Quasilinear hyperbolic systems and dissipative mechanisms

+ 6(y,t))U'(x

+ 6(y,t))
= a(y,t)U'(x

and r
+ S(y,i)) + W,

with V and W in L 2 . Furthermore we require ay + /?y - F 2 = 0,

+ 6(y,t))

+

Vx(x.y,t),

98

Quasilinear Hyperbolic Systems and Dissipative Mechanisms

which implies Stt - A2Syy + St - 2b2Syt + OCy + 0y = 0. Then we integrate (3.94) over (—00, x) to get
(3.95)

,

+Vy +/3U'5y + Wy +
_

-

.

It is also known that ¥>(*.», 0) = /

[uo(z, ff) - tf (* + S(y, 0)]dz,

J—oo

and l + (U- U-)8t + V^y + /

V>2y<^ = 0.

«/ —00

Thus, with the help of the formula on <$(y, 0) and 8t(y, 0), it follows that M*> y, 0) = -tf(*, y, 0) - [tf(a? + <S(y, 0)) - u-ft(y.O) - V2{x, y, Q)8y(y, 0) -/!oo^2y(*,y,0)
Relaxation

99

The next step is to perform- the energy and the weighted energy methods to get a time uniform estimate with which the stability theorem can be proved. Set N(t) = su P o < r < JIM-, r)||2 + | M - , r)\\l + | M - , r)\\2 + | | ^ ( - , r)\\l

+ll*(^)III+ll*(-,r)||a + sup 0 < r < t / fR2 \x\((pl + 0 A2 - a2 + Ar2(ai - s2) - a2 - 2ska > 0 (k(a1-82)-8
bl =

We give some estimates on a, /?, V and W in the next lemma, which enable us to handle the transverse waves. The proof of this Lemma can be found in [133]. L e m m a 3.2.11.

a2<0{l){u--u+)j\U'\
V2dxdy<0{l){u--u+)

P

//

W2dxdy <0(l)sup\r\2sup

-

f f


(fr
\x\

tp2xdxdy

here v = u~ — u+. With these preminary estimates, we can use energy and weighted energy methods to get the following time-uniform estimate. P r o p o s i t i o n 3.2.12. If v and v~2N(T)

are sufficiently small, then it holds

100 Quasilinear Hyperbolic Systems and Dissipative Mechanisms that

IM-,0111 + IM-,011! + IM-,*)lli +11^(^)111 +II*MII2 + II**MII2 + Ik* \x\{lv + ¥&)(*> V, i)dxdy

+ /o (irT'VII 2 + ii^m + ||^||i + IMH + ||^||l)rfr

(3.96)

+ /o(II^Hi + \\St\\l)dr
0
We prove Theorem 3.2.8 with the help of time-uniform estimate (3.96). In fact, if the initial perturbation N(0) is small enough, then (3.96) is true a little longer and hence forever. The smallness of N(0) can be insured by the assumptions in Theorem 3.2.8. According to the embedding theorem H3{R2) C C^R2), H2{R2) C C°{R2) and H^R1) C C2(R% we have y>x(.,t) G C\R2), ?**(•,<) € C°.(R2), 6(-,t) G C2{Rl) and J t (.,*) G C 1 ^ 1 ) for all t > 0. Then u(-,t) G C 1 ^ 2 ) , ti t (-,t) G

C°(tf2).

Next, we turn to prove the asymptotic stability. Due to / Jo

/ /
< +oo,

and

lo+0° \& nR*(pl(x>y>t)dxdy\
J Jl2

(pl(x.y,t)dxdy->0

as t —> -f oo. Similarly, we have

J

J2(Q,

as t —>■ + oo. On the other hand, su

P(*,y)ei 2 IM*>2/>*)| 2

< HkM

+ rfy)(*,y,*)^y]1/2[//Ea(^ + ^Ly)(*,y,0^
Relaxation

101

in virtue of

=

fl002
< 2 [ / _ + ~
^x{x,y,t)dx]^

< [//«»(^ +^tf)(*,y,<)d*rf»]1/WKa(^, + ^)(«,y,*)d^»] 1 / 2 Since ff^i^lx then

+
is bounded uniformly with respect to 2,

sup

\
->0,

(*,y)GK 2

as t —» + 0 0 . The same argument leads to

sup (\
(ar,y)€R 3

+ \vxy{x,y>t)\)

-*0

as ^ -> + 0 0 . With the same argument as above, we arrive at sup | % , t ) | - » 0 , yGM 1

as £ —» + 0 0 . Because ipx(x,y,t)

= u{x,y,t) sup

— U(x +

8(y,t)),

|ti(jc,y,f)-{7(a:)| -► 0

(3.97)

(a:,y)GlR2

as t —>• -foo. Recall the transformation of the coordinates. (3.97) gives, in our original coordinates, that ^

sup

\u(x,y,t)-U{x-st)\->0

(3.98)

as t —> + 0 0 . Similarly, it turns that sup

\ux{x, y, t) - U'{x - st)\ -+ 0,

(3.99)

(a:,y)Gffi2

sup (x,y)GM 2

as 2 —» + 0 0 .

\uy{x,y,t)\^0

(3.100)

102 Quasilinear Hyperbolic Systems and Dissipative Mechanisms Let p0(x, y, t) = u(x, y, t) - U{x - st) pi(x,y,t)

=

v1(x,y,t)-Vi(x-st)

p2(x,y,t)

=

v2(x,y,t)-V2(x-st).

It is easy to get POt +Plx +P2y = 0,

Pit + axpx = f(U + p0) - f(U) - pi, P2t + a2py = g{U + p 0 ) - ^(^) ~ P2Hence ^ ( x ^ ^ j ^ e - V i ^ ^ ^ J + r e - ^ ^ i f / ^ + P o J - Z ^ - a x p o , } ^ . (3.101) By using (3.98)-(3.100), this implies s u p ^ ^ a |j9i(x, y, <)| -» 0 as t -> oo. The same argument leads to sup,^ y \ GH2 |/>2(#>2/>OI ~^ 0 as t —>- oo, which completes the proof of Theorem 3.2.8. The nonlinear stability of elementary waves is an important subject and some progresses have been made for relaxation approximation in one space di­ mensional case as discussed in Section 3.1. However, few results are known on the stability of elementary waves for relaxation approximation in multidimen­ sional case. The results introduced in this section have made a good progress. 3.3

The zero relaxation limit Consider the hyperbolic system of conservation laws with relaxation

f ut + f(u, v)x = 0 \

_L /

\

V*(*)-v

(3.102)

We are interested in the zero relaxation limit r(u) -> 0. In many physical situa­ tions certain stability criterion holds and no oscillation is expected to develop in the limit. And the solution would converge to that of the equilibrium hyperbolic conservation law tit+/(ti,V;(ti)) x = 0. (3.103) A stability criterion for (3.102), under which (3.103) is expected to be the limiting equation, is obtained by Liu in [122] through asymptotic expansion of the Chapman-Enskog type in the kinetic theory. The first order expansion of (3.102) is (3.103) which is a scalar conservation law with the equilibrium characteristic speed A*(u) = -4-f{u, K.(w)).

Relaxation

103

For the second order approximation, we set v\ to be the deviation from equilibrium and obtain from the equation (3.102)2 that v = K (u) + Vd vd =

r(u)(vt+g(u,v)x)

=

T(u)(V*(u)t+g(u,V*(u))x)

where we have made a priori hypothesis that Vd is small and its gradients are even smaller due to the dissipation induced by relaxation. By virtue of dt + K{u)dx = 0, it follows that vd S T(U)[-\*(U)V;(U)

+ gu(u, V*(u)) + V:(u)gv(u,

V.(u))]ux.

Pluging this into the first equation in (3.102), we arrive at the second order viscous approximation to (3.102) u>t + / ( t i , V*(u))x =

[f3{u)ux]x.

The viscosity f3(u) is related to the equilibrium speed A* and the characteristic speeds Ai, A2 for (3.102), as follows P(u) = r(ti)[A,(ti) - Ai(ti, K(ti))][A 2 (ti, Vi(ti)) - A*(ti)]. The stability criterion is that the viscosity f3(u) is positive, namely, the equi­ librium speed A* is subcharacteristic with respect to the frozen speeds Ai and A2: Ai
1 (*-/(«)), + ^ M H l

=

0)/(0) = 0, / <>0,r>0,

( 31 ° 5 )

(3.104) holds when 0 < \i < 1 > / ' M > 0. In fact, (3.105) can be rewritten as <(ut h

+ (v =- 0v x (/i +- f(u)) l)f(u)

[vt = ±£blj^Jf

_

,, .

.

-

7 - / ( w ) , / ih > 0 , r > 0b , g,v = ( c

for which the corresponding equilibrium equation is

(3.106) f jghbvjh

u , + (/*/(«)), (3.107) It is easy to see that the characteristic speeds =Ai0.and A2 of (3.106) take the form Ai = 0,

A2 = / ' ( « )

104

Quasilinear Hyperbolic Systems and Dissipative Mechanisms

while the equibibrium speed A* = fif'(u).

Thus, the basic assumptions

0
/ » > 0

(3.108)

yield the stability criterion (3.104). It is proved by Chen and Liu in [15], for the stable case when (3.108) holds, that no oscillation is expected to develop for the system (3.106) in the zero relaxation limit and that solutions of (3.106) converge to those of (3.107). Consider the Cauchy problem for (3.106) with (u,v)(x,0) = (u0(x)1vo(x)) which satisfies 0 < v0(x) < f(u0(x))

< Mo < oo

(wo — u, vo — v) £ L°° fl L2(—oo, oo), for some equilibrium state (w, v) = (u, (/J, — The result in [15] states that

(3.109)

l)f(u)).

T h e o r e m 3 . 3 . 1 . The solutions (uT,vT) of (3.106) with initial data satisfying (3.109) converge to bounded measurable functions (t/, v)(x,t), a.e. in t > 0 as r —y 0. Moreover, u = u(x,i) is an admissible weak solution of the Cauchy problem of the equilibrium equation (3.107) with initial data u(x,0) = uo(x), and v(x,t) equals to the equilibrium value (fi — l)f(u(x,t)) for any t > 0. To prove the theorem, we first show the existence of L°° solutions of the Cauchy problem for (3.106). We construct the solution by the zero dissipation limit of solutions of the corresponding viscous system

{

u

t + (f(u) + v)x = euxx

vt+v-b(u,v){x,0)

WW =

(u<0(x)iv'0(x)).

L e m m a 3 . 3 . 2 . Suppose that (UQ(X),VO(X)) some smooth L2 approximation of (u0lvo), (u£,v£)(x,t) satisfying 0 < ve(x,t)

(3.110)

= £Vxx

satisfies (3.109). Let (u£0,v£0) be then (3.110) has global solutions

< f{u£(x,t))

<M
\\ve-{p-1)f{u<)\\L,
(3.111)

V^IIK,^),||L 2 <M, for some M independent of e and r . Moreover, there exists a sequence Sk -> 0, such that (u£k,v£k)(x,t)

-)■ (u,v)(x,t),

a.e.

Relaxation

105

and (u,v)(x,t) is a global weak solution for the Cauchy problem of (3.106) which has the same bound as the initial data in (3.109), 0 < v(x,t)

< f(u{x,t))

< M0

(3.112)

and satisfies the entropy condition r)(u, v)t + q{u, v)x + 77(tz, v)v V~W-

*) / W <

(3.113)

0?

T

and the estimate on the deviation from equilibrium \\v-(ii-l)f(u)\\L,
(3.114)

for some M independent of r where (77, q) is entropy pairs for (3.106). Proof. The initial data (WQJ^O ) ^s obtained through standard convolution with smooth modifiers j£ (x) : (u£0,v£Q)(x)=

/

^{u0{y),vo(y))je(x-y)dy. ~*

Suppose that there exists ue G (0,oo) with

(u-ue)f"(u)

> 0,

ue (0,oo)

then we can show that the following regions are invariant ones for the system in (3.110) for all e > 0 , r > 0 :

£ c

= {(",*) : 0 < v < ]—^C, f-'iv) V

where C > max<


P 1 _ °^ ^, f(ue)

f-\v

+ c)\ , )

\ (The definition of invariant regions is

given by Chuen, Conley and Smoller in [20].) The existence of global solutions to (3.110) follows then from usual local existence theory and the a priori sup norm estimate on the solutions by the in­ variant region argument. To show (3.111) we need certain estimates on entropy pairs for the model system (3.106). It is well-known that, for general conservation laws Ut + F(U)X = G{U), U € i T , the entropy pairs (77, g) satisfy V ? = V*7 V F,

(3.115)

106 Quasilinear Hyperbolic Systems and Dissipative Mechanisms so that the smooth solutions U of (3.115) satisfy v(U)t

+ q(U)x = \7vG(U).

An admissible weak solution U of (3.115) satisfies the entropy condition ri(U)t+q(U)x)r)uv = 0.

Thus, the general representation of entropy pairs is 17(11, *; H, G) = JU H(f(t)

- v)d£ + G(v),

f(u)-v

H(t)d£, / for any continuous functions H and G. Under the hypothesis that /'(0) ^> 1, the followings are particularly useful r)o(u,v;H)=

ij(u,v)=

f

H(f{{)-v)d£+

(/(o - «)<« + ft /

[

""

H(pf{Z))dt

/m,

•'/-'(jritr) •/ since the t; given above have the property that 7/w vanishes on equilibrium state v = (// — l)/(w) and that [^,,)/-^ ; l)/(")> C l [^-^-l)/(»)] 2 ,

j l ^ , , ) . ^ - ^ ; i)/(»)I < C2[y-(»-Ti)fW

(3-116)

for positive constants C\ and C2 depending only on the function H and M, which defines the bounded set {(u,v) : 0 < v < f(u) < M}. To show (3.111), we multiply the system in (3.110) by V*7*(w£> v£) with *7*(u, v) = rj(u, v) - rj(u,v) - Vrf(%v) I " _ ^ J , and obtain

+£^(we,ve)xx-£(w£,t;£)x v2^.(«c,«e) ("e J •

Relaxation 107 The estimate (3.111) follows by integrating the above equality over (—00,00) x(0,*) with the help of (3.116). Similarly, (3.112)-(3.114) follow from (3.111) and the above equality in the limit e —> 0. The convergence of (u£, v£) as e -> 0 can be proved by applying the theory of compensated compactness (see [15] for detail). Now, we turn to prove Theorem 3.3.1. For convergence and admissibility, we need to estimate the entropy inequal­ ity for the Cauchy problem of the equilibrium equalion ( « * + (f/(«)>• = ° {u(x10) = uo(x) based on the entropy condition (3.113), r)(uTjvT)t + q{uTlvT)x

(3.117) v

+ r)v(uT,vT) • —

;

*—— < 0 T

with V*(u) = (ii-l)f{u). It is not difficult to show that, given any entropy pair (77, q) for (3.106) with r)v(u, v) vanishing on the equilibrium curve v = V*(u) = (/i — l)/(u), the pair 17* (ti) = 77(1/, (fi - l ) / ( t i ) )

(3.118) g*(ti) = q{u, {fi- l)f(u)) forms an entropy pair for (3.107), namely,
+ q(uT, (1 -

/i)f(uT))g

= 7j(uTlVT)t + q(uT, VT)x + A r , where II A r HH-1

(n)


loc

/"{Mtlr,»r)-l?(«r,(l-/l)/(«r))]2

Jo Jn +[q{uT,Vr)-q{uT,{l-lL)f{uT))¥}dxdt I I [VT - (1 " fl)f(uT)]2dxdt Jo Jci < CT -* 0, as r -> 0.


108 Quasilinear Hyperbolic Systems and Dissipative Mechanisms Thus, the convergence of uT to u can be obtained by applying the theory of compensated compactness as in the proof of lemma 3.3.2. The admissibility of u can be guaranteed by the above estimate and (3.113). Finally, the convergence of vT(x,t),i > 0, to its equilibrium value (1 — fi)f(u(x,t)) follows from the convergence of uT to u and the formular below vT{x,t) = e-$vQ(x) +

/ e-l~r Jo

T

f(uT(xX))d(

which comes from the second equation in (3.110) and the convergence of (we,ve). Theorem 3.3.1 is then proved. Some new investigation on the model (3.105) is made by Luo and Natalini in [131]. They obtained the global existence of BV solutions by using Glimm scheme or modified Godunov scheme with a frictional step version in which the most significant part is the BV estimates on the approximate solutions by Godunov scheme. Since the BV and L°° estimates do not depend on relaxation time, the convergence to the equilibrium equation as r —> 0 is proved at the same time in [131]. By using the methods of compensated compactness, the rigorous justification of the relaxation approximation to equilibrium solutions containing shock waves has been verified by Chen, Liu and Levermore in [14] and [15] respectively with other special models. One such model is the following system f ut + vx = 0 \ vt +
- f{u))

(r > 0)

where a and / are some given smooth functions such that (T'(U) > v, (y > 0), /(0) = 0. For this model, the stability criterion (3.104) holds if |/»|

2

<
(3.119)

It is proved that the uniformly bounded sequences of perturbed solutions con­ verge strongly to some weak equilibrium solutions as the relaxation parameter tends to zero, for initial data close to the equilibrium, provided the subcharacteristic condition (3.119) is verified. The uniform boundedness of relaxing solutions is recovered by assuming the function / be constant out of a bounded set. However, a recent investigation on this model is made by Natalini in [150] to show that this uniform estimates are just a consequence of the stability con­ dition (3.119), without any further requirement on / . Another interesting model with relaxation effect is the Broadwell model in the kinetic theory of gases. It consists of the system of semilinear hyperbolic

Relaxation

109

equations

# - | l

= 2(/32-/i/2)

(3-120)

and derives from a six-velocity model when it is specialized to one-dimensional flows, for which the densities of particles moving in directions orthogonal to the flow are all equal. (The reader is refered to Broadwell [10] or Platkowski and Illner [161] for the derivation in the kinetic theory context). The function / = ( / i , hi h) is defined for ( i , i ) G l x M+ and describes densities of particles: / i for particles moving in the positive ^-direction, fa in the negative x-direction and fo in each of the positive or negative y-or z-directions. The parameter r stands for the mean free path, a measure of the average distance between successive collisions. The limit, when the mean free path approachs zero, is known as the fluid dynamic limit. For small mean free path the strong interactions of particles allow a macroscopic description of the flow to become meaningful. It is not difficult to identify the form of the induced macroscopic "Eular equations" as

S?+£(/»>=o §t(H + £(P9(«)) = o where g(u) := | [ 2 ( 1 + 3 « 2 ) 1 ' ' 2 - 1 ] ) and the macroscopic density and momentum of the fluid are given by p = Fi + 4 ( F i F 2 ) 1 / 2 + F 2 ,

m = pu = F1-

F2

with the notation (i<\, F2, F3) to denote the limit of / . By showing that a given piecewise smooth solution with noninteracting shocks of the limit fluid equations can be approximated by solutions of the Broadwell system as r —> 0, a recent study by Xin [185] gives a definitive answer to one direction of the zero relaxation limit problem. The converse problem, which shows that a given family of solutions to the Broadwell system converges globally in time to a fluid-dynamical solution, has not been solved yet. But, the approach of self-similar fluid dynamic limits is introduced by Slemrod and Tzavaras in [168] and further studies have been made by Tzavaras [176] [177], and Fan [42].

110 Quasilinear Hyperbolic Systems and Dissipative Mechanisms They consider a modified Broadwell system

^ - H

= H(/32-/1/2),

# = -2fe(/32-/1/2)> which preserves the invariance under dilation (x,t) —» (ax, at), a > 0. As far as the multidimensional case is concerned, a new progress is made by Katsoulakis and Tzavaras [91] recently for the scalar conservation law in sev­ eral space dimensions. They consider a class of semilinear hyperbolic systems of relaxation type which are contractive in the I, 1 -norm and admit invariant regions. The characteristic speeds of the conservation law satisfy, relative to the convective velocities of the relaxation system, a multidimensional analogue of the subcharacteristic condition. It is shown in [91] that solutions emanating from 51^-stable data are compact in a strong topology and converge, in the zero-relaxation limit, to a weak solution of the scalar multidimensional con­ servation law that satisfies the Kruzhkov entropy conditions. For solutions emanating from stable data in the L1 C\ L°°-norm, it is shown that the Young measure, describing the weak zero-relaxation limits, concentrates on the consti­ tutive manifold of the associated conservation law and gives rise to a measure valued solution.

Chapter 4 The influence of dissipation mechanism on the qualitative behavior of solutions For a system of hyperbolic conservation laws

mvnjbn where u = (i*i, • • •, un)> and the matrix /'(i«) has n real distinct eigenvalues Ai(u) < \2(u) < " < An(w), the nonlinearity of the system is reflected in the dependence of A,(u) on u and has a distabilizing effect on the qualitative behavior of the solution the result of which is the development of singularity at a finite time. The large time behavior of the solution for (4.1) with initial data u(ar,0) = uo(x) has been well studied also. For instance, suppose that (4.1) is a scalar equation, the flux function f(u) satisfies f"{u) > 0, and the initial datum uo(x) is a bounded measurable function, then it was shown by Diperna [33] that

IK,<)-^(-,0IUi W = O(r 1/2 ) if uo{x) has a compact support, where N(x,t) is called iV-wave which is intro­ duced by Lax in [102]; if UQ(X) takes the form as

{

u_, P(*). ti+,

x<-X -X <x<X x> X

with u- ^ u+, then it is proved by Dafermos [22] that

where UR(x,t) is the weak solution of Riemann problem with Riemann data u

, ,

«°W

=

fu_,

x< 0

{u+,

x>0.

Ill

112

Quasilinear Hyperbolic Systems and Dissipative Mechanisms

Suppose that (4.1) is a 2 x 2 system which is genuinely nonlinear in the sense of Lax [102], and the amplitudes of initial data are small, then it is proved by Glimm and Lax [46] that ||ti(-,J) - U\\L°°(R) = 0{t~l) if the initial data uo(x) is a periodic function, where u is the mean value of uo(x); \\u(-,t) — U*\\L°°(R) = 0{t~1/2) if u0(x) -> u* as \x\ -+ oo. For a general system (4.1) which is genuinely nonlinear, more results have been established by Liu [113] [115]. For instance, with the initial data which has small total variation and compact support, the solution of the Cauchy problem satisfy 7Vti(s*) = 0 ( * - 1 / 2 )

IK^)-E^(^)r«(0)^w = °( rl/6 ) i

where JVt- is the iV-wave corresponding to the i-characteristic field and r,- is the right eigenvector corresponding to the i-characteristic speed. For more detailed information, we refer the reader to Hopf [62], Lax [102], Diperna [33], Dafermos [22] [23] for a scalar equation, and to Glimm and Lax [46], Diperna [33], Liu [113] [114] [115] for systems. The situation will be different when dissipation is introduced. There are different kinds of dissipation in which the common three kinds are the follows. A. Lower order term, such as relaxation and damping which have been discussed in the first three chapters of the book. B. Lower order term, such as the materials with memory. Consider the equations of motion of a one-dimensional body, with unit reference density and zero body force in Lagrangian coordinates f ftii(ar, t) - dxv(x, t) = 0 \dtv(x,t)-dx
,

. V'Z)

where u-deformation gradient, v-velocity, cr-stress. When the body is elastic, the stress at the material point x and time t is determined solely by the value of deformation gradient at (x,i) via a constitutive relation
f(u{x,t)).

Under the standard assumption f'(u) > 0, (4.2) yields a strictly hyperbolic system. It is interesting to compare the behavior of elastic bodies with the behavior of materials with fading memory in which the stress (r(x,t) at the materials point x and time t is determined by the entire history TIW(X, •) of deformation gradient at x. For concreteness, let us consider the simple model
f J—oo

k(t-

r)g{u{x,

r))dr

The influence of dissipation mechanism...

113

namely, dtu{x,i) — dxv(x,t)

{

—0

dtv{x, t) - dxf{u(x, t)) + j

k(t-

r)gx{u(x, r))dr = 0.

It is known that the memory exerts a rather weak damping influence if k(t) satisfies appropriate assumptions. It has been shown that, when the initial data are smooth and "small", dissipation prevails over the destabilizing action of nonlinear instantaneous elastic response. As a result, a smooth solution exists globally in time. On the other hand, when /(«) is nonlinear and the initial data are "large" then the destabilizing action of nonlinearity prevails and solutions break down in a finite time. We refer the reader to Dafermos [25], Renardy, Hrusa and Nohel [163], and the references there. C. Higher order term, such as viscosity and heat diffusion. For instance, the basic system for the one-dimensional gas flow without any dissipation takes the following form in Lagrangian coordinates ( vt — ux — 0 \ ut+px = 0 I Et + (pu)x = 0 where v denotes specific volume, u denotes velocity, E denotes total energy, ?/ E = e -h %-, e is internal energy, and p denotes pressure. These equations are known as the conservation laws of mass, momentum and energy respectively. If the viscosity and heat diffusion can not be neglected, we have the system 2

{

vt — ux =0 v>t+Px- (l*ux)x = 0 (e + \

)

+

fr™)* ~ \PUU*]* + 9x = 0

where the constitutive relations are given according to the materials in consid­ eration: e = e{v,0),p = p{vJ0),q =

q{v,6,0x),e>Q,q{v,0,0x)0x<<),

fi > 0 may depend on v and 0, where 9 denotes the temparature. The dissipation mechanism appears in the form of higher order term and they do influence the smoothness and the large time behavior of the solution. It is shown in Section 4.1 that the combined dissipation of viscosity and heat diffusion will preserve the smoothness of the initial data without the restriction of the smallness. For the case when there is only heat diffusion as the dissipation, namely, p, = 0, the dissipation is able to preserve the smoothness of the initial

114 Quasilinear Hyperbolic Systems and Dissipative Mechanisms data only when initial data are smooth and small. This is also discussed in Section 4.1. Another emphasis of this chapter is to understand the influence of dissipa­ tion on the large-time behavior of solutions. Various phenomena may occur on the large-time behavior of solutions corresponding to different materials in con­ sideration. For instance, with the same boundary condition as stress-free and thermally insulated, the solution may go to infinity as t -» oo with certain rate, may approach to a unique state exponentially fast or may have phase transition phenomena, according to different kind of constitutive relations to be concerned. This is discussed in Section 4.2. The last section of this chapter - Section 4.3 is devoted to the study on various behavior of solutions corresponding to different boundary conditions. 4.1

The influence of dissipation mechanism on the smoothness of solutions

The intent of this section is to elucidate the role of dissipation, particularly the stabilizing effect, in the continuum physics. Within the framework of thermomechanics, the referential (Lagrangian) de­ scription of the balance laws of mass, momentum and energy for one-dimensional materials with reference density po = 1 is

{

Vt — ux = 0 ut -
(43) jfhg

while the second law of thermodynamics is expressed by the Clausius-Duhem inequality

*+(f),>°

(4-4)

where v, u, e, <7, 77, $ and q denote deformation gradient (specific volume), veloc­ ity, internal energy, stress, specific entropy, temperature and heat flux, respec­ tively. Note that v, e and 0 may only take positive values. For one-dimensional, homogeneous, thermoviscoelastic materials, internal energy, stress, entropy and heat flux are given by the constitutive relations e = c(v, 0), 0,
The influence of dissipation mechanism...

115

where ij) = e — Or) is the Helmholtz free energy. A typical problem is to consider a body with reference configuration the interval [0,1] whose boundary is stress-free and thermally insulated,

$;>:;<%=:

oS«»

<«,

and determine the thermomechanical process, described by the functions (v, w, Q)(x,t), 0 < x < 1, t > 0, under prescribed initial conditions v{x, 0) = vo(ar),

u(x, 0) = u0{x),

0(x, 0) = $0(x),

0 < x < 1.

(4.7)

In the absence of dissipation, when the body is a thermoelastic nonconductor, i.e., a = — p(v,0),e = e(v,0),g = 0, (4.3) reduces to a hyperbolic system of conservation laws (4.1) which does not generally possess globally defined smooth solutions, even when the initial data are very smooth. When the material is a thermoelastic conductor, it has been shown by Slemrod in [167] that smooth and "small" data generate globally defined smooth solutions to (4.3). However, it is generally conjectured that heat dissipation alone cannot prevent the breaking of waves of large amplitude and the forma­ tion of shocks. And it has been proved by Dafermos and Hsiao in [31] that this conjecture is indeed true. The question is whether the combined dissipative effects of viscosity and thermal diffusion, when the material is thermoviscoelastic, may counterbalance the destabilizing influence of nonlinearity and thus induce the existence of glob­ ally defined smooth solutions to the initial-boundary value problem (4.3) (4.6) (4.7) even for large initial data. This is first verified by Kazhikhov [97], Kazhikhov and Shelukhin [98] for the case where the material is an ideal, linearly viscous, gas with constant specific heats: e = c0,
— v

by means of an interesting analysis which depends crucially upon the special form of the above constitutive assumptions. For the case of ideal gas, the global existence of smooth solution to initial boundary value problems or Cauchy problem, with smooth and large initial data have been investigated also by Antontsev, Kazhikhov and Monakhov in [4], Kawashima and Nishida in [95], Nagasawa in [147] [148] and [149]. On the other hand, Dafermos and Hsiao in [29] and Dafermos in [24] consider the problem (4.3) (4.6) (4.7) for a fairly general class of solid like linearly viscous materials. They establish the global existence which will be discussed in this section. The global existence with a different boundary condition to (4.6) is obtained later by Jiang [86].

116

Quasilinear Hyperbolic Systems and Dissipative Mechanisms

The corresponding results for the material of real gases have been obtained by Kawohl in [96], Jiang in [87], Luo in [128], [129], Pan in [156], and etc. As far as the thermoviscoelastic material in more than one dimension is concerned, the global existence of smooth solutions for general domains has been investigated only in the case of sufficiently small initial data. We refer the reader to Matsumura and Nishida [141] [142] and Valli [178] [179]. For the material of thermoelastic conductor in multi-dimension, a survey of the study can be found in the book by Racke and the references cited there [162]. For the small initial data problem of nonlinear coupled hyperbolic-parabolic systems, there are a lot of works on both the global existence and blow-up of solutions, for instance, the book by Zheng [194]. This book also discusses the global existence and asymptotic behavior of solutions with arbitrary initial data for certain coupled hyperbolic-parabolic systems. The problem (4.3) (4.6) (4.7) is investigated by Dafermos and Hsiao in [29] for a fairly general class of solid like linearly viscous material e = e(v,0),cr = -p(v,0)

+ fiuXlq —

q(v,0,0x).

In this paper, shear viscosity \LV is inversely proportional to density. Conse­ quently, its dissipative effect is so weak at high density that existence of global solutions is to be expected only when the initial energy is not too large. To overcome this limitation, Dafermos [24] considers the problem (4.3) (4.6) (4.7) for linearly viscous materials e = c(«, 0), a = -p{v, 0) + jj.(v)ux, q = -K{v,

0)0X

(4.8)

where the viscosity p(v)v is uniformly positive, that is fi(v)v > /i 0 > 0,0 < v < oo.

(4.9)

R e m a r k . In the case when viscosity may depend on temperature, a typical problem to understand the effect, caused by the dependence of viscosity on temperature, has been studied first by Dafermos and Hsiao in [30]. We assume that e(v,9),p{v,0),fi(v) and K(v,0) are twice continuously dif­ f e r e n t i a t e o n O < v < o o , O < 0 < o o and are interrelated by ev(v,0)

= -p(v10)

+ 0pe{v,0),

(4.10)

so as to comply with (4.5). Furthermore, we require that, at any temperature, the elastic part of the stress be compressive at high density and tensile at low density, i.e. there are 0 < v < V < oo such that P(w, 0) > 0,0 < v < v, 0 < 0 < oo, (4.H) p(v, 0) < 0, V < v < oo, 0 < 0 < oo.

The influence of dissipation mechanism...

117

Employing an idea of Andrews in [2], we will show later in proposition 4.1.3 that, as a consequence of (4.9) and (4.11), the deformation gradient is a priori confined in a bounded interval 0 < v < v(x,i) < V and hence no restrictions are necessary on the behavior of e(v, 9),p(v, 0), K(v, 9) at v = 0+ and v = oo. As regards growth with respect to temperature, we assume that for any 0 < v < V < oo_ there are positive constants i/, Ar0 and N, possibly depending on v and / or V, such that, for any v < v
*o < K(v,9)

e(v,0) > 0, v < e0{v, 9) < N(l + 0 1 / 3 ),

(4.12)

\Pv(v, 0)| < N(l + 0 1 + * ) , \pe(v, 0)| < N(l + 0 1 ' 3 )

(4.13)

< N,\Kv{v,0)\

< W , | £ , ( t ; , 0 ) | < N,\Kvv{v,0)\

< N.

(4.14)

(A explanation on the above growth rates can be found in [24]). The existence theorem established in [24] reads: T h e o r e m 4 . 1 . 1 . Consider the initial-boundary value problem (4.3) (4.6) (4.7) under conditions (4.8)-(4.14). Assume that v0(x), v'0(x), u0(x), u'0(x), u%(x), 0o(x), 0'0(x), and 9%{x) are all in C a [0,1] and v0(x) > O,0o{x) > 0,0 < x < 1. Furthermore, let the initial data be compatible with the bound­ ary conditions (4.7) at (0,0) and (1,0). Then there exists a unique solution {v(x,t), u(x,t),9(x,t)} on [0,1] x [0,oo) such that for every T > 0 the functions ^ •> ^X 5 lit 5 ^XX ) " ) "x , Uxx lhkjbmnkj are all in Ca,a^2(QT) and vtt, u>xt, 9xt are in nvhbgj

V, Vx , l^t , Vxt)

L2(QT). Moreover, 0(x,t) > 0,i7 < v(x,t) < V, for 0 < x < 1, 0 < t < oo, where v and V are positive constants depending on the initial data, C^fO, 1] denotes the Banach space of functions on [0,1] which are uniformly Holder continuous with exponent a, while Ca,a^2(QT) stands for the Banach space of functions on QT = [0,1] x [0,T] which are uniformly Holder continuous with exponent a in x and a / 2 in t. nvbhfgy The theorem can be proved by the procedure devised by Dafermos and Hsiao in [29], namely, solutions to (4.3) (4.6) (4.7) are visualized as fixed points of a m a p P on the Banach space B of functions {V(x,t)1U(xit),Q(x1t)} kgjhnbjh with V,U,UX,Q,QX in C 1 / 3 ' 1 / 6 ( Q T ) and existence is established by means of the Leray-Schauder fixed point theorem. mbnjghyftgryhfgtf The m a p P carries {V(x,i), U(x,t),S(x,t)} into the solution of a compli­ cated linear "parabolic" system obtaned by linearizing (4.3) about {V(x,t), U(x,t), 0 ( # , t)}. Due to the smoothing action of linear parabolic systems, P is completely continuous and its range is contained in the set of functions {v(x,t),u(x,t),9(x,t)} with v, vx, vu vxU u,uXluuuxx,9,9X,9U9XX in C a ' a / 2 (QT) The construction of P and the precise statements and proofs of its aforemen­ tioned properties can be found in [29]. Here we only show that any possible fixed point of P , i.e. any solution {v(x,t), u(x,t),9(x,t)} of (4.3)J4.6) (4.7) satisfies the admissibility conditions 9(x,t) > 0,0 < v < v(x,t) < V and is contained

118 Quasilinear Hyperbolic Systems and Dissipative Mechanisms in an a priori bounded set of B. This will complete the list of requirements for the application of the Leray-Schauder fixed point theorem. Turn to the initial data, we normalize uo(x), by superimposing a trivial rigid motion if necessary, so that / uo(x)dx = 0 (4.15) Jo Let {v(x,t),u{x,t),6(x,t)} be a fixed solution of (4.3)(4.6)(4.7) on [0,1] x [0,oo) in the function class indicated in Theorem 4.1.1. Now, we derive the a priori estimates. Integrating (4.3) over [0,1] x [0,tf] and using the boundary conditions (4.7), we obtain the conservation laws of total momentum and energy: / u{x,t)dx= Jo I

U(x,t) + -u2{x,t)\dx=

I u0(x,t)dx Jo I

= 0,0
\e(x,0) + -ul(x)\dxd=

Eo,

(4.16)

0
Substituting a from (4.8), we write (4.3)2 in the form ut+p{v,e)x

= \ji(v)ux]g,

(4.18)

while combining (4.3)3 with (4.3)2 and using (4.8), (4.10) and (4.3)i we get e0(v,O)Ot + epe(v,O)ux-ji(v)ul

= [K{v,0)0x]x.

(4.19)

Applying the maximum principle on (4.19), recalling that 00(x) > 0,0 < x < 1, one deduces Proposition 4.1.2. 0OM) > 0,0 < x < 1,

0 < t < oo.

Now, we derive bounds on the deformation gradient. Using (4.3)i, the equa­ tion (4.18) can be rewritten as ut + p{v,0)x = M{v)xt,

(4.20)

where M(v) =

/

ji{u)duj.

Due to (4.9), M(v) is a strictly increasing function which maps (0,oo) onto (—00,00).

The influence of dissipation mechanism...

119

Proposition 4.1.3. It holds that v< v(x,t) < V,0 < x < 1,0 < * < oo, where v = M- 1 (M(min{i;,mint;o(-)}) - v ^ o ) i V = f M-HM(max{V r ,maxi;o(-)}) + \ / 2 ^ ) Prcx>/. Integrating (4.20) over [0, y] x [«, r], 0 < y < 1,0 < s < r, and using the boundary condition (4.7)i, we get M{v{y,T)) = M{v(y,s))+

rT

p(y,t)dt +

Js

ry

u(xJr)dx-

JO

rv

u(x,s)dx.

(4.21)

JO

By (4.16), (4.17) and (4.12), it follows r ry l2 1 Z*1 2 / u(xJt)dx\
1 < -E0 2

0
(4.22)

This, with the definition of v and V together, implies v < vo(x) < V, 0 < x < 1. Thus, if v(x, t) > v is violated on [0,1] x [0, oo), there are r > 0 and y E [0,1] such that v(x,t) > v, for 0 < x < 1,0
> M{v0(y)) - yj2E~o-

In the latter case, (4.21) together with (4.11) and (4.22) yield M(v(y, T)) > M(v) - \/2Eo~. In either case, it follows from the definition of v that M(v(y, r)) > M(v) which is a contradiction to v(y, r) — v. This shows v(x, t) >v,0<x< 1,0 0 and restrict our solution to the rectangle QT = [0,1] x [0, T]. In the sequel, A will denote a generic constant which may depend at most on T, no,v,ko,N and the upper bounds of the C a [0,1] norm of vo, v0l «o, w0, w o> 0o, #0* ^0- W e will show that {v(x,t), u(x,t), 0(x,t)} is a priori bounded in the Banach space B, that is

120 Quasilinear Hyperbolic Systems and Dissipative Mechanisms Proposition 4.1.4. It holds that ll v llc i / 3 . i / 6 (Q T ) ^

A

'

||ti||Cl/3,l/6(QT) < A,||wa:||Cl/3,l/6(QT) <

A,

||0||C1/3,1/6(QT) < A , | | ^ | | C l / 3 , l / 6 ( g T ) <

A.

To prove this proposition, we need the following lemma 4.1.5 - lemma 4.1.11. The first observation is that, in view of (4.17) and (4.12), max / 0(x,t)dx < A. [0,T]7 0

(4.23)

"

We now proceed to get estimates which are motivated by the second law of thermodynamics and embody the dissipative character of viscosity and thermal diffusion. Lemma 4.1.5. It holds that / / 0-Al3e2xdxdt Jo Jo

< A,

(4.24)

f f 0s/3dxdt
(4.25)

/

(4.26)

rT

max0 5/3 (.,<)cft< A.

f9 Proof. Define H(v,9) = / £~ 1/3 e0(v, £)
+ 0).

(4.27)

Moreover, we can show, with the help of (4.10) that He(vy0)=e-VZee(v,6)

where

These, with the setting H(x,t) e-1'3, imply

= H(v(x,t),6(x,t))

and multiplying (4.19) by

Ht + G(v, e)ux - e-1'5 ■ /*(„)«! - e-liz[k{v, e)ex)x = o.

The influence of dissipation mechanism...

121

By first integrating this equation over [0,1] x [0,T], and then integrating by parts with respect to x and recalling the boundary condition (4.7)2, we obtain / / JA(v)e"1f3uldxdt + l [ [ A > , 0 ) 0 " 4 / 3 -Oldxdt Jo Jo 3 Jo JO = / H{x,T)dxJo

f H{x,0)dx+ Jo

f f Jo Jo

G{v,0)uxdxdt.

By virtue of the definition of G(v, 0) and (4.13), it holds \G(v, 9)\ < 2N(l + 0). So, using (4.9), (4.14), (4.23), (4.27), Proposition 4.1.3, and the Cauchy-Schwarz inequality, we deduce fioV'1 [ f O-VSuldxdt+lko Jo Jo 3

I f Jo Jo

0-4'3'Oldxdt

< A + 47V 2 .\7^- 1 / / e^dxdt+^/ioW1 Jo Jo 2

f J0

f J0

Q-ll3u2xdxdt.

(4.28) On the other hand, it follows from (4.23) and the use of Cauchy-Schwarz in­ equality that / / Oldxdt Jo Jo

<\[ max05/3(-,*W* " Jo [0,1] < A+ A / / Jo Jo

e2f3\ex\dxdt

< A + 1 / / 08/3dxdt + A f f O-4/302xdxdt, Jo Jo Jo Jo (4.29) whence 4AT2F/i0"1 / / 07l3dxdt
/ / 0-Al302xdxdt. Jo Jo

(4.30)

Thus, (4.28) and (4.30) yield (4.24), and then (4.29) implies (4.25) and (4.26). As a corollary of Lemma 4.1.5, we have Lemma 4.1.6. / / u\dxdt < A. Jo Jo In the following lemmas we employ the bounds obtained in order to estimate by interpolation the square integral of various derivatives of the solution in terms

122 Quasilinear Hyperbolic Systems and Dissipative Mechanisms of low powers of 92{xA)dx,

Y = max / [0,T]7 0

Z = max / u2T(x,t)dx. XXK } [0,T]J0 It is not difficult to show, by (4.23) and Schwarz's inequality, that e3/2(y,t)
[ ^ ( ^ I M ^ O I ^ A + AY 1 / 2 , * Jo

0 < y < 1, (4.31)

whence m a x 0 < A + AY 1 / 3 .

(4.32)

QT

Similary, combining I u2x(x,t)dx + 2 [ Jo Jo

ul(y,t)<

\ux(x,t)\\uxx(x,t)\dx,0
with the standard interpolation estimate (see Agmon [1]) / u2x(x,t)dx<

108 I u2(x,t)dx+4S2<

j u2(x,t)dx\

I /

u2xx(x,t)dx\

and using (4.17), we arrive at max | ^ | < A + AZ 3 / 8 .

(4.33)

QT

L e m m a 4.1.7. max/

vl(x ,t)dx<

A + AY 1 / 9 .

Proof. Multiplying (4.20) by (M(v)x - u) and integrating over [0,1] x [0,*], 0 < t < T, we obtain \J^

[M(v{x,t))x-u{x,t)]2dx-^j

=

[M(vo(x))x-u0(x)]2dx

\PvVx +P$0x][M(v)x - u]dxdr. Jo Jo

By virtue of (4.13), the definition of M(v) and proposition 4.1.3, it follows / / pvVx[M(v)x I Jo Jo

- u)dxdr\ < A / / (1 + 6^3){[M{v)x I Jo Jo

- u}2 + u2}dxdr

The influence of dissipation mechanism...

123

5 3 < A / ( l + max0 / (-,r)l / [M(v(x, r))x - u(x, r)]2dxdr Jo I I0.1] J Jo

u2(x,T)dxdr.

+A / I l + max0(-,r)l j

Similarly, using (4.20) and applying Schwarz's inequality,

/ / p$0x[M(v)x - u]dxdr\ \Jo Jo I
f j l + m0 a1 x 0 5 / 3 ( . , r ) i / M(v(x, r))x - u(x, r))2dxdr Jo I I.! J JO

+ 7 v j l + max01/3j / ' /

0-^0\dxdr.

The lemma can be proved then by combining the above, applying GronwalPs inequality and taking account of (4.17), (4.24), (4.26) and (4.32). Lemma 4.1.8. Y < A + AZ 3 / 4 /

/

02dxdt
+ AZ3/4.

Jo Jo

Proof. Define


set Q(x,t) = Q{v(x,t), 0(x,t)), multiply (4.19) by Qt and integrate over [0,1] x [0, t], 0 < t < T. After an integration by parts with respect to x we arrive at

/ / {ieOt + 0peux - {ml}Qtdxdr+ Jo Jo

/ / K0xQxtdxdt Jo Jo

= 0.

By using (4.12), (4.13), (4.14), (4.24), (4.25), (4.26), (4.33), Proposition 4.1.3, Lemma 4.1.6, and the facts of \QV\ < NO, \QVV\ < NO, each term in the above

124 Quasilinear Hyperbolic Systems and Dissipative Mechanisms equation can be estimated as follows rt

pi

pt

t>\

/ / edK0?dxdT >vk0 / B2tdxdr, Jo Jo Jo Jo / / \Jo Jo

ee6tQvUxdxdr\ I / 02tdxdr + A + AZ3'4,

\ / {Opdux \Jo Jo

fjtul}Qvuxdxdr\ I


I {\ + 07l* + ul}dxdT

< /

AZ3/\ / {0peux -

p,ul}K0tdxdT\ I

\Jo Jo


f f {1 + 0 8 / 3 + u2x}dxdr Jo Jo

( f e^dxdr+{AmaxuDJo Jo Q

<\vk0

I

j

efdxdr+AZ3'4,

[ I kex[KOx]tdxdr>h0 Jo Jo " I /■* fl - I / / K6xQvuxxdxdT\ \Jo Jo I


I

I Jo

0%(x,t)dx-A,

+ AZ3'4, K0xQvvuxvxdxdr\

< iV 2 {maxK|} {max**/*} { j f £ • | j f max0 5 / 3 (.,r)jf

t£(*,r)cte(ir}

^^dxdr}^

The influence of dissipation mechanism...

125


I f* f1 ~ /

/

\Jo

Jo

~

I

K0xKvvx0tdxdT\ I

< lisk0 I f 02dxdr+A f f Jo Jo Jo Jo in which the last term can be estimated as A l l [KOx]2vldxdr Jo Jo

[K6x]2v2xdxdr,

< A / m0 a1x [ / ^ t ] 2 / Jo I - ] Jo

v2x{x,T)dxdr

<(A + Ay!/9) / /

\kex\\(kex)t\dxdT

Jo Jo

< (A + AY1'9) { m a x * 2 ' 3 j j f*

fV4'3e2xdxdr\

'{Jo L KK°'M2dxdT} < A + ^k0Y + A | /

/

{{kex)x)2dxdT\

To estimate the last term in the above inequality, we use (4.19) and the facts of / / e2ee2dxdr<[A Jo Jo

+ AY2'9]

f I {Opeux - fiulfdxdr Jo Jo

f f 02dxdr, Jo Jo

< A+

AZ3/4.

Thus, A

{ / / HK°*)*]2dxdr}

^ A + YQkoY + ^ * ° / / °tdxdT +

Combining all of the estimates obtained above, we obtain \vk0 2

/

/

02tdxdr+\ko

Jo Jo

2

f

$Z(x,t)dz < A + h0Y

J0

4

which finishes the proof of lemma 4.1.8. Lemma 4.1.9. A + AZ11'12

m a x / u!(z,t)dx.< [o,T]y0

~~

/ / ultdzdtKA + AZ11'12. Jo Jo

+ AZ 3 / 4

AZV4

'

126 Quasilinear Hyperbolic Systems and Dissipative Mechanisms Proof. Differentiate formally (4.18) with respect to 2, multiply by ut and inte­ grate over [0,1] x [0,2],0 < t < T. Integrating by parts with respect to x we obtain u2{x,t)dx-~

\ \ +

u2(x,Q)dx +

£L'(v)uluxtdxdTJo Jo

/

fi{v)u2xtdxdr

/ {pvux + pe0t}uxtdxdr Jo Jo

=0

in which each term can be estimated, with the help of (4.9), (4.13), (4.25), (4.32), (4.33), Proposition 4.1.3, Lemma 4.1.6, Lemma 4.1.8, as follows: / / fi(v)ultdxdr Jo Jo

/

/

< \voV

/

/ ultdxdr + A < m a x t ^ f /

< \^V~l

I

I u2xtdxdr + A + A^ 3 / 4 ,

/

/

u2xdxdr

I pvuxuxtdxdr\ I

\Jo Jo

< \noV~1

f

I

< \voV~1

f

f u2xtdxdr + A + AZ 3 ' 4 ,

u2xtdxdr + A lma.xu2x\

I f* f1 /

u2xtdxdr,

ft{v)uxuxtdxdT\

\ fl f1 /

> fi0V~1 / / Jo Jo

/

f

f (1 + 08'3)dxdT

I pgOtuxtdxdT\ I

\Jo Jo

<\»oV~l

j

I

<\noV~1

f

f u2xtdxdT+A

These yield Lemma 4.1.9.

u2xtdxdT+AU

+ max02'3\ +

AZn/12.

f

f

$2dxdr

The influence of dissipation mechanism...

127

L e m m a 4.1.10. max / u2(x,t)dx

< A,

[0,T]7 0

/

/

u2xtdxdt
Jo Jo 2 max / uxxK (x,t)dx [o,T]Jo '

< A. -

Proof. Due to Lemma 4.1.9 and the definition of Z, it suffices to show Z < A. We employ (4.18) whose terms can be estimated, by virtue of (4.9), (4.13), (4.32), (4.33), Proposition 4.1.3, Lemma 4.1.8, as follows: / fL-2fL2v2xu2xdx< A J m a x t ^ i j / ii~2fvv2xdx

< A + AZ 5 / 6 ,

v\dx

< 2N2 | l + max0 8 / 3 j C v2xdx < A + AZ 3 / 4 ,

f fi-2p2ee2xdx < 2N2 | l + m a x ^ 3 | f

Q2xdx < A + AZ 1 1 / 1 2 .

These yield Z < A + AZn/12, which implies Z < A. L e m m a 4.1.11. rrnix I e2(x1t)dx
(4.34)

/ 92xtdxdt
(4.35)

/ JO

m a x ^ 02rOM)dz
(4.36)

Proo/. We differentiate formally (4.19) with respect to 2, multiply by eoOt, integrate over [0,1] x [0,tf],0 < t < T, and integrate by parts. After a lengthy sequence of estimations, the Lemma 4.1.11 can be obtained. The detail can be found in [24]. Now, we finish the proof of Proposition 4.1.4. By (4.34), (4.35) and Schwarz's inequality, 0(x,t) is uniformly Holder con­ tinuous in t with exponent ^ and (4.36) yields that 0x(x,i) is uniformly Holder continuous in x with exponent A. It then follows, from a standard interpolation

128

Quasilinear Hyperbolic Systems and Dissipative Mechanisms

property (see Ladyzenskaja, Solonnikov and Ural'ceva [101]), that 0x(xjt) is also uniformly Holder continuous in t with exponent A. Hence ||^||c 1 / 3 ' 1 / 6 (Q T ) — A, which immediately yields | | 0 | | C I / 3 , I / 6 ( Q T ) < A. Similarly, Lemma 4.1.10 im­ plies that \\UX\\CI/*,I/*(QT)(qr) < A which gives that | | U | | C I / 3 , I / 6 ( Q T ) < A, | | V | | C I / 3 , I / 6 ( Q T ) < A. 4.2

T h e i n f l u e n c e of d i s s i p a t i o n m e c h a n i s m o n t h e l a r g e - t i m e b e ­ havior of solutions

This section investigates the large-time behavior of globally defined smooth solutions established in Section 4.1. For the material of ideal gas, it has been proved by Nagasawa in [147] that the solution (v*,u*,0*) to the problem (4.3) (4.6) (4.7) satisfies u*M)>C*log(l-M),

C* > 0 .

However, totally different phenomena may occur on large-time behavior of solutions for other kind of constitutive relations. In the case of isothermal viscoelasticity, the solution may approach to a unique state exponentially fast as shown by Greenberg and MacCamy in [52], or phase transition may take place as discovered by Andrews and Ball in [3] with nonmonotone pressure. It is proved, in [3], that the large time behavior of strain is described by a Young measure whose support is confined in the set of zeros of pressure. The goal here is to extend the analysis used in [52] and [3] to the nonisothermal case — thermoviscoelastic materials. For simplicity, we consider the kind of solid-like materials with the following constitutive relations e = Cv0y

(T = -f{v)6+fi(v)uXl

q = -k(v)0x

where Cy is a positive constant, k(y) and f(v) t i a t e for v > 0 such that k(v) > 0 for f{v) > 0,

0< v < v

(4.37)

are twice continuously differen­

v > 0

f(v) < 0,

V < v < +oo

(4.38)

for some fixed 0 < v < V < -foo as mentioned in (4.11), and the viscosity fi(v)v is uniformly positive, as assumed in (4.9), namely fi(v)v > /i 0 > 0,

0 < v < +oo.

(4.39)

Remark 4.2.0. The example proposed by Ericksen ([40]) shows that f(v) is not monotone in v and satisfies (4.38). For rubber, it is known that a good

The influence of dissipation mechanism...

129

model of pressure takes the form p(v, 0) = —~f0(v

«■),

7 is a positive constant

v namely, f(v) = —*f(v — \ ) , which satisfies (4.38) with v = V = 1. Now, we turn to assumptions on initial data. The initial velocity is normal­ ized as in (4.15) and the initial data are compatible with the boundary conditions (4.6). The global existence of (4.3), (4.6) and (4.7), under the assumptions of (4.37)-(4.39) and (4.15) has been obtained in Theorem 4.1.1., which concerns the solid-like material with more general constitutive relations than (4.37). The following results on large-time behavior of solutions have been estab­ lished by Hsiao and Luo in [76]. Theorem 4.2.1. Assume {u(x,tf), v(x,t),0(x,t)}, (x,t) solution of the problem (4.3), I. l|p(M)(-,t)||Li[o,i]

that (4.37)-(4.39) and (4.15) are satisfied. Let E [0,1] x [0,oo)) be the globally defined smooth (4.6), (4.7). Then = l|/W0(-,*)IUi[o,i] -> 0 as t -+ + c o ,

ll/(v)(->*)IU a [o,i] -> 0 as * -► +oo, IN-,t)|| L 2 [ 0 ) i] -> 0 as t -)- + o o , and 0(t) =

[ 6(x,t)dx Jo

-> ^ 5 . as t -> + o o ^v

f1 1 where E0 = / [Cv00 + -v$\(x)dx. Jo * II. There exists a family of probability measure {^ar}a:e[o,i] o n ^ (depending measurably on x) with suppi/ r C K - {z : f(z) = 0} such that if $ E C(M) and 0 * ( * ) d =
ae

-

then * ( v ( - , t ) ) A £<*>(•) in L°°[0,1] as * -> + o o . R e m a r k 4 . 2 . 2 . Theorem 4.2.1 extends the phase transition results in [3] to nonisothermal cases. Corollary 4 . 2 . 3 . Suppose the equation f(z) = 0 has exactly m roots, z\, z2, - • •, zm, m > 1. Then there exist nonnegative functions \i{ G £°°[0,1], 1 < i < m, such that m

*(«(-,'))

A

53*(Z*)W(-)

in

^°°[0,l], a s t - ^ + o o ,

130 Quasilinear Hyperbolic Systems and Dissipative Mechanisms for any $ G C(M). m

Furthermore Yjji,-(ar) = 1, a.e. . i=i

Corollary 4.2A. z = z\, then

Suppose the equation f(z)

v(-,t) —> z\

= 0 possesses only one root

L 9 (0,1)

strongly in

as t —>■ oo

for all ^, 1 < g < + o o , provided the conditions (4.37)-(4.39) and (4.15) hold. If f(v) is strictly monotone decreasing, namely, f'{v)<0

for

v€[t7,F]

(4.40)

it follows from (4.38) that there exists a unique v £ [£, V] such that f(v) = 0. Thus, we have further results in the next theorem. T h e o r e m 4 . 2 . 5 . Assume that (4.37)-(4.40) and (4.15) hold. Then there are positive constants / ? , T and A, independent o f t , such that IK-,*) - 1)11*1(0,1) + l|ti(-,t)||jyi (0> i) + ||0(-,<) - ^ - 1 1 ^ ( 0 , 1 ) < ^ " ^ for t > f. Theorem 4.2.5 generalizes the results obtained in [52] which discusses the case of isothermal viscoelasticity. Firs, we prove Theorem 4.2.1. ^From now on, {u(x, t), v(x, t),0(x, t)} will denote the solution described in the global existence theorem. It is known from Proposition 4.1.3 that o
v(x,t)

< V,

0(x,t)>O,

a? € [ 0 , 1 ] ,

*e[0,+oo).

(4.41)

(4.41) and (4.39) yield 0 < /ii < fi(v(x, t)) < ji 2 ,

x€ [0,1],

t e [0, + o o ) ,

*i < k(v) < k2

(4.42)

where //i,/i2, &i and k2 are positive constants, independent oft. In the sequel, A will denote a generic constant, independent oft. The conservation laws of total momentum and energy have been obtained in Section 4.1, namely (4.16) and (4.17). L e m m a 4.2.6.

jff Proof.

o

[^f

+ ^\{x,r)dxdr
<e[0,+oo).

Substituting
«« + lf(v)6]* = [£(»)««]«•

(4.43)

The influence of dissipation mechanism...

131

Combining (4.3)3 with (4.3)2 and using (4.37) we obtain CvOt + f(v)0ux - ft{v)u2x - {k{v)0x)x = 0.

(4.44)

Multiplying (4.44) by 0~l and integrating over [0,1] x [0, t], with the help of (4.6) and (4.3)i, we get jf £

P

^

+ ^ ]

(*>')***'
[J\og0{z,t)dz

- / \og0{x,O)dx\ + / G(v)(x,t)dxJo J Jo

f Jo

G(v)(x,0)dx

where G(v) = / /(£)<*£. This, with (4.41), (4.42), and the inequality log0 < J\F

0 — 1, for 0 > 0, implies the Lemma 4.2.6. Due to (4.16) and the mean value theorem, there exists a y(t) G [0,1] for every t > 0 such that ti(y(<),*)=0. Thus I />x

|ti(M)|= /

I

T /*1

i 1/2 r /»1

MU)dt\ < I e{x,i)dx\

"i 1/2

2

j ^(x,t)dx\

\Jy{t) I UO J VJo V which, combined with (4.17) and Lemma 4.2.6, yields / maxu 2 (-, r)dr < A, Jo [o,i]

t£ [0, +oo).

J

(4.45)

Lemma 4.2.7. / {u4 + 02){x,t)dx+ JO

f f [02x + u2u2x]{x,r)dxdT
*€[0,+oo).

2

Proof. Multiply (4.3) 3 with (CV0+ \ ) and integrate over [0,1] x [0,*]. With the help of (4.6), (4.37), (4.3), (4.4) and Young's inequality, we arrive at \ I

+^ <

[ C v 0 + y ] {x,t)dx + inj

j

u2u2x{x,r)dxdr

f f 0l(x,r)dxdT

2 A + A / maxu (-,r) / 02(x,r)dxdr 01 Jo t ' ! Jo

+A / / Jo Jo

u2u2x{x,r)dxdr. (4.46)

132 Quasilinear Hyperbolic Systems and Dissipative Mechanisms To estimate the term /

/ u2u2dxdr,

we multiply (4.3)2 by t/3, integrate the

Jo Jo

resulting equation over [0,1] x [0,tf], and use the boundary condition (4.6), (4.41), (4.42) and the Cauchy inequality. It then follows that \ I u4(x,t)dx + 2//i / J

°

\°

t 2

<

u2ul(x,r)dxdr

/

J J

(4.47) 2

A + A / maxu (-,r) / Jo [o>i] Jo

0 (x,T)dxdr. 2

TJT)2

By using the Cauchy inequality with the term (CyO + obtain /

62{x,t)dx+

Jo ft

<

/

/

u2ul(x,T)dxdr+

/

0l(x,r)dxdT

Jo Jo

fl

ft

fl

/ u2u2(x, r)dxdr-t- A / maxw2 ( , r ) /

A 4- A /

Jo Jo

+A / Jo

/

Jo Jo

in (4.46), we

Jo [M]

02(,T)dxdT

Jo

u4(x,t)dx.

(4.48) Multiplying (4.47) with a suitably large positive constant, and combing with (4.48), we get /

02(x,t)dx+

Jo

f u*{x,t)dx+ Jo

f

f u2u2x(x,T)dxdr+

Jo Jo

< A + A I maxw 2 (,r) /

e2x{x,T)dxdr

[ f Jo Jo

e2{x,r)dxdr.

(4.49) Applying GronwalFs inequality to (4.49) and using (4.45), we obtain the Lemma 4.2.7. L e m m a 4.2.8. / / ul(x,r)dxdr Jo Jo

< A.

Proof. Multiplying (4.3)2 by u and integrating over [0,1] x [0,<], it follows, with the help of (4.6), (4.41) and (4.42), that 2 J0 ^ M ^ + Z'iy J

u2x(x,T)dxdr
I

f

f{v)0uxdxdr.

The influence of dissipation mechanism...

133

Due to (4.3)i and (4.41), / / Jo Jo

f{v)0uxdxdr

=

11 f(v){0 -H)u*{x,T)dxdT + / / Jo Jo Jo Jo

<

^T I I U2x{x,T)dxdT+k Jo Jo

+ / / Jo Jo

f{v]6ux{x,T)dxdr (0-0)2{x,T)dxdT

I f Jo Jo

f{v)Bvt{x,T)dxdT

where 6(t) = / 0(x,i)dx,t 6 [0,-f-oo). Jo _ By the mean value theorem, there exists a z(t) £ [0,1] such that 0(t) = 0{z{t),i). Thus i

|0-0|(*,r) =

11*)

i

f

1

0x^T)dc\<\f el(xJr)dx\

\JZ{T)

I

,

L/o

r e [ 0 , + o o ) . (4.50)

J

Integrating (4.3)3 over [0,1] and using (4.6) and (4.37), we arrive at Cv([

0(x,r)dx\

= - ( /

\u2{x,r)dx\

,

re[0,+oo).

Namely, 0t(r) = - ^ - \J

i u 2 0 r , r)dx\

,

r e [0, +oo),

(4.51)

which, together with (4.17), implies 0(T) = ± . \ E

0

- J

\U2{X,

r)dx] ,

te[0,

+oo).

(4.52)

This, together with (4.3)i, (4.41), (4.17), (4.45), implies, upon integrating by parts and using Cauchy inequality, that / / Jo Jo

[f(v)9vt](x,r)dxdT

- A~JlJ! {f,{v)vvt [ih (E° ~ I + J!J!{f{v)v[^{J!^u2{x,r)d8)] = A - g . |jf \(v)(x,t)dx - jf A(»)(*,0)d*J

^2{s'r)ds)]}{x'r)dxdT }{x'T)dxdr

134 Quasilinear Hyperbolic Systems and Dissipative Mechanisms

+

I I y^VUx\J

+

& {jf [f{v)v Of r 2 ( M ) d s )] (x't)dx

- fQ

2u2^'T^ds)\^x'T^dxdT

[f(v)v Q f ' ^(s,

0)ds)] (*, 0)dx}

- ^ 7 j f fQ {[f'(v)v + /(«)] «• ( j f 5« 2 («, r)d«) | (*, r)dxdr <

A+^r / / ul(x,r)dxdr+Jo Jo

f 4 Jo

u2{x,r)dx

where

A(«)0M)= f *'' /'(OfdeJv Lemma 4.2.8 follows from (4.49) and all of the estimates obtained. Lemma 4.2.9. [f(v)6]2(x,T)dxdr
/ / Jo Jo

Proof. By integrating (4.3)2 over [0, x] for any x G [0,1] and using the bound­ ary condition (4.6), we obtain that [f(v)0](x,t) = [jji{v)ua!]{z,t)-(j

u(y,t)dy\

,

x G [0, l],t € [0, +oo).

(4.53) Multiplying (4.51) by f(v)0 and integrating it over [0,1] x [0,tf], we arrive at [f(v)0]2(x,T)dxdr

/ / Jo Jo =

1 1 JO

[fl{v)uxf{v)0}{x,T)dxdT

JQ

rt

~JoJo

/AKA\

fl

f r rx {[Jo

-i w(2/

'rH

We estimate each term separately.

>! / ( * ) ( ' - * ) } (*>r)d*dr

(4.54)

The influence of dissipation mechanism...

135

In view of (4.41), Lemma 4.2.8 and Cauchy's inequality, / / \Jo JO

[fi{v)uxf{v)e}{x,T)dxdr\ I

\jj[f{v)e)2{x,T)dxdT+K,

<

te [0,4-oo).

By (4.41), Lemma4.2.7, Lemma 4.2.8, (4.50), (4.53) and the Cauchy inequal­ ity,

III {[I^^H /(v)^-^)}(x>r)^r| u

(V'r)dy)

<

faj

JQ \(J*

] (*,r)dxdT + Aj

<

ij

I [ / W ^ ] 2 ( x , r ) ^ c f r + A,

J

Ol{x,T)dxdr

*e[0,+oo).

Integrating by parts and using (4.3)i, (4.17), (4.45), Lemma 4.2.8, (4.51), (4.52), (4.53), Holder's inequality and Cauchy's inequality, we obtain that \ j j <

\\j*

A+ A j

j

+

4

u{y,r)dy\ «*+/

/ ( » ) ? } (*,r)(tedr| u2(s,r)ds\{x,T)dxdT

u(y r)dy m

2(s r)ds {x r)dxdT

&+I/ r i [£ ' u{y r)dy\ trm+^u{y'r)dy nv)u \i ' A

* * ijf f {[or ' ) r or ' ) 2

.j

<

A

+ g /

-u {s,T)ds\{x,T)dxdr\

/

[f{v)0)2{x,T)dxdr.

Lemma4.2.9 follows from these estimates on each term in (4.54). Lemma 4.2.10. / u2(x, t)dx -* 0 as t -> +oo. Jo

*\

136

Quasilinear Hyperbolic Systems and Dissipative Mechanisms

Proof.

It is clear from (4.45) that /•+OO

/ Jo

*i

u2{x,t)dxdt
/ Jo

Namely, / u2(x,t)dx etfdOt+oo)). Jo Multiplying (4.3)2 by u and integrating over [0,1], we obtain, with the help of (4.37) and (4.6),

I f1

I

/ (uut)(x,t)dx\ \Jo I < A [ [(f(v)0)2 + ul](x,t)dx. Jo Combined with Lemma 4.2.8 and Lemma 4.2.9, it implies rii / u2(x,t)dx\dx \dt Jo I

/ Jo

< A.

Thus Lemma 4.2.10 follows. Lemma 4.2.11. 0(i) = / 0(x, t)dx -*-?r, Jo &v

as t -> +oo

I I [/(")]2(*> r)dxdr < A, t e [0, +oo) Jo Jo \\f(v)(; t)\\L'[o,i] -* 0, as t -+ +00 ||p(»,tf)(-,OIU'[o,i] = ll(/(«)tf)(-.«)llL«[o,i]-+0, a s i - j . + c o .

(4.55) (4.56) (4.57) (4.58)

PTW>/. (4.55) follows from (4.52) and Lemma 4.2.10 directly. It is known from (4.55) that there exists To > 0 such that

^ ) > 2 § 7 as<>T0. Together with (4.41), Lemma 4.2.7, (4.50) and Lemma 4.2.9, it implies /

[f(v)]2(x,r)dxdr

/

JTo JO

<¥$-[' t\f{v)e]2{*,T)dxdT &0

<

A.

JTo JO

The influence of dissipation mechanism...

137

This estimate and (4.41) yield (4.56). To prove (4.57), we use (4.3)i, (4.41), Lemma 4.2.8 and (4.56) to make the following estimate

r+oo

<

A+ A / Jo

rl

/ Jo

u2x{x,t)dxdt
This with (4.56) together implies (4.57) directly. It is known that II(/(»)*)(-,<)||LI[O,I]

< (jT[/(f)]2(:M)
.

This, combined with Lemma 4.2.7 and (4.57), gives (4.58). Thus, part I of Theorem 4.2.1 is established by the above Lemmas. Next, we employ an idea of Andrews and Ball (see [3]) and the above results to prove part II of Theorem 4.2.1. Suppose tf G £ 2 [0,1] with tf > 0 and $ G C2([v,V]) satisfying *'(*)/(*) > 0 for z G [v,V]

(4.59)

where v and V are the lower and upper bounds of v. Let

^t):=IomJ^§dy

fr

° **W*°-

Multiplying (4.3)2 with