° + M + 8«kr( G »-v! K "'' 1 ) 1 2
It involves three constants 90, 0\ and Gi,i, two of which, say #0 a n d 61, can be further determined by the boundary values of the gas temperature. G\:i, however, remains as an uncontrollable quantity. Comparison of the temperature field according to Eq. (3) with Ohwada's numerical results 4 under the same boundary conditions is made in Fig. 2, where Gi,i takes the optimum values estimated in our recent work 3 . Evidently, two groups of the curves agree quite well with each other, especially for smaller Knudsen numbers.
Figure 2. Comparison of our temperature profiles (solid lines) with Ohwada's numerical curves (dashed lines) (Ohwada's Knudsen number is regarded to be 0A255Kn ).
Furthermore, Fig. 3 presents all the 2nd and 3rd-order G-moments derived from Eq. (3), where the consistency of our solution is well demonstrated.
562
Figure 3. 0.1546).
T h e 2nd and 3rd-order G-moments derived from our series solution (Kn
=
3. Temperature Jump and Entropy Production at the Boundary As predicted by kinetic theory, there is a jump of temperature at the boundary, i.e., the difference of the gas temperatures at the boundaries (denoted by 6L and 6R) from the temperatures of the walls (denoted by TL and TR). We here analyze this interesting quantity by applying our series solution obtained in the preceding section. The calculation utilizes the so-called kinetic scheme 5 , where a Maxwellcondition as well as two conservation laws are considered. In the procedure, we assume a proper relationship between the uncontrollable value Gi ,i (heat flux) and the boundary values 6L and 9R3. The final results as well as Ohwada's corresponding numerical data 4 are listed in Table 1. Quite good
Table 1. Comparison of our gas temperatures at the boundaries with Ohwada's numerical data (wall temperatures: TL = 0.86 and TR = 1.14). 0.1546
0.3278
0.4564
0.7036
r\ present
0.891
0.910
0.918
0.930
^numerical
0.89
0.90
0.91
0.92
/^present
1.103
1.081
1.070
1.054
^numerical
1.11
1.09
1.08
1.07
Kn
563 agreement of the present result with that by numerical calculation provides a further confirmation of our consistent analysis. Meanwhile, we notice that the temperature jumps (A0 L ( J R) = @L(R) —TL(R)) become more pronounced in more rarefied gases. Lastly, let us deal with the entropy production at the boundary which is a measure of the irreversibility of the process occurring there. By taking into account the continuity condition of the normal component of heat flux q at the boundaries and assuming the entropy flux in the solid walls to be Q/TL(R), the entropy production rate a in unit area at the boundaries can be analyzed according to the following expressions: crL = hL
(4) 0fl
hR
Tk~ '
where the entropy flux h is determined by the kinetic theory of gases T h e final result is presented in Fig. 4.
o.oi
0.004
Figure 4. Entropy production rate a in unit area at the boundary for four Knudsen numbers: K„ = 0.1546, 0.3278, 0.4564, 0.7036.
As a fundamental requisite of the second law of thermodynamics, the entropy production in any case is positive. Simultaneously, this quantity, whether on the left or on the right, tends to be more and more considerable with the increase in Knudsen number.
564 4. C o n c l u d i n g R e m a r k s In this paper, through studying the heat conduction problem in a rarefied gas by the 3rd-order C O E T in detail, we have pointed out t h a t the "exact" solution for t h e basic equations of C O E T may suffer from the query of inconsistency. Taking this fact into account, we have proposed a solution in a series expansion form, which has been proved t o be fully consistent. T h e validity and usefulness of such a solution have also been demonstrated by analyzing explicitly t h e t e m p e r a t u r e j u m p and entropy production at the boundary. Although the present analysis is based on the 3rd-order theory, it is not difficult to extend the basic idea presented in this paper to higher-order theories. As can be seen in the preceding section, the existence of the t e m p e r a t u r e j u m p indicates a free b o u n d a r y value problem. Systematic analysis for the t e m p e r a t u r e j u m p is now under study. It can be shown t h a t C O E T has a good theoretical structure for studying such a problem. T h e details of the analytical results will soon be reported elsewhere. Acknowledgement This work was supported by t h e J a p a n Society for the Promotion of Science (1604076).
References 1. I. Miiller, D. Reitebuch and W. Weiss, Continuum Mech. Thermodyn. 15, 113 (2003). 2. I. Miiller and T. Ruggeri, Rational Extended Thermodynamics (Springer, New York) Springer Tracts in Natural Philosophy, Vol. 37. (1998). 3. M. Sugiyama and N. R. Zhao, J. Phys. Soc. Jpn 74, 1899 (2005). 4. T. Ohwada, Phys. Fluid 8, 2153 (1996). 5. H. Struchtrup, Phys. Rev. E 65, 041204 (2002). 6. H. Grad, On the Kinetic Theory of Rarefied Gases, Communications on Pure and Applied Mathematics 2, Wiley, New York, 331 (1949).
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