Mathematical Topics in Fluid Mechanics: Volume 2: Compressible Models (Oxford Lecture Series in Mathematics and Its Applications , Vol 2, No 10)

(5.22)

where K1 = St in the periodic case or if 12 = RN, N > 3 and K1 is an arbitrary compact set in 12 otherwise, where ST = SZ except in the case when St = R2 where IT is an arbitrary bounded open set in R2, and where K2 =11 in the periodic case or in the case of Dirichlet boundary conditions and K2 is an arbitrary compact

set in RN if 0 = RN. Remark 5.6 As we shall see below, the heart of the matter is really the convergence given in (5.20) that will be proved in the next section. Then, (5.21)-(5.22) follow from (5.20) in a rather straightforward way as shown in the next section also. On the other hand, part (1) of the above result is much simpler and we give its proof in this section. It is simply based upon the bounds we assumed and the equations (5.1)-(5.2) : such passages to the limit are in some sense results a la compensated compactness.

Remark 5.7 In view of Part (1), we see that the only real difficulty in passing

to the limit in (5.1)-(5.2) is the pressure term p1. Of course, since (t -- t'') is strictly convex on [0, oo], passing to the limit weakly in the term pr' means in fact that pn converges in L7 to p. The above result shows that this is the case if we require that the initial conditions po converges strongly to po. The considerations developed in the next two remarks show that the requirement is essential: otherwise, the strong convergence of pn to p is not true in general and, in fact, (p, u) is not in general a weak solution of (5.1)-(5.2)! At this point, we wish to emphasize the difference between this type of behaviour and what is known for the non-viscous case at least when N = 1: indeed, and this will be recalled and detailed in chapter 8, when N = 1, p = = 0, St = R, f n - 0, bounded sequences in L°° (SZ x (0, oo)) of (entropy) solutions of (5.1)-(5.2) satisfy automatically for all R E (0, oo), T E (0, 00), p E [1, oo) Pn - P, Pnun -n pu in LP ((-R, R) x (0,T)),

un n, u in LP ((-R, R) x (0, T) n {p > 01). This result is shown in P.-L. Lions, B. Perthame and E. Tadmor [360] for -y > 3, and P.-L. Lions, B. Perthame and P.E. Souganidis [358] for 1 < -y < 3. The strong convergence of pn at t = 0 (i.e. of po) is not needed and thus a compactification takes place which is lost when we add viscous terms! A "physical" explanation of this rather surprising fact (the non-viscous system is "better behaved" than the viscous one!) is as follows: if oscillations are present at t = 0

Compactness results

10

then shocks (i.e. discontinuities of (p", u")) form faster and faster as the frequency of oscillations increases and these shocks destroy the oscillations. On the other hand, in the viscous case, the velocity field u is smoother and shocks cannot appear for u. In this case, the initial oscillations are just being transported (with-

out any significant damping) by the flow, i.e. by the transport equation (5.1). Finally, (5.20) shows that if there are no oscillations at t = 0, no oscillations can

form at later times. . Remark 5.8 (Propagation of oscillations and homogenization). The situation we describe here will be analysed in detail elsewhere. We consider a model homogenization problem corresponding to the system (5.1)-(5.2). Let n = Q be the unit cube in RN; we consider (5.1)-(5.2) with periodic boundary conditions and we choose

po = po (nx) ,

mo = 1 mo (nx) in Q

(5.23)

n where po > 0, mo are periodic (of period 1 in each xi for 1 _< i < N), smooth

and satisfy -p Duo - 1;V div uo + Vpo = 0 in RN. As shown in chapter 8, it is then possible to find a unique periodic smooth solution (p, u) of + div (pu) = 0,

-yAu - eVdiv u + Vp'r = 0r in RN x [0, oo),

(5.24)

fQpudy=J mo dy for all t>0, Q

where p > 0 and pIt=o = po in RN.

We then consider p"(x, t) = p(nx, t), un(x, t) = (1/n)u(nx, t) and we check easily that (pn, u") solve (5.1)-(5.2) with the initial conditions (5.23) and with f replaced by fn given by

f"(x, t) = i ( au + (u 0)u

na

(nx, t)

on RN x (0, oo).

(5.25)

Obviously, (5.18) and (5.19) hold. However, (5.20) does not hold (as soon as p is not constant in x which is the case as soon as po is not a constant). In other words, oscillations can persist. Related phenomena are presented in D. Serre [4871.

It is possible to study much more complex situations where po = po (x, nx), mo = mo(x)+(1/n)mi(x, nx) and po(x, y), mi(x, y), mo(x) are periodic in (x, y) and in x respectively. Let us only mention that if, for instance, f " - 0 then the solution (pn, u") of (5.1)-(5.2) corresponding to the initial conditions (5.24) does indeed behave like p(nx, t), (1/n)u(nx, t). In this example, the weak limits (p, u) are very simple: u - 0, p - fQ pody. And it turns out that (p, u) is a solution of (5.1)-(5.2). However, we shall explain in the next remark why it is not always the case.

Compactness results and propagation of oscillations

11

Remark 5.9 (Weak limits are not in general solutions) . We wish to explain here why weak limits are not in general solutions. In fact, we are going to show that, roughly speaking, if the weak limit (p, u) is a solution of (5.1)-(5.2) in the case when 11 = RN then necessarily po converges strongly to po. More precisely, we assume that (p, u) is a solution of (5.1)-(5.2) and that we have sup n>1

f po (x) dx - 0

as R tends to + oo.

(5.26)

(IxI>R)

Then, we claim that po converges to po in LP(RN) for 1 < p < 7. Indeed, in view of part 1) of Theorem 5.1, the fact that (p, u) solves (5.1)(5.2) yields the convergence of V (p" )l' to V pry in D' (RN x (0,T)). Recalling that (pn)" is bounded in L°° (0, T; L1(RN)) and that p"r E L°° (0, T; L1(RN)), we deduce easily that (pn)7 converges to p7 weakly in the sense of measures

on RN x (0, T). Next, since 7 > 1 and thus (t '- try) is strictly convex on [0, oo), we deduce from standard functional analysis results that pn converges to p in L" (K x [0, T)) for any compact set K C RN. In particular, extracting a subsequence if necessary, we can find to E (T/2, T) converging to some t E [T/2, T] such that pn(tn) converges in L C(RN) to p(i). In addition, as we shall see in the proof of Theorem 5.1, (5.26) yields sup sup n>1

J

pn (x, t) dx ---> 0

as R tends to + oo.

(5.27)

(IxI>R)

Hence, pn (tn) converges to p (0 in LP (RN) for 1 < p < 7.

We then introduce p'n(x, t) = pn(x, to - t), un(x, t) = -u(x, to - t) for t E [0, tn], p(x, t) = p(x, t - t), u(x, t) = -u(x, t - t) for 0 < t < ) . We deduce from the above that pro = p`"It=o converges to po = Alt=o in LP(RN) f1 L 10C(RN)

for 1 < p < 7, Vun converges weakly in L2 to Du, and pn converges in V to p. Finally, we have obviously

ap 8t

+ div(lbn`n) = 0 in RN x (0, tn).

(5.28)

These facts imply our claim, namely An converges (uniformly in t) in LP(RN) to p for 1 < p < 7 and thus in particular po = pn(tn) converges in LP(RN) to po = p( ). This uniform convergence will be in fact proved at the end of the proof of Theorem 5.1 where we deduce "the uniform convergence of pn from the convergence in (x, t)" .

Remark 5.10 In the case when SZ = RN, it is possible to relax slightly the convergence of po to po in L1(f) replacing it by a convergence in Ll C(cl). Then, all convergence in (5.20)-(5.21) become local ones. In other words, the condition (5.26) is not needed neither for part 2) of Theorem 5.1, nor for Remark 5.4. This minor extension will be clear from the proof of Theorem 5.1.

Compactness results

12

Remark 5.11 The convergence result stated in Theorem 5.1 is precisely what was needed by D. Hoff [251] in order to extend the existence result of global solutions (close to an equilibrium) to more general barotropic laws p(p) (for instance p(p) = ap7 for a > 0, -y > 1) than p(p) = p. In fact, as indicated in the next remark, very general laws p(p) can be treated as well. 13

Remark 5.12 As we shall see after the proof of Theorem 5.1, Theorem 5.1 can be extended to general barotropic laws p = p(p) (instead of p = ap"). Our main assumption is that p is increasing on [0, oo): a natural assumption from the physical viewpoint-see chapter 1.

Remark 5.13 As we shall see in the next section, where we prove part (2) of Theorem 5.1, one of the key ingredients is the following fact (established in the next section)

Q(p") {(p + l;)div u" - a(p")^1}

- Q {(µ + t;)div u - ap}

in D'

(5.29)

for any continuous functions 3 on [0, oo) such that /3(t)t(q-,') and j3(t)t(g/2) goes to 0 as t goes to +oo, where we denote by p, 0 the weak limits (in the appropriate L'' spaces) of respectively. This property was shown by D. Serre [487] in the one-dimensional case, allowing a rather complete description of the propagation of oscillations (see also WE [167]). In [487], this descripsion of the propagation of oscillations was also obtained in higher dimensions postulating (5.29) (and various other more technical conditions). We shall come back to this interesting question in chapter 9.

We now turn to the proof of part (1) of Theorem 5.1. It is in fact a simple consequence of the following general lemma.

Lemma 5.1 Let g", h" converge weakly tog, h respectively in L°' (0, T; Lr2 (11)), L°1 (0, T; L'2 (51)) where 1 < pl, p2 < +oo,

-+-=-+-=1. pi 1

1

1

1

Qi

P2

Q2

We assume in addition that agn is bounded in Ll (0, T; W"1 (52)) for some m > 0 independent of n

8t

-

11h" - h"( +e,t)1ILgl(o,T;Lg2(n)) -* 0 as ICI -+ 0, uniformly in n.

(5.30) (5.31)

Then, gnh" converges to gh (in the sense of distributions on 52 x (0,T)).

Remark 5.14 In the statement above, weak convergences are weak-* convergences whenever some of the exponents are infinite. In (5.31), hn(. + C, t) is not really defined on fl x (0, T) but we can either extend h" to R" by 0 or restrict the T,a3 norm to 1' = fx E 0 /dist (x,852) > Ihl}.

Compactness results and propagation of oscillations

13

Proof of Lemma 5.1. Let e > 0. We are going to prove the above claim in fZEO x (0,T) where 11,0 = {x E 0/dist (x,811) > eo } and we simply denote from now on Sl instead of ftE. Then, we consider he = f hn(y, t) rcE(x - y) dy is a regularizing kernel: is > 0, Supp fore E (0, co) where r., = ,c c Bi, fRN rc dz = 1. The condition (5.31) ensures that we have 11h n

-

hE I I L91

(o,T;L42 (ft)) w

0 as a ---> 0, uniformly in n.

In addition, hn converges weakly to hE = h*rKE as n goes to +oo and hE converges strongly to h in L91(0, T; Lq2 (f2)) as a goes to 0. Hence, writing gnhn = gn(hn hE) + g"hE, we deduce easily that it is enough to show that gnhE converges in

-

D' to gh,. In other words, we may assume without loss of generality that hn is bounded in LQ1(0, T; Wk,q2 (Il)) for all k > 0 and converges weakly to h in Lg1(0, T; Wk,g2 (ft)).

Next, we set H" = fo hn(x, s) ds. The bound on hn when qi > 1 implies easily that Hn converges to H = fo h(s) ds in Wk,q2 (el) uniformly in t E [0, TJ for all k > 0 at least when 12 is bounded (otherwise the same convergence holds on compact subsets). The same claim holds when qi = 1. In fact, for any multiindex

a, Di hn is clearly bounded on ft x (0, T) and for all t, s E [0, T], for all x E 1 I Di Hn (x,

t) - DiHn (x, s) I < Cb + I

k

11

f DaH' x (y, t) dy a(x,a)

f Dz Hn (y, s) dy a(z,S)

t

dy

< CE -}I

BaI

(x,d)

dTDz hn(y, o) a


Il x [0, T] (or on K x [0, T] for all compact subsets K of 0 when 11 = RN) to

DzHforanylal>0.

-

It is then easy to conclude writing gnhn = 8(g"H")/at (ag"/at)H" (this equality of course requires to be justified but there is no difficulty here in view of the spaces to which belong gn, H", agn/at, h"). Indeed, g"H" obviously converges to gH (in fact weakly in Lp1(0, T; LP2 (f2)) while (8gn/8t)Hn converges in the sense of distributions to (8g/8t)H since Hn converges, in particular, to H in C' MI) uniformly in t E [0, T] . We conclude observing that

gh = a(gH)/at - (ag/at)H. 0 Proof of part (1) of Theorem 5.1. We begin by proving that m = pu, e = pu ®u. First of all, we recall that un is bounded in L2 (0, T; Hl). In this section, we do not specify on which domain we are working in order to avoid recalling each time the various cases we consider: let us simply indicate that in the proofs

Compactness results

14

that follow all function spaces concern a spatial domain which is 91 in the case of Dirichlet boundary conditions, the periodic cube in the case of periodic boundary

conditions or a ball of an arbitrary radius R E (0, oo) in the whole space case. Therefore, (5.31) holds with hn = un, q1 = 2, q2 arbitrary in [2, 2N(N - 2) (Q2 arbitrary in (2, oo) if N = 2).

We then wish to apply Lemma 5.1 with g' = pn and gn = pnun. If this is possible, we conclude: m = pu and e = m 0 u = pu 0 u. The bounds on 8pn/8t, 8pun/8t are straightforward since we have for instance in view of (5.1), (5.2) and the bounds on pn, un 8pn

W-1,27/0(+1))

is bounded in L°O (0 , T;

C L1 (0 , T;

W-1,1)

apnun is bounded in L' (0, T; W-1'1) + L2 (0, T; H-1) c L' (0, T;

W-1,1 )

8t Therefore, in order to conclude, we just need to check that pn and pnun are bounded in L2 (0, T; LP) for some p > 2N/(N + 2). This is clear for pn which is bounded in view of (5.19) in L°O(0,T; L'') and r > N/2 > 2N/(N + 2) (N > 2). Since un is bounded in L2 (0, T; H'), we also deduce from (5.19) that pnun is bounded in L2 (0, T; LP) where

2N2 ifN>3, p
p = r1 +

We conclude choosing p E (1, r) if N = 2 and observing that 1

<

p

2

N

+

N-2-N+2ifN>3. 2N

2N

-

It just remains to show that v = f u. This equality is proved exactly as we did for m and e, the only modification being that we need to obtain a bound on 8 pn/8t. Let us first of all argue formally: we just observe that (5.1) yields

(div u) = 0.

(Vlp-) + div (uv ) +

(5.32)

2

Hence, if the equality (5.32) holds, we deduce easily that 8 p /8t is bounded in L°° (0, T; W-1,2) +L 2 (0, T; L2/(1'+')) and we conclude as above. Therefore, we only have to show that (5.32) holds for pn, un in place of p, u. In order to justify (5.32), we need p or pn to belong to L2 (St x (0, T)) and this is the case because of (5.19). Indeed, using the regularization lemma shown in chapter 2, we deduce

that

+ div (up,) = rE

,

1 `1+1 p

2

q

where pE = p * ?E and kE is the regularizing kernel already used in the proof above.

Proofs of compactness results in the whole space case

15

Hence, we have for any Q E Ci ([O, oo)) such that 31(1 + t)-* is bounded on (0,oo) where

, E) + div (u/3(pf)) + (dlv u) {$(PE) - 13'(pE)I E} = and thus letting e go to 0+

aa(p) + div (u,(3(p)) + (div u) {,3(p) -,Q'(p)p} = 0,

(5.33)

an equation which holds when p E Lq(K x (0, T)) (for all compact set K C 0) for some q > 2. The equality (5.32) follows from (5.33) choosing Qa = b -+t for b > 0, letting 6 go to 0+ while remarking that,(3a (p) - ,3, (p) p = VT -+P - 2 p/ vw-+p converges as 5 goes to 0+ (in Lz i for instance); and the proof of part 1) of Theorem to a 5.1 is complete. Remark 5.15 We observe for future purposes that if Q E C'(0, oo), 3'(t) = o(t ) as t goes to 0+ and ,(3'(t)(1 + t)-a is bounded on (1, oo) where a = (q - 2)/2, then (5.33) holds with .3(p), u replaced by ,3(p11), un (we agree that 3'(p)p = 0 if p = 0). In particular 84(pn)/8t is bounded in L2q/(q-1) (0,T; Lq/(q-1)) +

Ll (0, T; L1) ((C L'(0,T; Ll)) and thus (5.30) holds with gn = ,6(pn). Since ,Q(pn) is bounded in L2 (0, T; L2), we can use Lemma 5.1 as in the preceding proof to deduce that 8(pn)un converges (in the sense of distributions) to ,Qu if ,3(pn) converges weakly in L2 to some ,0.

5.3 Proofs of compactness results in the whole space case This section is devoted to the proof of part (2) of Theorem 5.1 and more precisely of (5.20). But before we begin this rather delicate proof, we wish to present first a formal argument that yields the compactness of pn and which is based upon the energy equation. The reason why we present this argument, which can be justified modulo various technical bounds that we do not know how to obtain, is that we believe it is a more direct one than the actual proof while it contains some of its essential features. Furthermore, we think this argument could lead to some interesting analysis of numerical schemes for the system of equations (5.1)-(5.2).

Formal argument for the strong convergence of pn The main assumptions we need for this formal argument are that p, u (the weak limits) have to satisfy (5.9) (or a variant of it written below), (div u)- E

L1(0, T; L°O(f )), and (pn, un) satisfies (5.9) (with (p, u, f) replaced by (pn, un; f,)) or even the following weaker version Iunl2 2

d

d

R

pn

+

a 'Y

< J p"un. fndx. n

pIDun12 + l;(div un)2 dx

1(Pn)ry dx + n

(5.34)

Compactness results

16

Without loss of generality, we may then assume that (e)" converges weakly in Lq/7 x (0, T)) to some p : since (t - t1) is convex, we deduce that p > p7. Using part (1) of Theorem 5.1, (5.34) yields 2

dt

in

p

+'Y a l p dx +

-

I

Jn µI DuI2 + t;(div u)2

dx <

r u pf dx.

(5.35)

n

On the other hand, we have, using part (1) of Theorem 5.1,

+div (pu) = 0, a Pu

+div (pu®u) - piu - l;Odiv u +O(ap) = pf.

This yields, at least formally, the following analogue of (5.9): dt

,

+

p

(2

7

2

+

P ry

l dx + J,DltI2+(divu)2dxl

J in

f

_

(5.36)

n

where pe = p - p7 > 0. Combining (5.35) and (5.36), we obtain

d

jPe dx + (-y - 1) j(div u)pe dx < 0.

(5.37)

In particular, fn pe (t) dx < exp [('Y - 1) fo I I (div u) I I Loo ds] fn pe (0) dx. Therefore, if po converges to po in L'7 (a), pe (0) = 0 and we deduce that pe = 0. This shows in fact that p" converges in Ll' (SZ x (0, T)) to p and the main assertion of

part (2) of Theorem 5.1 is shown. 0

Remark 5.16 At this stage, it is worth looking at the conditions which are needed in order to justify (5.9) or (5.36). Typically, if we assume that p (or p7) E L2 (n x (0, T)), we have in addition to (5.1) the following equation

+ div (pu (9 u) = T E L2(0, T; H-1(1))

(5.38)

(considering for instance the case of Dirichlet boundary conditions, the other cases can be treated in a similar way). And it is then possible to justify (5.36).

Proof in the case when St = RN, N > 3 and when pn is bounded in L-f+1(RN x (0,T)) n L°°(O,T; L°(RN)) with s> N We begin with this case in order to explain some of the main ideas developed in this proof, leaving out all the technical difficulties associated with boundary

conditions (or N = 2) or the fact that "good" bounds on p" are not always ava.i I able.

Proofs of compactness results in the whole space case

17

Step 1: Formal proof. Furthermore, we wish to begin with a formal presentation of the argument that we shall develop and justify below. We use two main ingredients that both follow from (5.1)-(5.2) : indeed, we first observe that we have 19P Log

p

+ div (u p Log p) + div u p = 0

(5.39)

and taking the divergence of (5.2) (-0)-'div

a

(pu) + (-A)-laij(Puiuj) + [(µ + )div u - app'] (-A)-'div (Pf)

=

(5.40)

From (5.40), we extract div u and obtain

(µ + )div u = ap7 +

- (-0)

(=A)-ldiv

(pf) 8j(Puiu.i) -

5j(-0)-'div

(5.41)

(pu).

Then, inserting this relationship into the right-hand side of (5.39), we deduce, using (5.1)

(P + 0 + 5

aP &g + div (up Log p)] {p(_A)-'div (pu)} + div

_p(_p)-'div

+

(Pu)(-0)-'div

(pu) +

(Pf ) p(-0)-laij(Puiuj)

hence finally (µ

+

e)

aP

P

&g

+ div (up Log p) + apl+7 =

_ p(_A)-'div (pf) + +p {(-0)-'aij (Puiuj)5i- u

[p(-0)-'div (pu)]

+ div

[pu(-0)-'div (pu)]

V(-0)-ldiv (pu)}.

(5.42)

We are going to use systematically these (and related) identities. First of all, we can write (5.42) with (p, u, f) replaced (p", un, fn) and we pass to the limit in (5.1), (5.2) and (5.42). Part(1) of Theorem 5.1 and a systematic use of Lemma 5.1 then yield (we shall come back to these points later on): (5.1) holds and (5.2) at the limit is replaced by a

(pu)+div (pu(9 u) - pAu - Vdiv u +Vp= pf,

(5.43)

where P is the weak limit of a(pn)-. In addition, we deduce from (5.42) (p + l;)

+ div (us) + apl+7 =

- p(_A)-'div (Pf) +

(9

at + p {(-A)-1a=j(Puiuj) - u

[p(-z)-'div (pu)]

+ div

[pu(-0)-'div (pu)]

V(-z)-'div (pu)}

(5.44)

Compactness results

18

where p1+7, s denote respectively the weak limits of (pn)1+7, ppLog pn (extracting subsequences we can always assume they converge weakly). The first three terms on the right-hand side of (5.44) are obtained easily using Lemma 5.1 as we shall see below. The fact that we can pass to the limit in the term pn 81j(pnui un V(-A)-ldiv (pnun)} and recover the expression p{(-0)-1 8ij (puiuj) (pu)} is also aconsequence of Lemma

81j(puluj)u - V(-0)-ldiv (pu)} as the commutator of uj and aj(-0)-1div acting upon pu, i.e. [uj, Rij] (put) where Rij = (-A)-1 81j is a "nice" Calderon-Sygmund 5.1 and of the following crucial observation: one can write

singular integral operator which is nothing else than the composition of two Riesz a1(-0)-1/2 and aj(-A)-1/2 ). Next, we have a (limited) transforms (namely

"Sobolev" regularity of u", u (in L2(O,T; H1)) and this is enough, as we shall explain below, to make the operator [uj, Rj] a smoothing operator which allows us to pass to the limit. Next, we deduce from (5.1) and (5.43) exactly as we deduced (formally) (5.42) from (5.1) and (5.2)

(µ + )

+ div (u s) -p(-0)-ldiv

_

+ pi [p(-0)-ldiv (pu)]

(pf)

+N {(-0)-1a1j(puiuj) - u

+p

+ div [pu(-0)-1div(pu)]

V(-0)-ldiv (pu)}

(5.45)

where s = p Log p. Comparing (5.44) and (5.45), we obtain 19

(s

- s) + div

[u(s - s)] +

a

[p1+7

- p 7] =

0

(5.46)

where 7 = p/a is the weak limit of (pn)7). The conclusion follows from rather standard functional analysis and .onvexity

considerations since we know that s < s a.e. and p1+7 > pp7 a.e. In particular, integrating (5.46) in x, we deduce d

dtINs-sdx < 0 R

while fRN s -

s dxlt=o = 0 since we assumed that po converges strongly to po

and fRN s - s dx < 0 for all t > 0. Hence, s = s a.e. and, since t Log t is strictly convex, we deduce finally the strong convergence of pn towards p which is the crucial information required to complete the proof of Theorem 5.1. Of course, the above argument is just a sketch of the actual proof we may now begin.

Step 2 : Preliminaries. We justify here some of the considerations developed in step 1 above.

Proofs of compactness results in the whole space case

19

First of all, if u E L2(O,T;H1 c) and p E Li c satisfy (5.1) then using the fundamental regularization lemma stated in Part 1, chapter 2, we see that we have

/3(p) + div [u3(p)] + (diva) [p3'(p)

- Q(p)] = 0

(5.47)

for any C' function /3 from [0, oo) into R such that

3C > 0,

dt > 0,

(5.48)

Ii3'(t)I < C(1 + t°1)

with a = (q - 2)/2 (if p E Li ; if q = 2, (5.48) reduces to assuming that

/3'

is bounded) . Next, approximating /3(p) = p Log p by /36 (p) = p Log (p + 6), we write (5.47) with 0 replaced by Q6 for b > 0. Then, we observe that we have p Q6 (p)

-,06(p) = p2 (p + b) -' -'pin LQ (or L2) as b - 0+.

In this manner, we justify (5.39) passing to the limit as b goes to 0+. In particular (5.39) holds both for (p", u") (replacing of course (p, u) by (p", u")) and for the

weak limit (p, u) provided we show that (5.1) holds-a fact that we establish below.

Next, we justify (5.42) (with (p, u) replaced by (p", u")). The equations (5.40)-(5.41) are immediate (in view of the properties of -A on RN, N > 3). Then, in order to justify (5.42), we have to make sure that each of the terms of the right-hand side of (5.41) cafr be multiplied by p (or p") and to justify the manipulations on the term p J(-A-1div (pu)). Multiplying pry by p is meaningful since we assumed that p" is bounded in D+1. Similarly, the assumptions made upon f (or f") namely (5.18) ensure that we can multiply (-0)-'div (pf ) (-0)-' aij(puiu3) and we observe that, by by p. Next, we consider the term assumption, puiu. E L°° (0, T; L' (RN)) n L' (0, T; L'(RN )) where 1

1

N-2

r

s

N

(p"ui u,' is bounded in that space) since uiuj E

L'(O,T;LN/(N-2)(RN)) by

Sobolev inequalities. In particular, the operator (-z )''a{j (=RJR;) is bounded in all L' spaces (for 1 < r < co), so we see that

(-A)-'ai3(puiui)

E L' (O, T;Lr(RN))(nL'(O,T;Lq(RN))

where 1 < p < oo, 11q = 1/pr + 1 - 1/p. Hence, we can multiply this term by p since p E L°O (0, T; L3 (RN)) and

s+r = s+N N 2

2

N-2 N =1.

N The only fact left in order to justify (5.42) is the multiplication of p by

(-j)-'div (pu) and writing

Compactness results

20

pa [(-0)-'div (pu)] {p(-A)-ldiv (pu)}

_

_ _

19

- L (-L)-'div (pu)

{p(-A)-'div (pu)} + div {p(-A)-ldiv (pu)} + div

(pu)(-0)-ldiv (pu) (pu(-o)-'div

(pu)) - pu

V(-A)-ldiv

(pu).

This is not difficult to justify so we only sketch the argument. Indeed, smoothing Ic E Co (RN x R), p by convolution in (x, t) (pE = p * ice, 'E _ (1/EN+1)(,/e),

Supp rc C B1, e E (0,1]) so that (5.1) becomes W + div (pu)E = 0 we can perform the above manipulations replacing p by pE since pE is smooth. Then we let a go to 0 and we recover the equallity between the first and the last terms. In doing so, we essentially only need to check that p . [(-A)-ldiv (pu)] , p(-A)-1

div (pu), pu(-0)ldiv (pu) and pu - V(-0)-ldiv (pu) make sense (in Ll ,,): in fact, the last of these four terms is very much similar to the term studied above namely p(-A)-181j(puiu1) and the two preceding ones are even simpler. Finally, per(-A)-ldiv (pu) makes sense because of (5.41) and because p div u makes sense since div u E L2 (RN x (0,T)) and p E Lf+' (RN x (0, T)) (-y + 1 > 2) . We may now begin the passage to the limit as n goes to +oo in (5.1), (5.2) and (5.42). Without loss of generality, we may assume that pn Log pn, pn)7, pn = a(pn)7, (pn)7+1 converge weakly respectively to s, pti, p = a 7, p'r+l. Observe that pn Log pn, (pnyt (and thus pn) are bounded in some L" space for some p > 1 and the weak limits are taken in these spaces while (pn) T+1 is bounded in L' (RN x (0, T)) and the weak limit to_+ is in the sense of measures (or in the sense of distributions). In particular p1+7 is a bounded non-negative measure on RN x [01T). We have already shown in the preceding section that (5.1) holds at the limit

while (5.2) leads to (5.43) (i.e. (5.2) with p = ap7 replaced by p = a 7). In order to conclude this step, we explain how one can pass to the limit in pn)un,pn(-i)-'div

the expressions

pn(-A)-ldiv (pnfn),

pnun(_O)-'div (pnun). For(pnfn), all these terms, we are going to use Lemma 5.1. pn(Log

First of all, pn(Log pn)un goes to s u for the same reasons as given in the proof of part (1) of Theorem 5.1 (see section 5.2 above). Next, for the remaining three terms, we use Lemma 5.1 with g' = pn, pn, pnun, hn = (-A)-1div (pn fn), (-A)-'div (pnun), (-A)-ldiv (pnun), respectively (recall that we already know that pnun converges to pu). The three terms are treated similarly and we just

detail how to use Lemma 5.1 for the last one (which is the "worst" of the three). Indeed, pnun is bounded in L°° (0, T; L1 n Lq(RN)) with 1/q = 1/2s + 1/2 (write pnun = pn pnun) and in L2(0,T; L'' (RN)) with 1/r = (N - 2)/2N + 1/s < 1/2. In addition, (5.2) immediately yields the fact that (5.30) holds. And

Proofs of compactness results in the whole space case

21

(-A)-ldiv (p"u") is thus bounded in L°O(0,T;W1l (RN))nL2(0,T;W1o (RN)) so that (5.31) holds with q1 = 2, q2 E [2, Nr/(N - r)). And we conclude easily. Step 3 : Passing to the limit in the commutator. Since (5.45) (and thus (5.46)) is deduced from (5.1) and (5:43) exactly as we deduced (5.42) from (5.1) and (5.2), we only have to show that pn { [u7 , Rjj](pnui)

} w, p {[uj, R=j](pu=)} in D'

(5.49)

in order to conclude that (5.46) holds. This is precisely what we prove here. Since Dun is bounded in L2 (RN x (0, T) ) and p"u" is bounded in L2(0,T; LP(RN))(nL°°(0,T; L1 (RN))) where 1/p = (N2)/2N + 1/s < 1/2, we may use the general results of Bajsanski and R. Coifman [25], and R. Coifman and Y. Meyer [109] on commutators of the form [uu , R,=j] to deduce that [uu,Rj ](p"ui) is bounded in L1(O,T;W1,4(RN)) where 1/q = 1/2 + 1/p. Next, we wish to deduce (5.49) from this regularity and by using Lemma

5.1 twice. Indeed, we first use Lemma 5.1 with gn = pnui and hn = u, (more precisely with Ri j (p"ui) since we already know that p"ui u, -and thus R,j(p"ui u )-converge weakly to pu8uj-and thus to R=j(pusuj) respectivelyby the proof of part (1) of Theorem 5.1). Then, pnui (or R= j (pnui)) is bounded in L2(O,T; LP(RN)) with p > 2, (5.30) obviously holds in view of (5.2) and uj satisfies (5.31) with q, = 2, q2 = 2 since Dud is bounded in L2(RN x (0,T)). converges Therefore, applying Lemma 5.1, we see that Un = [u7 , Rij] weakly to U = [uj, Rjj] (pui) (for instance in L1(O,T; L°2 (RN)) for q2 > 1

satisfying 1/q2 < 1 + 1/s - 2/N while, as shown above, Un is bounded in L' (01 T; W1,-?

(RN)).

Hence, we may apply Lemma 5.1 with g" = pn, p1 = oo, p2 = s, hn = Un,g1 = 1, q2 = s/(s - 1) since (5.30) clearly holds in view of (5.1) and (5.31) follows from the bound on Un together with the inequality 1 2 1 Nq 1 -N=1-N+S q2q 1

i.e. s > N. This completes the proof of (5.49) and thus of (5.44)-(5.46).

Step 4 : Pointwise convergence. We start from (5.46) and we wish to prove that r = s - s , where s = pLog p, vanishes identically on RN x (0, T). From this fact, we shall then deduce the pointwise convergence of p" to p. We begin by observing that, since p" I Log p" I can be bounded by C [(p" )1+b+

(p")1-a] for any S E (0, 1), s, s and thus r belong to LOO (0, T; LP (RN)) for all 1 < p < N (for instance). In addition, using once more the Sobolev inequalities, L2N/(N-2) (RN)) we know that u E L2(0, T;

We next claim that r > 0 a.e. and that p1+7

-p7>0

(in the sense of measures). This is a simple consequence of the convexity of the functions

Compactness results

22

(t i- t Log t), (t '-+ t(1+7)/7) and (t H t1+7) on [0, oo). Indeed, we obviously have

(pn)l+7 =

{(pn)7}(1+7)/7

therefore pl+7 > p7 (1+7)/7 and pl+7 > (p)1+7 7/(1+7)

1/(1+7)

and p < { (p1+7) reg } Therefore, p7 < {(pi+ir),eg} where µreg denotes the regular part of the measure it with respect to Lebesque measure. Hence, Formally, the sign of r is clear but the (weak) singu7P < (p1+7) Ceg < larity of Log at 0 requires some justification. One possible argument is to write the following convexity inequality pl+7,

pn Log pn > V Log cp + (Log cp + 1) (pn -W) = (Log cp)pn + pn -W a.e.

for all cp E Lt bounded away from 0. Letting n go to +oo, we deduce

> (Log cp) p + p - cp and we conclude choosing cp = p + 6 for 5 > 0 and letting S go to 0+. Next, in order to prove that r vanishes, we first deduce from (5.46)

+ div (ur) < 0 in RN x (0, T).

(5.50)

And we simply need to integrate this inequality over RN x [0, t] and to use the sign of r. Once more, we need to justify all this. First of all, we remark that a (pn Log pn) ,

are bounded in L°° (0, T ; W -1,1(RN)) for example. And using

the result shown in Appendix C of Part 1, we deduce that pn and pnLog pn converge weakly respectively to p, s weakly in LP(RN) for 1 < p < s uniformly in [0, T]. In particular, p, r E C[0, T]; LP - w) where LP - w means LP endowed with the weak topology. Furthermore, since po converges to po in L' (RN) and thus in LP (RN) for 1 < p < s, we deduce that p(O) = po and r (O) = 0 a.e. in RN. Finally, in order to justify the integration on RN of (5.50), we only have to show, using cut-off functions of the form c,(./R) where cp E Co (RN), 0 < co 1, co = 1 on B1, Supp cp C B2, R E (1, +oo), that we have

1T1

luIr

R I Vcp ()I dx - 0 as R --> +oo.

(5.51)

N L2N/(N-2) (RN))

, r E Loo (0, T; LP(RN)) for 1
1 IIUIILI(0,T;L2N/(N-2)(RN))

-

IIrII Loo(o,T;L2N/(N}2)(RN))

RIIvWIILoo(RN)

Next, we wish to conclude, from the fact that r = 0, the strong convergence in LL (or pointwise) of pn to p. This is a straightforward and classical consequence

Proofs of compactness results in the whole space case

23

of the strict convexity of the function (t E [0, oo) '-- t Log t). One possible proof consists in observing first that, extracting a subsequence if necessary, we may assume without loss of generality that ((pn + p)/2) Log ((pn + p)/2) converges

weakly (say in LP(RN x (0, T)) for 1 < p < N) to some s and (by convexity) s > p Log p a.e. . Thus, we have

0 < On = z p Log P+ pn Log pn -

(Pfl

p

2

Z

P"

Log

2

P

p Log p - s < 0 a.e. (weakly). Hence, On

n 0 weakly and since On > 0, Ann 0 strongly in L' (BR x (0, T))

for all R > 0. From the strict convexity of On, one deduces immediately that pn converges to p in measure on each BR x (0, T) (for all R E (0, oo)). Therefore, pn converges top in Lpl (0, T; LP (BR)) n Lq(BR x (0, T)) for all 15 pl < oo, 15

p2<s, 1
Step 5: Conclusion. We prove here the convergence of pn in C ([0,T];LP(RN))n Lq (RN x (0, T)) for all 1 < p < s, 1 < q < ry + 1. It is clearly enough to show the convergence of pn to p in C ([0,T]; L1(RN)). To this end, we introduce do = pn which clearly converges to Vp- in L2 (0, T; L2P2 (BR)) n L'9 (BR x (0, T)) to d = for all R E (0, oo). We next remark that p E C ([0, T]; Ll (RN)) and thus d E C ([0, T]; L2(RN)) . Indeed, using once more the regularization lemma shown in Part 1, chapter 2, (Lemma 2.3), we obtain the existence of a bounded pE E C ([0,T]; L1(RN)) smooth in x for all t satisfying

ft+div(upE)=rE-+0 PE - p in L1(RN x (0,T)),

in L}(RN x (0,T)) as e -- 0+. PEI,=o -' pIo_o in L1(RN) as e -- 0+.

From these facts, it is straightforward to deduce that a

8t

1p,

- Pn I + div (ulpE - Pn I) = rE - rn

and thus

T

I PE - p,, l dx =

T [,1

JR

N

I rE - rn I dx.

Since p E C ([0, T]; LP (BR) - w) (for all R E (0, oo), 1 < p < s), we may then deduce that pE converges to p in C ([0, T]; L1(RN)).

Next, we observe that we can justify as we did in step 2 above that do and d satisfy n

div (undn) = z (div un)dn at +

,

+ div (ud) = 2 (div u)d.

Therefore, using once more the result of Appendix C in Part I, do converges to din C Q0, T]; L2(RN) w).

-

Compactness results

24

Thus, in order to conclude, we just have to show that whenever to E [0, T],

to ,f

t, then dn(tn) n d(t) in L2(RN) or equivalently that fRN dn'(tn)2

dx = fRN pn(tn) dx w* fRN d(t)2 dx = fRN p(t) dx. This is the case since we deduce from (5.1), integrating this equation over RN and justifying the integration exactly as in step 4 above, JgN

e(tn) dx=JRNP0 dx

0

JRNp

dx=p(t) dx. fgN

Proof in the case when fl = RN, N > 3. We now explain how we have to modify the preceding proof in the case when we only assume (5.19) and we no longer postulate further bounds on pn. The main idea is to replace p' Log p' by (pn)e (for instance) where 0 E (0,1) will be chosen small enough later on. Then, as in steps 1 and 2, we obtain for all 0 E (0,1)

a at (pn)e +div [u'(Pn)e] = (1 - 0)(div un)(Pn)e.

(5.52)

Next, we use the same manipulations on (5.2) as in steps 1-2 above to deduce

for 0 > 0 small enough (so that all terms below make sense (namely 0 < 0 < 2v

min(q - -y,

- 1))

Cu + )(div un)(Pn)e = a(p')p'+A + (pn)A(_0)-ldiv (Pnfn)

_

=

[(pn)0(_Q)-1d1V (pnun)]

(pn)ry+A

-

(Pn)A(_0)-l0.

+

(Pn)e

(Pnui uj ) (-0)-1div (Pnun)

+ (pn)A(_0)-1diV (Pnfn) - (Pn)A(_0)-laii(Pnui uj )

n A (-0)-div (pnun )] - div [u n (p) n A (-0) - ata [(P) 1

+(p')eu'

V(-A)-'div

I div (pnun)]

(p'u') + (1 - 0)(div

u")(Pn)A(-0)-'div

(Pnun)

or finally

(p + )(div 'un)(P")e = a(pn)4A +

(Pn)e(-0)-'div (Pnfn)

_ [(pn)9(_0)-1div (p'u')] [u'(P")e(-A)-ld1V (pnun)] - div + (Pn)e

[un V(-0)-ldiv

+(1 - 0) (div

(pnun)

(5.53)

-

un)(P")e(-o)-1div

(-A)-1ai.(Pnui

uj )]

(p'un).

We next pass to the limit in (5.52)-(5.53). To this end, we denote by 7,7;79and Q the respective weak limits (in I," (RN x (0, T)) for p = q/0, p = q/ (^I+())

Proofs of compactness results in the whole space case

25

and p = 2q/(q+26) respectively-recall that by assumption q > 2 and q > -y-of (pn)9, (pn)7+e, (div un)(pn)e. Exactly as in steps 2 and 3, we obtain

(pe) + div (upe) = (1 - A) Q,

(µ + OQ = ap'l+e

+ p(-p)-'div (P.f) _'

(5.54)

[(_y'div (Pu)] -div [u pe(0)-1div (pu)] +7[u V(-0)-idiv (pu) - (-0)-18ij(Puiuj)] + (1 - 6)Q(-A)-'div(pu). 8t

(5.55)

In addition, (5.43) still holds, from which we deduce in view of (5.54)

(p + e)div upe = ap" pe +

-

pe(-0)-idiv

(Pf) - Pe(-j)-'

5i(_Pe(-A)-'div

(-0)-'div

(pu)) +

= a p pe +

pe(-L)-'div

(Pf)

(Pu:uj) (pu)

[pe(-,&)-'div (pu)]

-rat

- div [u pe(-0)-idiv (pu)] + Pe [u

- (-0)-laij(P uiuj)] + (1 -

V(-0)-'div e)Q(-0)-idiv

(Pu)

(pu).

Comparing this equality with (5.55), we deduce

(p + )Q - ap7+9 = (µ + )divu pe - app pe

a.e.

(5.56)

Next, we observe that we have (by convexity, see also step 4 above)

(p7+91 e/(7+e) >

(pe/

(pir+e)1"8)

>- p7

a.e.

hence we deduce from (5.56)

Q > (divu) pe

a.e.

(5.57)

In particular, this inequality combined with (5.54) yields A

pe + div (u pe) > (1 - 6) (divu) pe.

(5.58)

We then wish to conclude about the pointwise convergence of pn as in step 4 above by proving that (p9)1/e _ p. We observe that, on one hand, (pe)1/9 < p

Compactness results

26

and (p)1/el,.o = PI, = Po while, on the other hand, we may deduce at least formally from (5.58)

{iIe} at 49

+div

{U1/e}

> 0.

(5.59)

This looks like a rather innocent manipulation but it turns out that a proper justification requires a bound on p in L2,t: in other words, it is precisely at this point (and at this point only) that we need to assume q > 2. Indeed, we may regularize (as usual, by Lemma 2.3, Chapter 2, Part I) (5.58) and find for all 6 E C'([O, oo)) with say /3' E Co ([0, oo) ) a at

{/3()} +div {u/3()] ? (1 - e)(divu) 01(-p-6)7 +(div u) [i3G) - 701(Pe)]

_ -A (div u) 6'(7)7 + (div u) 6(P9). {coM(pe)}1/e

where cpM = MV(./M), M > 1, cp E

We then choose Q =

C0 ([0, oo)), V(x) = x on [0,1], Supp cp C [0, 21, and we obtain 1/e 1/e a {co(p)} + div u {coM(P8)} Cat

> -(divu) {caM (pe))}

_ (divu) {caMp8}

1 a -1 W(

1/e-1

[caM(pe)

{WM(pe)}1/e

)P8 + (divu)

- cP'M(P9)7]

1(Pe>M)

-Co Idiv ul MO' 1 (7>M) where Co = sup { I'(x) I co(x) - cp'(x)xl /x E [0, oo) } . We then deduce easily (5.59) provided we show that Idly ul Ml/e 1(Pe>M) converges to 0 in L1(RN x (0, T)) as M goes to +oo. First, we observe that pel/e < p hence Idiv ul M11e 1 (Pe>M)<- Idiv ul M1/e 1(P>M'/e). Next, divu E L2(RN x (0, T)) and p E L2 (R" x (0, T)). Therefore, we have 11/e-1

f 0

T

dt

JRN

dx Idivul M1/e 1(P>Ml/e)

as M -4 +00 At this stage, we have shown that p" converges to p in L}(BR x (0, T)) for all p E [1, q) and in LP1(0,T; LP2 (BR)) for all pi E [1, oo), P2 E [1, r) and for all R E (0, oo). The argument made in step 5 is still valid here and we complete the proof of Theorem 5.1 in the case when 11 = RN and N > 3. 0 < Ildivu 1(P>M1/e)IIL2(RNx(o,T))IIP1(P>M1/e)IIL2(RNx(o,T))

0

Proofs of compactness results in the whole space case

27

Remark 5.17 We wish to observe that we used in the proof of Theorem 5.1 bounds on pn in LOO (O, T; L'' (RN)) for some r > N/2 and in L4 (RN x (0, T) )

for some q > y. We assumed that q > 2 and we used the L2 bound in the last argument above: in fact, when y < 2, inspecting the above argument we see that we only need a bound on pn in L2'°O(RN x (0,T)). In fact, we only need to know

that the limit p belongs to L2i0O(RN x (0, T)). Indeed, if this is the case, we remark that we have

fTdtj C

dx Idiv ul Mlle 1(p>Miie)

II(divu) 1(p>M1/e)IIL2(RNx(O,T))Mi/e meas (p> M1/e)1/2

< C I I (div u)1(p>M1,e) I IL2(RN x

(O,T)) -'

0

as M -- +oo.

Remark 5.18 Let us now prove the statement announced in Remark 5.13. First of all, let us observe that, even if we no longer assume the strong convergence of po to po, the property (5.29) holds. In fact, the proof made above immediately yields, replacing (pn)e, pe by /3(p'1), 8 respectively, the fact that (5.29) holds for any function /3 E Cl([O, oo)) such that Q'(t) goes to 0 fast enough as t goes to +00 (say, 3' E Co ([0, oo)) or If3'(t)I t1-6 is bounded for some 8 as in the preceding proof). Next, if 0 is an arbitrary continuous function on [0, oo) such that Q(t) 0-7 and ,Q(t) tq/2 go to 0 as t goes to +oo, we approximate it by a sequence of functions Qk E C1([0, oo)) such that f3k E Co ([0, oo)), Qk converges to 83 uniformly on each [0, R] for all R E (0, oo) and Qk (t) V-7, /3k (t) tq/2 go to 0 as t goes to +oo uniformly in k. Without loss of generality, we may assume that f3k (pn) converges weakly (in w - L°° *) to 13k. Then, (5.29) holds for each 13k. We deduce that (5.29) holds for 3 by letting k go to +oo. Indeed we have IIi3k(Pn)(divun) - 3(pn)(divun)II Ll(RNx(0,T) < I Idiv un1IL2(RN ),(O,T)) I I Qk(Pn) - I3(Pn)II L2(RN x (O,T))

IIQk(Pn)(Pn)7 - 8(Pn)(Pn)7II L1(RNx(O,T)) IIP"IILQ(RNx(o,T))II /3k(Pn) -,Q(Pf)I'Lq/(q_7)(RNx(o,T))

and for each e E (0, 1), there exists a constant CE > 0 independent of n such that Iak(Pn) - /3(P' )I

c Min [(Pn)q/2e

+ SUP {100)

(Pn)q-71

- Q(t) I /0 < t < CE }

.

0

Remark 5.19 (Connections with compensated compactness). We wish to explain the part of Remark 5.13 and assertion (5.29) that is related to the div-curl

Compactness results

28

lemma of the compensated-compactness theory (F. Murat [401], [402], L. Tartar [530], [531]). Indeed, on the one hand, we have some information (and LP bounds, hence W-1,p compactness for appropriate p) on div t,x (00(P)un) (see equation (5.33)) for convenient 3. On the other hand, we can project equation (5.2) on the space of gradients decomposing (orthogonally) "arbitrary" vector fields v as Pv + Qv where curl (Qv) = 0, div (Pv) = 0. We then find

at

Q(pu) + Q (div (pu ® u))

- (p + e)Vdiv u + V ap = Q(pf);

(5.60)

notice indeed that Du = V (div u) - curl curl u. Therefore, we get some informaun -a(pn )-y -,Gn) where Oon = Qdiv (pnun tion (i.e. bounds) on curlt,y ((p'+C) div Q(Pnun) un). Using, at least formally (one needs to work out the appropriate functional setting, which is a bit delicate here but nevertheless can be done), the div-curl lemma, we deduce that /3(pn) [(p+ )divan -a(pn)ry - Y ')n+unQ (pnun)] weakly passes to the limit. f3(pn)(,)n This is precisely the quantity given in (5.29) with the extra term unQ(pnun)). Next, we observe that 7pn = -(-0)-1 div [div (pnun 0 un)] _

-(-0)-10ij(pnui ujn) and that Q(pnun) = -V(-0)-'div (pnun). Hence, the additional term is nothing but V(-0)-'div (pnun)

/3(P") {un

_

uj )}

which is the crucial term we had to analyse in detail in the above proofs; it is the term that involves the commutators [ui , Rij]. It turns out that here the div-curl lemma sheds some light on the argument but unfortunately on the easy part of

the argument. 0 We wish to conclude this section with a few facts. First of all, the general structure hidden behind the product &n) {(p + e)div un - a(pn)" } is studied briefly in Appendix B, where we also show the following fact. Let P be a general

pseudo-differential operator of order -1 (like for instance (-0)-1/2, (_A)div,

(-0)-'curl, (-A)-'D). Then the following convergence holds in D'

O(pn) {P(iuu' + Vdiv un

-

a0(pn)-Y)}

n , {P(-pOu + Odiv u - aV

pry }

for the same class of non-linearities 3 as before. In particular, choosing P = (-A)-lcurl, we find

/3(pn) curl un n /3 curl u in V. Similarly, choosing P (v) _ (-0)-' ak vi and P = Rki(-0)-'div (Rki = RkRi) we deduce for all 1 < i, k < N 8(pn) {p C3kun + eRki div un - aRki(pn)' } n Q {pakui + Rkidiv u - aRki Pry} %

Proofs of compactness results in the whole space case

29

and 3(PP) {Rkt { (µ

+ )div un - a(pn)" } }

n Q {R, i {(µ + l )div u - a py}}

.

Therefore, we also have for all 1 < i, k < N

fl(pn) {8kui - Rki div un} W. 8 {aku - Rkidiv u}. All these properties obviously contain information about the possible behaviour of sequences of solutions. However it is not clear how one can use them and this is the reason why we did not incorporate them in Theorem 5.1 (nor in its proof). Our final observation concerns the decay of "oscillations" in the context of Theorem 5.1 when we no longer assume that po converges strongly to po. In this case, as we have seen above, oscillations on pn may persist for all t > 0 but we want to show that in some sense their strength decays as t increases. Indeed, we claim that we always have

at{P-(Pe)pie}+div{u[p-(pe)pie]} ry + 1, we have the following precise identities

tIe

8 cat{P-(Pe)

}+div{u[p-(p9)

p7+e Cpe) n

pie

a

+µ+

1

6-1

_ p7(pe)lie j=OinV', for all 0 < 8 < 1,

{ p Log p - p Log p J + div {u p Log p - p Log p] } +

a u+t; [pT+71-pp =0 in D'.

The relationship between our "claim on the strength of oscillations" and the preceding identities becomes clear once we recall that p - (pe)lie, pLog p p Log p are non-negative and vanish if and only if pn converges (in Lt)

to p.

We briefly sketch the proof of these identities. First of all, it is enough to consider the case of (p - (pe)lie) since the other case can be deduced from it.

Indeed, we just need to observe that lie (p - (pe)'/e) converges, as 8 -- 1, to p Log p - p Log p. Next, our proof relies upon a truncation of pn and we shall simply use pn A R (R E (0, oo)) to simplify notation. In fact, this is not absolutely correct since (t - tAR) is not C' and we need in fact a further layer of approximation (smoothing t A R or directly working with an increasing, concave

truncation function): we ignore this irrelevant technical detail here. Then, the proof of Theorem 5.1 yields the following identity (for all 0 < e < 1). 5j (p A R)e + div (u(p -A R)8) = (div u) {(1

- 8)(p A R)e + RelP>R}

Compactness results

30

-µ+c(1-6) [PPAR)e_iIPAR)eJ a +µ + {Rep'1lP>R - P7 Re1p>R} in V. C

In fact, the equality also holds with 1P>R (depending on the type of smoothing we use for t A R). Next, if q > -y + 1, we let R go to +co and easily deduce

(1 - A) [p7+e - p7 pe] in V.

(pe) + div(u pe) = (1 - 6)(div u) pe + +C

5j In the general case, we use the fact that p7(p A R)e > p7(p A R)e, p'Y1P>R > 7 lP>R. These inequalities are very particular cases of general inequalities that we show in the next section. Therefore, we have

5j (pe) + div (u pe) > (1 - 6)(div u) pe in D'. At this point, we may follow the proof of Theorem 5.1 and recover the desired identities. In fact, with a bit more work (a similar proof will be given in chapter

6), one can show that the above equalities hold if q > -y (and q > 2). Let us also finally mention that the term µ+C [p7+1 - 7p] creates some damping of oscillations measured by (p Log p - p Log p) (for instance). It is worth noting that,

roughly speaking, the damping increases as a goes to +oo which corresponds to the incompressible limit (low Mach number limit) and as p + l; goes to 0 which corresponds to the inviscid limit (to the Euler equations), which are two asymptotic regimes where "some compactness" is to be expected.

5.4 Proofs of compactness results in the other cases We are going to conclude in this section the proof of Theorem 5.1 first in the periodic case, next in the case of Dirichlet boundary conditions and finally in the particular case when Sl = R2.

Proof in the periodic case. The proof is essentially the same as in the case when 1 = RN and N > 3 (see section 5.3) except for one modification concerning the inversion of -A with periodic boundary conditions. More precisely, for each

periodic function g on RN such that fn g dx = 0 we denote w = (-A)-'g the unique periodic solution of

-Ow = g in RN, w periodic,

fo

wdx=0.

(5.61)

This operator makes sense not only for periodic functions g E LoC (RN) , 1 < p < oo) but by duality for any distribution g = E «j =1 act gcx where m > 1, ga E is periodic. In addition, whenever it makes sense, (-A)-1 commutes with derivatives.

Proofs of compactness results in the other cases

31

With this convention, we deduce from (5.2) and (5.43) the following relationships

(p + ) div un = a [(pn)7 -

fi2(Pn)7 dx]

+ (-0)-idiv (pn fn)

(pntf )] - (-0)-iai,,(pnui uj) - -5ia-0)-'div [( I

and

-f

A) i div (pf) ( , a + ) div u = a p 7 a p7 dx] + (-0 (5 . 63) (-A) iaa7(puiuj) ) i div (pu)] . [( at The proof then follows step by step the argument made in the preceding section and we do not repeat it here. We simply have to observe that (pn)e (fn(pn)7 dx) converges (weakly) to pe(fn p7dx) (use for instance Lemma 5.1, noticing that fn(pn)7 dx is independent of x and thus is smooth in x!). Proof in the case of Dirichlet boundary conditions. In the case of Dirichlet

-

boundary conditions, the proof given in the preceding section has to be modified

in two places. First of all, one has to localize the argument which yields the following limit

{(p + l;)div un - a(pn)1'} (pn)e {(p + ) div u - a

71 pe

in D'(S2 x [0,T]).

(5.64)

Next, the conclusion about i) the pointwise convergence and then ii) the full convergence in C([O,T]; LP(11)) for p < r has to be carried out taking care of boundary difficulties. First of all, we deal with the localization of the proof of (5.64). This is rather

straightforward (but somewhat tedious). Indeed, we can still write

i div (p'u') + 8ij (pnut uj) - (p + )0 div un + 0(a(pn)7) = div (pnfn) in D'

(5.65)

and

div (pu) + ai,(puiuj) - (p+t;)Adivu+ A(ap7 = div (p f) in D'

(5.66)

Therefore, letting cp be an arbitrary cut-off function namely cp E C00001), 0 < cp < 1, Supp cp D K for an arbitrary fixed compact set K C Il we deduce at div (WPnun) + ai; (WPnui uj)

- (p + e)0 (V div un) + A(aW(Pn)7)

= div (cp pn fn) + Fn

a div (Wpu) + aij (vuiui) - (p + ) A (w div u) + A(cpap7

Compactness results

32

=div(cppf)+F where we have

Fn =

div un at (Pnun VV) + (atjW)Pnui uj + 2aicp aj (Pnu uj) - (µ + C)Acp fn

-2(µ + )VW Vdivun + Acpa(pn)7 + 2aVcp . V (e)"

- P"

. Vcp

and

F=

a

(Pu V W) + (aij cP)P'uiuj + 2a,cp aj (Puiuj) - (µ + C)OV div u VW

or equivalently Fn = ascp 8j (Pnunju7) + (aij w)Pnun uj

- (µ + e)dcp div u'

- (2µ + C)Ocp V div un + p un Vcp + AV a (pn)7

(5.67)

+aV cp V (e)-' and

F = a=cp aj(Pujuj) + (a0jcP)Pujuj - (µ + )Ocp div u

7.

(5.68)

We may then follow without further changes the argument developed in the (pn)e(-A)-1Fn preceding section and establish (5.64) once we observe that converges weakly (say in D') to pe (-A)-1 F for small enough 6 > 0. Once more, we only need to apply Lemma 5.1 remarking that Fn is bounded in LP(0, T; W ',P)

for p > 1 close enough to 1 (depending only on q, r and ry) and choosing 9 satisfying in particular q > A p/(p - 1). Let us only mention that the operator (-0)-1 can be taken as the inverse of -A on the whole space since all functions considered are supported on Supp cp or on fl with Dirichlet boundary conditions.

We then deduce from (5.64) the following inequalities as in the preceding section

a

1/9

(pe)

+ div

1/9

u (pel

while (-6)11E) < p and (pe)1/eIt=o

=

> 0 in D'(St x (0, T))

(5.69)

plt=o = po in 11; and in order to be able

to conclude that p = (pe)1/e, we need to integrate (5.69) over ft Formally, this is clear since u vanishes on aQ. To make it rigorous, we recall that since u E L2(0, T; Ho (11)) then u/d E L2(SZx (0, T)) where d(x) = dist(x, aQ). We then

denote by (E = ((-) where ( E C' Q0, oo)), ((t) - 0 if 0 < t < 1/2, 0 < C < 1

Proofs of compactness results in the other cases

33

on [0, oo), C(t) - 1 if t > 1 and e E (0, 1) is fixed. Multiplying (5.69) by (,(d), we see that we only need to show that we have

f

T

r

dt

Jn

dx p I u V (E (d)

0

as a - 0+.

(5.70)

This is the case since

pu VCE(d) I = p

Jul

C'

(d)

(IVdl =1 a.e.)

< t>0 IC'(t)I 1(.
w u - VCE(d)I < 2 p IiI 1(d<E)(t>o IC'(t)I ), and (5.70) is proven since p E L2(fZ x (0, T)).

At this stage, we have shown that pt3 converges to p in LP(CZ x (0, T)) fl LP1 (0, T;172 (e)) for all 1 < p < q, 1 < pl < oo, 1 < P2 < r. The convergence of p? in C ([0, T]; LP (11)) for all 1 < p < r or equivalently in C ([0,T] ; Ll (fl)) is shown exactly as in the preceding section keeping in mind the above justification of the integration over Q. The only new ingredient is the proof of the fact that p E C ([0,T] ; L' (a)) since the argument in the preceding section uses a mollification and thus is not clear near the boundary. This difficulty is solved by combining the above truncation argument together with the regularization argument. More precisely, we obtain as in the preceding section for c E (0, 1), pE E C ([0,T]; L1(cl )) where 1ZE is, say, defined by {x E fZ /d(x) > e/2}, smooth in x for all t E 10, T], satisfying

ap,

div (up,) = rE in Q, x (0,T), PCIt=o = p° at +

where II p° - pE I I LI (n!) -' 0

, I I rE I I LI (n!

(5.71)

x (o,T)) ` 0 as e - 0+. It is rather

straightforward to show that PC - PIIc((o,T1;1,1(fl)) __1'O as E - 0+. We then deduce from (5.1) and (5.71) d

T

n

jIPP6IC6(d)dx

< .fn

IrEIdx+J

IVCE(d)I Jul Ip - PEI dx

hence

(I (t)

T

sup oT fn , 1

I P -PEI dx <

Joo

dt

n!

IrEI dx + 2(sup I t>_O o

I)

Jo

T

dt f

e

dx Jul (p + PE) d

Compactness results

34

<

jTj dt

IrEI dx

+ 2(sup K'(t)I)II Idi 1(d<E)UL2(nx(0,T))2 IIPIIL2(nx(O,T))

since I1PEIIL2(nx(o,T)) <- IIPIIL2(0x(0,T)) We then conclude easily.

Proof in the case when SZ = R2. First of all, the main difficulty is the fact that we no longer have global L" bounds on u71. On the other hand, as we have seen from the preceding proof in the case of Dirichlet boundary conditions, most of the proof is in fact local and we do know that u" is bounded in L2 (0, T; LF (BR) )

for all p E [1, oo), R E (0, oo). This is why we can adapt the proof in the case of Dirichlet boundary conditions and prove that (V.69) hold (of course with SZ = R2).

Inspecting the proof given in the case of Dirichlet boundary conditions, we immediately see that the only point left to check is the justification of the integration over R2 of terms like div ((peu) or div (pu) and more precisely that

"the integral vanishes". This is in fact straightforward provided we use the bounds on p and pIuI2 (and the fact that 0 < (pe)1/e < p a.e.). Indeed, we know that p E L°°(0,T; Ll(R2)) and pItI2 E L°O(0,T; L1(R2)). Therefore, p u E L°O(0,T; Li(R2)). Then, letting cp E Co (R2), 0 < W:5 1, cp . 1 on B1, cp = 0 on B2 and setting

f f

V (

for R > 1, we have

T

dt

dx PI ul I V 'R(x) I

1

II Ocp II L-(R2)

II Pu II L1(R2 x (0,T)) --> 0

as

This allows us to conclude as in the preceding cases and thus to complete the proof of Theorem 5.1.

Proof of the convergence assertions (5.21)-(5.22). We begin with the convergence of pnu". In order to prove it, we use once more a mollifier rcc = rc(E) where r. E Co 00 (RN), 'c > 0, fRN rc dx = 1, Supprc C B, and we let gE = g * cE for an arbitrary function g: notice that, in the case of Dirichlet boundary conditions, if g is defined on Si x (0, T), gE is defined on iE x (0, T) when SzE = {x E Si / dist (x, 8S2) > e}. We first observe that we have for all

N/2
11(P"un)E - P"u"}(x)I = J [Pt 1(y, t) - P" (x, t)l of (y, t) rcE (x

+

P" (x, t) (uE

- u") (x, t)

I

- y) dy

Proofs of compactness results in the other cases

f

IPn(y,t) - p"(x,t)IP

35

i6(x-y)d

N

(lunlp/(p-1)) (P-1)/P

+

/E

Pn IuE - un)

hence for all t > 0

r I(Pnu")E -

pnun'I

J

/' dx < [f J dx J

1/p N

I Pn(y, t) - Pn(x, t)I p Ke(x - y) dy V-1 [Iunlp/(P-1)],'jj

P + IIP"IIL,IIu. - unllLp/P-1

ISUPIJPn(.+Z)_PIILP]

IIunI

ILp/(p-1)

lzl<e

+ IIP"IILP IIuE - unIILP/(p-1)

where all integrals in x (and all L' norms) are taken over RN in the case when ,Q = RN, over the periodic cell 11 in the periodic case or over 11E in the case of Dirichlet boundary conditions. Next, if we choose p > N+2 , pp l < N z and thus IIuE - un I I L2 (o,T;LP/ (P-1)) converges to 0 as e goes to 0+, uniformly in n. In addition, (5.20) implies that sup, .1<E I IPn(. + e) PnIILP converges to 0 as e goes

-

to 0+, uniformly in n. Therefore, in conclusion, (p"u")E - pnun converges to 0 in L2(0,T; L1) as a goes to 0+, uniformly in n. Next, (p"u")E is obviously smooth in x , uniformly in n and in t E [0, T]. Therefore, remarking that, because of (1.2), ac (p"u")E is clearly bounded in L2(0,T; H') for any m > 0, we deduce that (p"u")E converges to (pu)E as n goes to +oo in L1(0, x (0, T)) (say) for each e > 0. Then, using the bound on pnun in LOO (0, T; L27/(7+1)(SZ)), we deduce from those facts that pnun converges to pu in

L' (f x (0, T)) and the first part of (5.21) follows easily. The second part of (5.21) is an immediate consequence of the first part together with the bound on un in LPN/(N-2) (K) ) L2 (K x (0, T))-in fact the convergence is even true in LP(0, T;

for all 1 < p < 2 by the same argument. The L2 convergence in (5.21) of un is easily deduced from the fact that p"Iu" uI2 converges in L' to 0 (and the

-

strong convergence of p" uniform in t) since

p"Iu" -uI2 =

p"IunI2

-2

(p"un) . U + pnIul2

converges weakly to 0.

In particular, (5.21) yields the a.e.-convergence of p"uEuj (or of a subse-

quence) to puiuj as n goes to +oo (V 1

_<

i, j _< N) and (5.22) follows

in view of the bounds on p"u; in LOO (0, T; L1 (St)) fl L1(0,T; La(11')) where 1/a = 1/r + (N 2)/N < 1 and f' = 0 except in the case when S = R2, where S1' is an arbitrary compact set of R2. 0

-

Compactness results

36

We conclude this section with a simple observation about the bounds which

have to be satisfied by f" in order to apply the above arguments. Inspecting the proof closely, we see that we only need p" f " to belong (and be bounded in)

Liar and to be able to pass to the limit in the product (p")e (-A)div (p" f") using, for instance, Lemma 5.1. This is clearly the case if we replace (5.18) by

f" = fl' + f2, fl' (resp. f2) converges weakly to fi (resp. f2) in La (!a x (0,T)) (resp. Lb(0,T; Lc(SZ))

(5.72)

where a > q/(q - 1), b > 1, c > s/(s - 1). This is how the above proofs yield the following

Corollary 5.1 Theorem 5.1 still holds if we replace (5.18) by (5.72).

General pressure laws

5.5

In this section, we shall investigate a variant (or extension ) of the preceding results and proofs. We are concerned here with the case of a general barotropie

fluid, that is the equations are still (5.1)-(5.2) but in (5.2) the term apt is replaced by p = p(p) where p is a general "pressure laid', i.e. p is assumed to be a continuous non-decreasing function on [0, oo) vanishing at 0 (this is an irrelevant normalization of p). We already explained in chapter 1 of part I how the monotonicity of p follows from thermodynamics and we shall see here that our mathematical analysis relies in an essential way on that property. In other words, we study the system of equations consisting of (5.1) and

+ div (pu (9 u) - p0u - l;V div u + Vp(p) = pf

(5.73)

and we consider the same three possibilities for boundary conditions (whole space, periodic case, Dirichlet boundary conditions). Let us only briefly mention the analogue of (5.8) in that general situation or in other words the local energy identity

at

pI2I2

+ div

pu 22

+l;(divu)2 +

µA 22 - l; div (u div u) + plDu12 (5.74)

(q(p)) + div (u[q +p]) = pu f

where q is defined (up to a constant) by dt(q(t)/t) = p(t)/t2 for t > 0 . From this identity, we can deduce various bounds as we did in section 5.1. We do not want to give too many details here but let us just make a few remarks in that direction. There are two cases worth considering: first of all, if p(t) is such that fo ds < oo then we can choose q(p) = p fo' tt dt and then only the behaviour of p at infinity matters in order to obtain a priori bounds. For instance,

if lim,-+oo p(s) Is" > 0 when y > 1, then the same bounds as in section 5.1

General pressure laws

37

I2

Next, if p(t) t-' goes to a positive constant as t goes to 0+, we can choose q(p) = pfi tt dt and q behaves like p Log p as p goes to 0+. In the case of Dirichlet boundary conditions or in the periodic case, the change of sign in q does not affect the a priori bounds since q is bounded from below and we easily reach a priori estimates for plui2, plLog pI in LO°(0,T; L'(11)), Du E L2(St x (0, T)). In the case when 1= RN, we have to be slightly more careful. We can recover some

bounds using the existence of some p E L1(RN) such that Vq'(p) E L°O(RN): take for example p = e-IzI, assuming for instance that p E C1([0, oo)). We then write

a +div {u[P__++P_PQ'(/3)]} at PI22+{q(P)-q(p)-gV)(P-P)} ediv(u div u) + pl Du12 + t; (div u)2 = pu f - pu Vq'(p) µD 12

-

-

hence, if f E L'(0, T; LOO (RN)), we have for some a > 0

[f1+ 2

dt

{q(P) - q() - q'(P)(P - )} dx] + a < IIp IUI2

JR

IDuI2 dx

IILI(RN) IIPIILI(RN) IIf - Vq'(P)II LO(RN)

Recalling that do fRN p dx = 0, we deduce bounds for Du in L2 (RN x (0, T)) and

for pIuI2 and {q(p) - q(p) - q'(p)(p - p)} in L°°(0,T; Ll(RN)). Notice indeed that this last quantity is non-negative because q is convex since q'(t)t-q(t) = p(t) and thus q"(t)t = p(t) > 0 (in the sense of distributions).

We now consider a sequence of solutions (pn, u') of (5.1), (5.73) and we make the same assumptions on this sequence as in section (5.2) except that we need to modify the assumptions on pn. We assume that (pn)n>1 is bounded in C ([O,T];L'(S2)), (p(pn))n>1 is bounded in L°°(O,T; L1(SZ)) and we replace (5.19) by (pn)n>1 is bounded in LQ(S1 x (0,T)) nL°O(0,T;Lr(S2))

in the periodic case or if it = RN, N > 3, (pn)n>1 is bounded in L9 (K x (0, T)) n L' (0, T; Lr (11)) in the case of Dirichlet boundary conditions or if S1= R2,

(5.75)

for some q > 2, r > N/2, and we also assume that we have ((pn)sp(pn))n>1 is bounded in L'(K x (0,T))

(5.76)

for some s > 0, where K is an arbitrary compact set included in S1.

Theorem 5.2 Theorem 5.1 still holds and, in addition in part (2), p(pn) converges to p(p) in L'(K x (0,T)) for any compact set K in SZ.

Compactness results

38

Proof. Most of the proof of Theorem 5.2 is the same as that of Theorem 5.1. In particular, we obtain the analogue of (5.29), namely

,8(P") I

div u" - p(p' )} n j {(µ + C) divu - fi} in D'(1 x (o, T)) (5.77) 0(t)t_9

go to 0 for any continuous function 3 on [0, oo) such that ,Q(t)t- /2 and as t goes to +oo. The rest of the proof is exactly the same as in the proof of Theorem 5.1 using Q(p) = pe for 6 small enough (9 > 0) and the following crucial lemma that we

apply with pi(t) = te, p2 = p. Lemma 5.2 Let P1,P2 E C([O, o0)) be non-decreasing functions. We assume that pi(p"), p2 (p") and pi (p") p2(pn) are relatively weakly compact in L'(K x (0, T)) for any compact set K C St. Then, we have Pi (P) P2 (P) ? Pi (P)

P2 (P)

a.e.

(5.78)

Remark 5.20 The assumption on pi (p"), P2 (P") is equivalent to requiring that we have {Ip1(P")I + IP2(P")I + IP1(P")P2(Pn)I } 1(P..>M)

--- 0 in Li(K x (O,T))

as M - +oo,

uniformly in n > 1, for any compact set K C Q. Remark 5.21 In fact, we do not need the full strength of (5.78); we only need it when pi (t) = to and P2 = p. But, it turns out that there is no gain in generality (for p) since we claim (as is quite standard) that requesting that (5.78) holds for such a choice (and for an arbitrary weakly convergent sequence p") is in fact equivalent to requesting that p is non-decreasing! In order to prove this claim, in view of the proof below of Lemma 5.2, we observe that if we choose a sequence (pn) n> i such that p" oscillates between two values a and b (a # b > 0 are

fixed) with probability 1/2 (pn = a if t E (2k/n, (2k + 1)/n), = b otherwise, for 0 < k < [(n + 1)/2] -1 when St = (0,1), N = 1 for example) then (5.78) reduces to

2 aep(a) + beP(b)

-

2

(2 ae + 2

-

be)

(2p(a) +

Zp(b))

or equivalently, (ae be) (p(a) p(b)) > 0. Since a, b > 0 are arbitrary, this means that p has4 to be non-decreasing on [0, 00).

Proof of Lemma 5.2. We first make a few preliminary reductions. Without loss of generality, we may assume that pi (0) = P2 (0) = 0 (subtracting a constant from either pi or P2 does not affect (5.78)) and thus P1, P2 > 0. Next, we observe that it is enough to prove (5.78) when pi and P2 are bounded. Indeed, if this is the case, (5.78) holds for pi A K, P2 A K for all K E (0, oo). Then, we have for

i = 1,2 and for all ME (0,00)

Other boundary value problems

0 : P, (P,) - pi(Pn) A K -< pi(P")1(pn>M) if K >

39

sup lpi(t)) = pi(M)

0
0 < P1(Pn)P2(Pn) - (Pi (e) A K) (p2(Pn) A K) <- PI(Pn)P2(Pn) 1(p">_M)

if K > p1(M)p2(M). Hence, in view of Remark 5.15, we see that (pi (pn) A K) (p2 (pn) A K) , pi (pn) A K, p2(pn) AK converge respectively in L1 to p1(pn)p2(pn), pi(pn),p2(pn) as K goes to +oo, uniformly in n. Therefore, we have as K goes to +oo

pi(Pn) A K - pi(P)

,

(pl(Pn) A K)(p2(pn) A K) --o pi(P) p2(P)

in L'

and (5.78) is shown in full generality. Next, we remark that it is enough to prove (5.78) when P2 (for instance) is strictly increasing: indeed, we simply consider p2 (t) + 8 T+-t, apply (5.78) for each 8 > 0 and let b go to 0+. Then, introducing hn = p2(pn), pi(pn) may be written as P(hn) for some bounded continuous non-decreasing function P on [0, oo) (set

P(t) = P(sup,>0 p2(s)) if t > sup,>0 p2(s)) such that P(0) = 0. In other words, we may assume that pn converges to p weakly in LOO - *, that p2(t) - t and that pi is bounded. Furthermore, the same argument as the above truncation argument shows, by mollification, that we may assume without loss of generality that pi is smooth on [0, oo), say C1.

Finally, writing pi (t) = fo 1(t>,) p' (s) ds (pi (s) > 0, V s > 0), we see that we can furthermore simply look at pi (t) = 1(t>a) for an arbitrary positive a. Then we write

Pi P = P 1(p>a) = p - a 1(p>a) + a

1(p>a)

(p - a)+ 1(p?a) + a 1(p>a) < (p - a)+ + a 1(p>a) = (p - a)+ + a 1(p2:a) = p 1(p>a).

5.6

Other boundary value problems

We consider in this section two situations which are rather different from the three cases considered in the preceding sections. These situations are relevant for practical applications and realistic flows and they involve boundary conditions somewhat different from those studied and used until now. The first situation-which we shall call below the exterior case-we wish to investigate consists of an exterior problem with a stationary (but non-trivial)

40

Compactness results

flow at infinity. More precisely, let SZ = 01 where 0 is a bounded, open, smooth domain in RN (N > 2), we still consider (5.1)-(5.2) in SZ x (0, oo) but in order to simplify the presentation and keep the ideas clear we ignore the force terms (which can easily be reincoporated if necessary). In other words, we look for a solution (p, u), defined on SZ x [0, oo), of lop

+ div (pu) = 0

5i apu +div (on®u) - uEu - tVdiv u + V ap' = 0 at in SZ x (0, oo)

}

(5.79)

with p > 0 on Sl x [0, oo), p > 0, p + e > 0, a > 0, 7y > 1. In addition, we require (p, u) to satisfy the following boundary conditions:

u = 0 on aS2 x (0, oo) (p, u) (x, t) - (p°O, u°°) as jxj --+ +oo,

,

(5.80)

for all t > 0,

(5.81)

where p°° > 0, u°° E RN. Also, we wish to include the case when SZ = RN as a special case of the

preceding framework and we shall thus agree that when 0 = 0 or 1 = RN we simply drop the meaningless boundary condition (5.80). It turns out that in the analysis below the arguments (or at least some of them) become somewhat simpler but we shall not detail these easy points. The second situation we wish to study is a tube-like problem and we shall refer to it below as the tube case. We set 12 = R x w where w is a smooth, open,

bounded domain in RN-1 (N > 2) and we look for a solution (p, u) of (5.79) satisfying the following boundary conditions u = 0 on

(R x (9w) x (0, oo)

(p, u) -' (p+, u+) as x1 -- +oo,

(5.82)

83)

(5 (p, u) -- (p , u-) as x1 - -oo, for all x' E w, t > 0 0), ui, u1 E R and we write where p-, p+ > 0, u+ = (ui 10)? u- = (U- ,O), .

x = (x1, x') E R x w a generic point in S2.

Of course, in both cases we complement the equations and the boundary conditions with the initial conditions (5.6). We first want to explain how it is possible to obtain natural a priori bounds which correspond to energy-like identities. Then, we shall address the issue of the compactness of sequences of solutions. The main difficulty with a priori bounds is due to the fact that quantities like p, pJu12, plf cannot belong to L1 in view of the boundary conditions at infinity. This is why we need to introduce some tricks,

which we believe to be new and that are reminiscent of the analysis performed in P.-L. Lions [353] for related questions concerning Boltzmann's equation.

Other boundary value problems

41

We begin with the exterior case. We then introduce u(x) satisfying the following conditions

u - 0 on a ball BR. D SZ , u = u°O for lxj large , Du E Co (St).

(5.84)

The construction of such a function u is a trivial exercise that we leave to the reader. Next, we write the following (formal) identities 1 1

-

{Pry

ry

(Poo)ry-

'Y(P°°)1-1(P

-

P0o)}

'Y

+ div u ry

1(Pry

- (P,)"-,P)

= u V (Pry),

Pa(u

_

_

2

2u1

2

2ul

+a(u-u)'OPry =

therefore we find

u-ui2

a at

a + -Y - 1

2

P

+ div

u

{Pry -'Y(POO)ry-1P + ('Y - 1)(POO)ry}

-

a2 1(o 'Y

+pz

(5.85)

p) + IuP2 u'2

(POO)If

We may then integrate (5.85) in x on Q. Then (5.80) and (5.81) yield the following (formal) equality for all t > 0

at

f

u P

2u12 +

I

a ry

{pry -'Y(P0°)ryP + ('Y

1

- 1)(POO)ry} dx

+J

(5.86)

n

=

Jn

-(pu V)u + (div u) p' dx.

In particular, we deduce for all t > 0

st p

+

2

a -Y - 1 {pry

'Y(POO)ry-1p + ('Y t r

< fn PO

2

fo

a

n

t r +CoJ dsJ dx plu-ui+p+pry, + -y o

a

1

(t)

ds J dxIDuf2

+v

uo -

- 1)(p')') dx

{Po -'Y(POO)ry-1Po + ('Y

(5.87) 1)(PO°)ry} dx

x

for some v > 0 which depends only on p and t;, some constant Co > 0 and some compact set K C Sl which depend only on U.

Compactness results

42

Bounds then follow easily from (5.87) upon noticing that, by convexity of the function (t i- t-1), { pry - ry(p°O) ry-1 p + (ry -1) (p°O) ry } = j7 (p) is non-negative and that we only need to bound fK p + pry dx by C(K) fK j7 (p) dx. If this is the case,

we then deduce from (5.87) the following a priori bounds valid for all t > 0:

f P(t) Iu(t) -

ul2

n

t

dx

+ j7(P(t)) dx + fo

CleC,t f PoIuo n

r

Jn

dxIDuj2 (5.88)

- u12 + j7(Po) dx

for constants Cl > 1, C2 > 0 which are independent of t, p, u, po, mo = Pouo The above claim on j7 (p) is straightforward since we have

JK7H J pry dx +

-

2meas(K)(7-1)

fx

pdx

r pry dx + 1meas(K)-i7-1) f pd2 J K K 2

7

ry(P°O)7-1

K

pdx

-C

for some positive constant C which depends only on 'y, meas(K) and p°O.

In the tube case, one can argue in a similar fashion introducing p(x), u(x) satisfying the following requirements: p, u = (u1, 0) are functions of x1 only, P', ui E Co (R), (p, ul) _ (p+, ul) for x1 large enough, (p, ul) = (p-, ul) for

-xi large enough and p > min(p+, p) > 0 on R. Then, we perform similar

computations to the ones above writing now j7 (p) = p7 - .y p -1 p + (-y - 1) per' and

a

1 1

j7(p)

+ div

{u1(p_r1p)}

= -'IPU .

Vp(P)7-2

= -'yPu1P

(xl)(P)7-2.

In this way, the analogue of (5.85) is given by the following identity

a 1.7-Y(P) +div u at Plu 2u12 + -Y-

ary1(P7-,r-1P)+p

2

ry

- µ0u- (u - u) -l;Vdivu (u - u) -(Pu V)u - u VP7 -

.yPu1P'(xl)(P)7-2

(5.89)

from which we deduce _ 2 d Pfu 2ul + 'Y a 1j-t(p) dx + n liVu V(u - u) dt n dx = nf -pul ui + (ul)'Pry -

f

f

div (u - u) dx

ryPulP'(P)7-2

(5.90)

Other boundary value problems

43

and this identity leads, exactly as before, to the bound (5.88) where of course j7 is defined as above. We see that, in both cases, we obtain some a priori estimates on Du (or equivalently on D(u-u)) in L2(O,T; L2(0)) and on jy(p), plu-uI2 in L°O(0,T; L1(0)) for all T E (0, oo). If N > 3, this implies in particular that we obtain an estimate L2N/(N-2)(S2)) and on p in L°(0,T; L-1 (BRnO)) for all R, T E on u-u in L2(O,T; (0, oo) and thus on u in L2 (0, T; L2N/(N-2) (f n BR)) for all R, T E (0, oo). If N = 2, we claim that we can obtain an estimate on u in L2 (0, T; LP(BR n 1k))

for all R, T E (0, oo), 1 < p < oo. Indeed, denoting p = p°O, in the exterior case, we deduce from the bound on jry(p) in L°° (0, T; Ll(S2)) that there exists a constant C > 0 such that, for all t E [0, T], meas {x E S2 /p(x, t) < 1 p} _< C; notice that j.y is bounded away from 0 on [0, 2 Therefore, for R large enough (R > Ro): fBRnn dx p(t) > Z (Inf p) {meas(BR n 0) - C} , for all t E [0, T].

Let us observe that R0, C depend only on T and on the bound on j.,. Next, we observe that we have for all R E (0, oo)

J Rnn Iv12 dx < CJB

for all v E H'(BR n fl) (5.91)

Rnn lDvI2 + hIvI2 dx

and for all h E L7(BR n n) (recall that -y > 1) such that fBRnn h dx > v > 0. Furthermore, the constant C apppearing in (5.91) depends only on v, R and on bounds upon h in L1'(BR n 12).

The proof of (5.91) is easily done by contradiction: if fBRnn Ivnl2 dx = 1 and fBRnn I Dvn 12 + hn I vn 12 dx n 0 where fBRnn his dx > v > 0, h7L is bounded in L"(BR n 11), then vn converges in Lp(BR n 12) for all 1 < p < 00 to meas(BR n SZ)-1/2 while we can always assume that his converges weakly in Llf (BR n fl) to some h satisfying fBRnn h dx > v > 0. We then reach a contradiction since on the one hand, fBRnn his Ivnl2dx w+ 0 and on the other hand, fBRnn his I vnl2dx (fBRnn h dx) meas (BRn2)-1, and (5.91) is shown.

n

We may then conclude, using (5.91) with h = p(t), v = u(t) for all t E [0, T], R > Ro, that when N = 2, u is bounded in L2 (0, T; L2(1 n BR)) and thus in L2 (0,T; Lp(S2 n BR)) for all R, T E (0,00), 1 < p < 00. Let us finally observe that all the estimates on p follow from the LOO (0, T; L1 (S2 n BR)) bound on j.r (p) = p1' - -tjr-1 p + (-y - 1)j5 '-recall that in the exterior case p - p°° and that in both cases Mina p > 0. We shall need later on for the proof of compactness results an equivalent (and simpler to use) formulation of that bound. The formulation is shown in the next lemma.

Lemma 5.3 j.y(p) E L1(12) if and only if (p p)1(1 p- ,al >6) E

-

E L2(11) and (p -

Ll (11), for any 6 E (0, Mina p).

Proof. Obviously, on the set fl p - pI < 6} both p and p are bounded by 0 and bounded from above (Mina p < p < Maxa p, p > Minis p - 6 > 0, p < Maxis p + 6). Since -y > 1, we thus deduce that j.y(p) is equivalent to Ip the set {I p pl < S}.

-

-

pl2

on

Compactness results

44

Next, on the set {ip - Al > b}, we just have to observe that we have for some v E (0, 1), C E (1, oo)

i IP - All <_ jl(P) <_ CI P - All where v and C only depend on Minn A, Maxn p and 6. In fact, we have shown the following inequality

a Jn IP - PI2

1(1p-v1:56)

< J1(P) < 0 in IP

+

lp

- All 1(Ip-#I>6) dx

- PI2 1(Ia-vI<6) + l p -PIT 1(Iv-PI>6) dx

(5.92)

for some positive constants a E (0, 1), Q E (1, oo) independent of p. 0

In Appendix A, we consider and study the set L2 (1) of all h E L oC(SZ) such that h 1(Ihl<6) E L2(11), h 1(lhl>6) E L-1(11). It turns out that this set is a separable, reflexible Banach space (of Orlicz type) and various facts about it are shown in Appendix A; and, we have shown above that we have an a priori estimate on p - p in L°° (0,T; L2 (Q)) for all T E (0, oo). We may now turn to our main compactness results. First of all, we consider sequences of solutions (p", u") of (5.79) in the exterior case or in the tube case corresponding to some initial conditions (po , mo). We assume that J., (p0') and po I p -uI2 are bounded in L1(SZ) and that po converges weakly in Lz (St) (equivalently in V'(SZ)) to some po: in view of Appendix A, j1(po) E L'(11). We next assume that j,,(pn), pnlun uI2 are bounded in L°°(O,T; L1(SZ)), D(u" - u) is bounded in L2 (fl x (0, T) ), and un is bounded in L2 (O, T; H' (SZ fl BR)) for all R, T E (0, oo). Extracting subsequences if necessary, we may assume without loss of generality that p", u" converge weakly respectively in L''((SZ n BR) x (0, T))

-

(say), L2(0, T; H' (fl n BR)) to p, u for all R, T E (0, oo). We also extract subsequences for which pnun , pn un , pnun ® un converge weakly exactly as in section 5.2. The H1 regularity of un allows us to define in a meaningful way the boundary condition (5.80) or (5.82) and we thus request that they hold. Notice that the conditions at infinity (5.81) and (5.83) are implicitly contained in the facts that j7(pn) and pnlu - uI2 E L'(Q); indeed, we deduce in particular that p"- p' E L2(SZ)+L7(Sl) and thus meas{Ipn -pl > b} < oo for all b > 0. Therefore, we also have for all b > 0

-

meas{Iun ill > b} < meas{lpn - PI > !Min-. p'} + mews{Iun ul > b, P" > lMinn p} < meas{Ipn - pI > 2 Mini p'} + 1(Min p)62 f pn l un ul2 dx < 00

-

-

In addition to the above "natural" bounds and assumptions, we need some further bounds on p" which, of course, we shall need to establish in the course

Other boundary value problems

45

of proving existence results at least for some range of exponents 'y. We assume that for all R, T E (0, oo) and for all compact sets K C fl p" is bounded in L4 (K x (0, T)) n L°O (0, T; L''(SZ n BR))

(5.93)

for some q > 2, q > ry, r > N/2. Finally, we need some (technical) condition in the exterior case: p" is bounded in L2(0,T;

L2rr/(N-2))

if N > 5, for all T E (0, oo).

(5.94)

Notice that we already assumed that p' is bounded in L°° (0, T; L2 (Q)) ; therefore (5.94) automatically holds when N > 5 if 'y > 2N/(N + 2).

Theorem 5.3 (1) Part (1) of Theorem 5.1 also holds here. (2) If, in addition to the above assumptions, we assume that po converges in Ll (St n BR) (for all R E (0, oo)) to po then we have for all R, T E (0, oo)

p" w pin C ([0,T]; LP (11 n BR)) n L' (K x (0,T)) for all l < p < r, 1 < s < q,

p"u"

(5.95)

pu in Lp (0, T; L°t(1l n BR))

for all 1 < p < oo,

< cc <

2r

r+1

(5.96)

u" w u in LP (0,T; LQ(1 n BR n {p > 0}))

f o r all 1
(5.97)

where K is an arbitrary compact set included in 11.

Proof of Theorem 5.3. The proof of part (1) is exactly the same as the proof of part (1) of Theorem 5.1 and (5.96)-(5.97) follow from (5.95) exactly as we did in the proof of Theorem 5.1. Next, the proof of the crucial assertion (5.95) follows the strategy of the proof of Theorem 5.1. Since the proof of (5.29) is purely local (see section 5.4), it still holds here and in particular we have for small enough e > 0 (p")e {(µ + e) div u" - a(p")I} pe {(µ + e) div u" - a p'F}

in D'(Q x (0, oo))

at(pe) + div (upe) > (1 - 6)(divu)pe in D'(SZ x (0,oo)).

(5.98) (5.99)

In addition, since (pe)1/e) E L2 (K x (0,T)) for any compact set K C St and for all T E (0, oo), we may follow the proof of Theorem 5.1 to deduce

Compactness results

46

0 {()1/9} +div {(()1/e} > 0

in D'(SZ x (0,oo))

(5.100)

while we have (5.1), (pe)1/e _< p a.e. in SZ x (0, oo) and (pe)1/elt=o = plt=o in SZ.

Subtracting (5.99) from (5.1) and setting f = p - (pe)1/e, we obtain

8 +div (u f) < 0

in D'(SZ x (0, oo)),

f > 0 a.e., f It=o = 0 in Q. (5.101)

We then claim that f E LOO (0, T; Lz (Q)), flu - ul2 E Loo (01 T; Ll (Q)) for all

T E (0, oo). The second fact is obvious since 0 < f < p and pl u - ul2

E

LOO (0, T; Ll (Q)). In order to prove the first claim, we only have to show that (Pe)1/e - P E LO°(0, T; L2 (SZ)). But, (p" )e (p)e = (p + (p" - p))e (p)e is bounded in L°° (0, T; L2 /e (11)) in view of the results of Appendix A. There-

-

-

fore, (pe) - (p)e E L°°(0, T; L2/e(SZ)). Next, we write (pe)1/e _ p = [(p)e + 1/e (pe and using once more the results of Appendix A, (p)eJ we deduce that (pe)1/6 - p E LO°(0,T; L2-'(Q)) for all T E (0, oo). Let us

-

remark that, if N >-5, (5.94) implies in the same way that f E L2 (0, T; L2N/(N+2)A),

Next we wish to deduce from (5.100) that f - 0. Once this is proven, the proof of Theorem 5.1 applies and yields the L' (for s < q) convergence of p" in (5.95) and (5.96)-(5.97). Formally, the fact that f vanishes follows from (5.100) simply by integrating the inequality "by parts" over Q. In order to justify this integration by parts, we use a cut-off function. In fact, we really need two cut-off functions: one to truncate the integration at infinity and the second to truncate the integration near BSZ. We skip the second one since this is the same argument as in the case of Dirichlet boundary conditions (even if it is really needed on each BR n SZ for R fixed in (0, oo)). Therefore, we concentrate on the more delicate

truncation at infinity. In order to do so, we introduce cp E Co (Rk), 0 < cp < 1, Supp cp C B2, cp - 1 on B1 and we set cpR = cp(R) for R > 1. In the exterior case we take k = N while in the tube case we take k = 1. Multiplying (5.100) by cpR(x - ut), we deduce

x Rut jfccR(x-ut)dx=jf(u-u).VP( dt R

-

P

U j 0k co

x Rut

t 83 uk dx,

for t>0. }

j,k

(5.102)

Next, if T > 0 is fixed, we see that Vcp (x R t) # 0 implies that R < lx - utl < 2R and thus R - l l ul l L,,,,T < l xl < 2R + I l ul l Therefore, for R large enough, (5.100) reduces to L,,.T.

Other boundary value problems

dt

a Rut

fWR(x-ut)dx= J

n R

/'

Jn

47

dx

n

(5.103)

flu - ul1(R_c<JzI<2R+c) dx,

for t E (0,T),

where, here and below, C denotes various positive constants independent of R and t E (01T). Of course, in the tube case, {R - C < lxl < 2R + C} n 1 C {R - C < Ixi < 2R + C} n St.

In order to conclude that f - 0, we only have to show that T

1 1

R J0

dt

.

Jst

flu - ul 1(R-c
But we have shown above that

lu - uI E LOO (0, T; L2 (S2)) while f E L°° (0, T;

L2 (f)) and thus, in view of the results of Appendix A, f E L°°(0, T; L2(St)) C L°°(0,T; L2(S2) + L' (0)). Therefore, V7 E L°°(0,T; L4(11) + L2(f )) and thus we have for some eR going to 0 as R goes to +oo : 1

IT

Ro

dt

f f lu _'al n

< ER

1(R-c
+

[meas(S2 n {R - C < fix) < 2R + C}]1/4

Then, in the tube case, meas (St n {R - C < lxl < 2R + C}) < CR while, in the exterior case,

meas(1n{R-C< (xl <2R+C}) <meas(BR+c) - 5, we use the fact that f E L2 (0, T; L2(Q)+ L2N(N+2)(1))

while u - u E L2(0,T;

L2N/(N_2)(St)).

Therefore, flu - ul E

L1(0, T; LN/(N-1) (S2) + P(Q)) and we conclude as before.

At this stage, it only remains to show that, for instance, pn converges to p in C([0, T]; L'(11 n BR)) for all R, T E (0, oo). In order to do so, we just have to localize the corresponding argument in the proof of Theorem 5.1. Therefore, we choose for R,T E (0, oo) fixed , cp E Co (RN) such that cp - 1 on BR, 0 < cp on RN. Then, we observe that we have

a

(cp2 pn)

+ div [un (cp2pn)] = pnun Ocp2 div [tin (cPPn)]

at

(cpVp-)

,

!

(cp2 p) + div [u(cp2 p)] = pu Ocp2

= 2 (div un)cp pn + pnun ' OcP,

+ div [u(cp / )] = 2 (div u)cp j +

u VV.

From these equations, we deduce as in the proof of Theorem 5.1, that cp2 p E C([0, oo); L1(f )), cpf E C((0, oo); L2(S )) and that cp pn converges weakly in

Compactness results

48

L2 (St), uniformly in t E [0, TI. Therefore, in order to conclude, we just have to show that we have dx w*

in whenever t7L E [0, T], to

dx

t, and this is straightforward since we have, in view

of the above equations,

p(t) dx = J

to

2Pn dx +

n

J

dx

ds

dxpun n

Op2

I wp( dx. 0

s

STATIONARY PROBLEMS 6.1 Preliminaries In this chapter, we study the stationary problems associated with the compressible Navier-Stokes equations (mainly in the isentropic regime) (5.1)-(5.2). More

precisely, we shall study two (related) types of stationary problems namely: by 1) those obtained from (5.1), (5.2) through time-discretization (replace of (cp - cp) where cp is given) and 2) the real stationary problems associated with (5.1)-(5.2) (simply set to 0 all time derivatives in (5.1)-(5.2)). We begin in the sections below with time-discretized problems of the following form: let a > 0; we look for p > 0, u solutions of

a p + div(pu) = h

in 52

(6.1)

(6.2) a pu + div(pu 0 u) - pAu - eVdiv u + V(ap") = pf + g in 52 where y > 1, p > 0, p + > 0 and f, g, h are given function on 11, h > 0 in 52, h00.

In sections 6.2-6.7 below, we shall be concerned with our "usual" boundary conditions, namely i) Dirichlet boundary conditions in which case 52 is a bounded,

open, smooth set in RN (N > 2) and we require u to vanish on 852, ii) the whole space case in which Il = RN and (in some weak sense) p, u vanish at infinity, iii) the periodic case in which case (6.1)-(6.2) hold on RN and p, u are periodic in each x_ (1 < i < N) with a fixed period Tt > 0-we then agree that

--

= The above problem is naturally obtained from (5.1)-(5.2) through timefN1(0,T).

-

discretization replacing all time derivatives ap/at, apu/at respectively by of (pn pn-1), (pn,un_pn-l,un-1) and in all the other terms p, u respectively by pn, un.

of

In this way, we obtain (6.1), (6.2) with a = ot, h = of pn-1, f = fn, g = 1 of p n-1Un-1

In section 6.2 below, we state our main existence and regularity results while

we prove the key a priori bounds in section 6.3. These bounds are obviously crucial and involve two kinds of estimates: first of all, LP bounds on p where N P = N-2 (^f - 1) ) are obtained and next, if 'Y > 1 when N = 2 or ifY '> 3 when N = 3, further bounds and regularity estimates are shown in section

Stationary problems

50

6.3. Section 6.4 is devoted to compactness issues on sequences of solutions: of

course, the results are of a similar nature to those in the preceding chapter but the proofs are much simpler. Finally, in section 6.5 we prove the existence (and regularity) results stated in section 6.2, thus completing our study of timediscretized problems when p = apy, ry > 1, a > 0. Section 6.6 is devoted to the rather particular isothermal case, that is the case when ^y = 1 which we can study in two dimensions (N = 2). It will involve an adaptation of the concentrationcompactness theory (see P.-L. Lions [347], [348]). Next, in section 6.7, we study real stationary problems, namely

div(pu) = 0, div(pu 0 u) - µ0u - V div u + V (ap'') = pf + g in 1

(6.3)

with the same boundary conditions as before. Existence and regularity results will be obtained using the results and methods developed in the preceding sections and by letting a go to 0+ in (6.1)-(6.2) once h is conveniently chosen as we explain now. First of all, we wish to point out that solutions of (6.3) are non-unique (and have to be non-unique!). Indeed, we can always take p = 0 and solve for u (u = 0 if g = 0) at least in the case of Dirichlet boundary conditions. Furthermore, if f is a gradient and g - 0 (resp. f = 0 and g is a gradient), then we can take u = 0 and we simply have to solve aV pr' = p f

aVp'' = g If f = VO (resp. g = V O), then

p=

in 1l , p

> 0 in SZ

(6.4)

in 1 , p > 0 in Q.

(resp. 6.5)

(_lcyi

is a solution of (6.4) for any C E R (resp. p = (± + C)11_1 for any C E R such that a + C > 0). In other words, the system (6.3) is underdetermined. From a physical viewpoint, this is to be expected since we have to prescribe the total mass of the gas, in other words we may (and have to) prescribe

in

p dx = M, p>0 in

11

(6.6)

where M > 0. The case M = 0, i.e. p = 0 is easily settled since it leads to the elliptic equation: -µ0u - CV div u = g in &I-notice that in the periodic case, we need fn g dx = 0 and that if SZ = R2 we need g to decay fast enough and fR2

g dx = 0.

We shall precisely show in section 6.7 the analogues of the results shown in the preceding section for the system (6.3) and (6.6). The constant (6.6) will be enforced by letting a go to 0+ in (6.1) while choosing h such that « fn h dx = M: notice indeed that if p solves (6.1), then fn p dx = a fn h dx.

Existence and regularity results for time-discretized problems

51

In section 6.7, we also consider a different normalization of solutions (p, u) than (6.6), namely

in

p'dx

= K, p>Oinll

and we show that, if p is chosen large enough, then there exists for any ry > 0 and for any K > 0 at least a bounded solution (p, u). Furthermore, by a topological degree argument, it is possible to assert that these solutions belong to a continuum containing p - 0, u - 0 (for instance when g - 0). This will allow us to deduce that there exists a maximal M E ]0, +oo] such that there exists a bounded solution (pM, um) of (6.3) and (6.6) if 0 < M< M and, if (or FPM Lp (,,)) goes to +oo as M goes to M. We do not M < oo, pM know whether M can be finite in some situations. Let us finally mention that the reason why we can enforce a constraint like (6.7) comes from the fact that we can approximate the equation div(pu) = 0 by

div(pu) + a p' = ah in SZ

(p > 0 in 1k)

,

for a > 0,

and if fn h dx = K, than we have automatically fn p' dx = K. Section 6.8 is devoted to problems set in unbounded domains but where we no longer assume that p vanishes at infinity, like for instance exterior problems. In section 6.9, we investigate the regularity of solutions of the above stationary problems. We show there that if infess p > 0 and p E LO° then p and Du are Holder continuous assuming that all the data are smooth and we briefly discuss situations where these conditions are met. We also present a class of examples that show that the assumption on a bound from below on p is crucial: indeed, solutions (p, u) such that p E LO' (Q) may not belong to C x C' (or to Wo e x w2 ) 10C 1

even if the data are COO.

In section 6.10, we consider related problems such as the case of a general pressure law. Finally section 6.11 is devoted to general compressible models with an addi-

tional unknown scalar function which corresponds to the temperature. We show there how we can extend at least some of our results to "full compressible models".

6.2 Existence and regularity results for time-discretized problems We shall make the following assumptions on the data f, g, h : h > 0, h : 0 and h E L~ (S2), h E L1 (f)

if Q = RN

,

(6.8)

g E L2N/(N+2) (SZ) if N = 3 ;

gEL9(f)forsomep>1 if N=2andS1R2; if SZ =R 2 , g E L9 (11 2)

jgu dx

C

r

(f

for some p > 1 and 3C > 0, 1/2

hJuI2 + IDuI2 d2

for all u E Dl,' (R 2),

(6.9)

Stationary problems

52

where Di/2 (R2) is the space of functions u E Li , (1[22) such that Du E L2 (1122 )see Appendix B of part I for more details on that space. In particular, in view of the results shown in this appendix, the above conditions hold on 1R2 if IgI < Cvfh- + gi where gi > 0, g1+6 1x16 E L'(R2) for some 6 > 0. Next, we shall

assume that f satisfies

f EL4(SZ)with q>

(N + 2Ny - 2N

if N>4 or if N=3,y>2;

q> 3) if N=3, y<2; > max(21 y

y-1

if N = 21 St

1E22;

q=

22

y

(6.10)

if St = R2.

In the particular case when 11 = R2 , we will use also the additonal assumption hlxl6 E L' (R') for some b > 0 , f E L° (1[22) with q > max(21y)

(6.11)

Y

Our first result concerns the existence of a solution of (6.1)-(6.2). First of all, we have to define what we mean by solution. We look for a solution (p, u) of (6.1)-(6.2) in the sense of distributions, such that p > 0, p E L7 (a), p E Li (St) if SZ = RN; p, it are periodic in the periodic case; u E H' in the periodic case; L2N/(N-2)(RN) if St = RN with N > 3; u E H11.c(R2) in the Vu E L2(RN), u E case when S2 = R2; u E Ho (SZ) in the case of Dirichlet boundary conditions; and pIul2, hlul2 E L'(fl). Our main existence result is then the

Theorem 6.1 We assume (6.8)-(6.10) or (6.8)-(6.11) and y > N/2, if N > 2 , y > 5/3 if N = 3. Then, there exists a solution (p, u) of (6.1)-(6.2) satisfying pEL2-,

(f) if N=3, y>3orif N=2, 1154R2;

p E Lloc (1[22) if 1 = R2 ; pE

L2-f (122) + L4 (R2) for any q E (2y, +oo] , pl xI b E Ll (1[22)

(6.12)

if SZ = R2 and (6.11) holds ; (71) (St) if N=3, y<3orif N>4, pEL

and if N > 4, or if N > 3, 5/3<,y<3 D curl u, V

div u - +

py

E L° ( K )

( + L2N/(N+2) if c = RN )

if y>N-1 D curl u, V di v it

- µa

pl

E 1-C (

K)

(+ L2N/(N+2)if

i

= RN )

if N=3, 4 (6.13)

where

Existence and regularity results for time-discretized problems

q

N(-y -1)

N(7 - 1)

(N-2)+(N-1)(7-1)

(N-1)7-1'

53

and K = SZ except in the case of Dirichlet boundary conditions where K is an arbitrary smooth open set such that k C Q. In addition, if N _< 5, 7 > (N + 2)/(6 - N), there exists such a solution satisfying in addition the following local inequality

h122+ap122+ a71

-µ0I

u

PI22+ a71

7 7 2 2 + plDu12 - t div (udiv u) + e(div u)2 < pu f + U. g, (6.14)

while, if N = 2 or if N = 3, y > 3, any solution as above satisfies in fact the 0 local energy equality, that is the equality holds in (6.14). Remark 6.1 Let us make a few simple remarks about this result. First of all, ,H° (SZ) stands for the classical Hardy space-observe that q > N(N + 1) if 7 > N/2. Notice also that the above regularity result (6.13) is stated for N > 4 or for N > 3, 5/3 < 7 < 3 since the remaining cases are studied in a regularity result given below. Let us also observe that (6.13) implies in particular, using Sobolev's

embeddings (also valid in ?-l° for q > N/(N+1)), that diva-a/(µ+l;) p'r belongs to Lq' (SZ) + L2(11) where q* = Nq/(N - q) = (N/(N - 2)) ((7 - 1)/7): since Vu E L2(SZ), this consequence is in fact contained in (6.12). Next, we with to point out that, when N > 3 or N = 3, -f:5 3, then q:5 2N/(N + 2). Finally, let us observe that (6.14) implies easily that we have

I h 122 + pI 22 + 7 1(ap - hp"-1) + pIDuI2 + 6(divu)2dx Js2 7 <

(6.15)

fopf

with equality if N = 2 or if N = 3, 7 > 2. This integral (in) equality is indeed easily deduced from (6.1) except in the case when fl = R2 where we have to worry about the integration over R2. The only (real) difficulty being in the justification of the integration of div IU [p 12 + p r] } over R2, we solve it by using a cut-off function (as usual) cpR(.) = cp where R > 1, co - 1 on B1, Supp cp C B2i 0 < cp < 1 and we write, denoting < x >= (1 + IX12)112, P2

div

2

u p11 + 7

1 pry

co, dx

<- IIVcIIL'(R2) 12 JR2 p<113 1(I_I>_R) x>

dx+

47

7-1 f2

ICI

< x >p71(I=I>R) dx P-1

IuI

< 2IIVWIIL-(R2) {2IPILP IIR2

P

iT

1(IZI,R) dx

Stationary problems

54

P

4-(

+ 7=1 Ilp IILP (L2

IuI

P-+

.E.i 1(I=I_ R)dx

for some 1 < p < min(2, ry), and we conclude easily letting R go to +oo, using the results of Appendix B of part I, since we have for 6 E (0, pp 1 2)

-

P-1

uI

I

P-1

P


1(I=I,R) dx

fR2

IIIUIII151.2(A2)

1

P

Re

p-1 P

p-1

ICI (fR2



dx

C

Re

Remark 6.2 Let us explain the restrictions upon the exponent -y. First of all, the main restriction is y > N/2. In fact, we believe that the above result still holds for y > 3/2 when N = 3 but our existence proof (the bound (6.12) is still valid in this case) breaks down when 3/2 < -y < 5/3 and N = 3 for a rather technical reason: we then need, as in chapter 5, p to belong to L2 and this, in view of (6.12) requires -y > 5/3. We need to assume that -y > N/2 in order to make sure that p u 0 u is "better behaved" than p''1: indeed, if we assume that p E L7 and u E H', then obviously p7 E Ll and by Sobolev's embeddings p u ® u E La where = 7 + N 2 = 1 + 7 - N < 1 hence p > 1. This allows us to consider the term div(pu (9 u) as a perturbation of the essential terms namely [-pAu - CV div u + aVp7]. Even for 'y = N/2, we do not know if it is possible to obtain some Lio, bound on p for some p > N/2: if it were the case, our existence proof would apply.

Remark 6.3 The bound (6.12), shown in the next section, can be understood as follows. First of all, when N > 4 or N = 3, -y < 3, we shall show that p7 "behaves" like pIuI2 or equivalently that p7-1 "behaves" like Iu12. By Sobolev's

embeddings, we obtain (6.12). Next, if N = 2 or N = 3, y > 3, N/(N - 2)(^y 1) > and in these cases (6.2) will imply that p7 "behaves" like Du E L2. However, we shall see in the regularity results which follow that the bounds in LN/(N-2)(7-1) are still valid in those two cases. We do not know whether the exponent N/(N - 2)(-y - 1) is optimal.

Remark 6.4 The proof of Theorem 6.1 shows in fact that we can estimate all the norms of p, u, pIuI2, hI uL2, curl u, div u - a/(µ + C) p7 occurring in the definition of solutions or in (6.12)-(6.13) in terms of bounds on the data corresponding to (6.8)-(6.11).

Existence and regularity results for time-discretized problems

55

We now turn to regularity results. In these results, we assume that f E L° (0), g E LP (0), h E Lr (ft) where p, q, r do not necessarily correspond to the exponents given in (6.8)-(6.11) and we postulate the existence of a solution as in Theorem 6.1. Combining the two types of results, namely existence and regularity, is straightforward and we do not detail it here. We begin with the two-dimensional case.

Theorem 6.2 (N = 2). Let f E L° (SZ) with q > 2y/(2-y - 1), g E

LP (SZ)

with p > 1, h E Lr (1) with r > 'Y. We assume there exists a solution (p, u) of (6.1)-(6.2) satisfying (6.12). Then, we have

PE L"' (K); DUE L'(K) if s
p>2;s2=

-

1

(6.16)

2 if p < 2, s1 < oo if p = 2, si = +oo if

ifq<2, s22;s3=r

.

In addition, we have

D (divu

- µ+

pry

,

D curl u E Lt (K),

where t=min(tiit2,t3) and t1 =pifp<00,ti <00ifp=00;t2=

(6.17) gsy Q

ors
if q

In (6.16), (6.17), K = 11 in the periodic case and K is an arbitrary compact set included in SZ in the case of Dirichlet boundary conditions or when 11 = R2. 0 We now turn to the three-dimensional case.

Theorem 6.3 (N = 3). We assume that -y > 3. Let f E LQ (11) with q > ,61 5,y-31 let g E LP (St) with p > 5, let h E Lr (St) with r > y. We assume there exists a solution of (6.1)-(6.2) satisfying (6.12). Then, we have letting K = S except in the case of Dirichlet boundary conditions where K denotes an arbitrary compact set included in SZ

p E L'' (K) ; Du E L' (K) if s < oo,

Du E BMO, div u E L°° (K), curl u E L° (K) if s = +00

(6.18)

if p < 3, si < oo if p = 3, si = +oo if

where s = Min(si, S2, S3) and si = S2<

1+1

In addition, we have

D div u -

a

µ+

p'

E

L` (K), D curl u E Le (K)

(6.19)

where t = min(t1, t2, t3) and t1 = p if p < oo, ti < oo if p = +oo; t2 = qq+-'-'y if q

ors < oo, t2 < oo ifq and s = +oo; t3 =rifr < oo, t3 < oo if r = +oo.

Stationary problems

56

In addition, if N = 3, 'y= 3, andgELPnL615 withp> 5, hEL'nL'' with r > 'y, f E LQ n L2 with q > 2 then there exists a solution of (6.1)-(6.2) satisfying (6.12), (6.18) and (6.19).

Remark 6.5 When N = 3, ry = 3, the assumptions made upon f, g, h in the case of Dirichlet boundary conditions or in the periodic case reduce to g E LP with p > 5, h E L' with r > -y and f E L' if a < q:5 2, f E LQ if q > 2.

Remark 6.6 In the case when n =1R2, it is possible to show global regularity results on R2: we then need to assume a sufficient decay of p (which is in fact ensured by a similar decay of h). We shall not consider here such extensions or variants for the sake of simplicity.

Remark 6.7 Of course, (6.16)-(6.19) really mean that we obtain a priori bounds in terms of the data f, g, h, except, however, in the case N = 3, 'y = 3 which is a critical case.

Remark 6.8 In order to understand the above regularity results, it is worth taking an example: let us take the case when f and h are smooth, then Du E L' (St) where 1 = 1 - .1 and we obtain the same regularity as for the elliptic equation: - pAu - CV div u = g. However, we do not quite obtain the same regularity for the second derivatives of u, since instead of obtaining D2u E LP (11),

we prove that D curl u E L' (11) and D(div u - µ+E p') E LP (11); and we shall build examples in section 6.9 where D2u (and thus Dp") and where p is not continuous! The very particular example of gradient forces already shows that there are clearly some limits to the regularity we can expect.

Example 6.1 We take g - 0, h - ap and f = Dqi. Then, it is easy to check that necessarily u - 0 and thus aVpry = pO0. Obviously,

(7 -1) +

Pay

,-1

im+A

if-f>1' p=c°

if ry=1

is a solution for all A E R. Let us take the example of 'y = 3, N = 3: then, as we shall show in the next sections, it is possible to extend the existence theorem to the case when f E L312 , hence 0 is not in general in LP for p > 3 and thus p does not belong (in general) to LQ for q > 6: notice that Theorem 5.1 yields precisely the fact that p E L6!

It is worth looking at all solutions of aV p" = pVg. To this end, we first consider the case when -y = 1 and assume that V0 E L oc, p E Llo, for some a, /3 > 1, a + 1 < 1. Then we deduce that Vp E Llama and thus VLog(b + p) _

V¢ for any 6 > 0. Since P0¢ converges to V1(>0) in Li (if Q < oo), we deduce that, unless p - 0, Log (b + p) converges as 6 goes to 0+ to some 10

element of W, hence Logp E W10 and VLogp = a \70. Finally, we deduce that p=c on each connected component for some A E R2. Next, we consider

A priori estimates

57 p

ry-1

the case when ry > 1 and deduce as before that fo a ds converges as 6 goes to 0+ to some element of Wlo which has to be 1 p' -1. Therefore, we find: -1 _ ti-i Op

= a7

1(p>0)V q.

If ,8 > N, then this implies that p"-1 is Holder continuous. Therefore, {x/p(x) > 0} is an open set and we deduce that on each connected component of this open set

p= (7_1+)+J for some A E R.

Remark 6.9 As we shall see from the proofs made below in the next sections, it is possible to improve slightly the assumptions on f for the existence thorem (Theorem 6.1) combining in fact the proofs of Theorems 6.1 and 6.2. Indeed,

in theqcase when y > 3, we shall see that it is possible to assume only that

-

f EL

(Il) where q > 2-y/(2-y 1) when N = 2 (and 11 34 R2, the whole space case SZ = R2 requires as usual some modifications we do not wish to detail here),

q = 6-y/(5-y - 3) when N = 3 and q = (2N(-y - 1)) /((N + 2)7 - 3N + 2) when N > 4. o

Remark 6.10 All the regularity results, including assertion (6.13), are only local in the case of a bounded domain i and Dirichlet boundary conditions. It is an important open question to determine whether they can be obtained globally in Q. This issue is very sensitive on the type of boundary conditions we impose since, as we shall see in the next sections, it can be solved positively in the case when we impose the following natural but somewhat more complicated boundary

conditions: u n = curl u = 0 on 8c if N = 2, u . n = 0, curl u x n = 0 on all if N = 3. Other types of boundary conditions allowing for regularity results up to 0 the boundary will be discussed at the end of the next section.

6.3 A priori estimates We split the a priori estimates into various categories. Of course, all these estimates are purely formal at this stage and will have to be justified later on.

Step 1: Energy-like estimates. We multiply (6.2) by u taking into account (6.1) and we find h

122

-µAI 22 +pIDuI2 -e div(udivu)

pu f Using (6.1), we see that we have

u Op = div U __L p7 'Y-1

- --!-(h ry-1 - cep)

(6.20)

Stationary problems

58

This equality conbined with (6.16) and (6.1) yields 2

hl

2 +ap

IZ

2

2

aryl

+

2

(ap7 - hp7-1) +div u

pi

2 + 7 1p 'Y

'Y

- µ0I 2 + plDul2 - .div(u div u) + (div u)2 = pu f + u . g

.

(6.21)

Hence, integrating over fl, we deduce

l22 +

Jh--+a-----n I

+ e (div u)2 dx =

a2 (a p,r - hp1) 7

+ pI Dul2 (6.22)

Jn pu f + u g dx .

In particular, we obtain fn hIuI2 + pI u12 + (DuI2 + pr' dx < C 1 + fn Pl ul If I + Jul191 dz

(6.23)

where, here and below, C denotes various positive constants which depend only on bounds upon the data f, g, h. Let us begin with the case N > 3. In the case of Dirichlet boundary conditions or if St = RN, then we can bound fn Iullgldx by IIuIIL2N/(N_2) and thus, in view of Sobolev's inequality by II9II LZN/(N-2) IIDuI IL2 In the case of periodic boundary conditions, a similar bound holds with II91I L2N/(N-2) I IullH1 II9IIL2N/(N-2)

and we observe that since h E L", ry > N/2

,

h # 0, the quantity (fn hIuI2+

I Dul2dx)1/2 is an equivalent norm on H1. Therefore, we find

PIu12+e dx+IlullH1
plul ifldx

unless SZ=RN

In,

I

hIU12

+ PInl2 +

dx +

IIUI12

2N/(N-2) + IIDul12

f

< C 1 + n plul if I dx _

if SZ = RN

(6.24) Next, if y > 2, we bound (fn pl ul If I dx) by I IPI IL-Y IIuI I LsN/(N-2) I if I ILa if q = 22)--2N N-y (notice that 9 + ,1-y + 2N2 = 1). If 'y > 2, f = RN and f e LQ

(N-

2N2

+7=1 7-2N, we replace IIPIIL, by lIPlLL. where r + and 1 < r < dy. We then need to observe that integrating (6.1) over RN yields fRN p dx = « fR, h dx. These considerations show that, in the case when ry > 2, we obtain bounds on p in L-Y (f1L1 if Q = RN), u E H1 (or L2N/(N-2) with Du E L2 if S2 = RN) and in particular plul2, hIuI2 are bounded in V. with q >

N2

If N/2 < -y < 2 (i.e. N = 3 and 2 < ry < 2 or if N = 4, -y = 2), we bound fn pl ul l f I in the following manner: we detail it only in the case when

N=3, !<-y<2,

A priori estimates

f p Jul Ifldx <-

1

59

a/2

_Lga

IIuII

IIPI ILI

IIfIILq

where 0 < a < 1 is to be determined and ti (1 - 2) + 2 + 1 sa + q = 1. It turns out that, when all the calculations have been made, the optimal choice for a is 2 -'y which leads to q = 3/(7 - 1) and f PIul Ifl <- IIPIIL;

Observe also that if q > 3/(7 -1) we can take a > 2 - 7 and thus 1 - a/2 < 7

In this case, the bounds follow easily. If f E Lthen we can decompose f into fi + f2 where f2 E L"'(0) and f, is arbitrarily small in L3/('-1). In this case, we find

f

PI uI Ill dx <_

IIf2IIL°O

IIPIILI

IIV UIIL2 +IIPIIL, IIv uIIL2 hull

IIfl

llL3/(,-1)

hence

in

e + pIuI2 dx + IIuIIL6 < C Ii/ UIIL2 +

1 llfilIL3/(,-1)

IIPIIL,' II,'

ulIL2711UII Lg

and we deduce bounds on p in L'Y, pIuI2 in L' and u in L6 (Du in L2) taking IIf1IIL310-1) small enough. When N = 4, 7 = 2, q = 4, the analysis is exactly the same, the critical case being q = 4. In conclusion, we have shown, when N > 3, bounds on p in L' (L n L1 when SZ = RN), u in H1 (u in L2N/(N_2), Du in L2 when 12 = RN) and thus on hIuI2, pIuI2 in V. Notice that these bounds, in the case when N = 3, 2 < 7 < 2, q = 3/(7 - 1) or when N = 4, 7 = 2, q = 4, depend upon bounds on IIfE II LQ where

e E (0,1) is arbitrary and f = f 1 + f 2 with f'

E

L00

and I l f i I I LQ < e.

The analysis when N = 2 is exactly the same (and somewhat simpler since no critical case is to be treated) and leads to the analogous conclusion except when does not S1 =R2 where certain difficulties arise. Indeed, (fg2 hIuI2 + IDuI2 yield any LP bound on u but only a bound on u in D1'2 (R2) (see Appendix B of part I) and thus in H oC . The assumption made upon g in (6.9) circumvents that difficulty since, together with the exponent q = 27/(7 - 1) in (6.10), it allows us to deduce from (6.23) dx)1/2

I hIuI2 + PIuI2 + p7 + lDul2dx 1/2


(fR2

hIuI2 + IDuI2

dX

+ CIIfIIL2,/(i-1)II\ ullL2IIPIILly 1/2

< C fR hIuI2 + IDuI2 dX

Cl + 2 fR PIuI2 dX + IPI IL1'

Stationary problems

60

In this case, we obtain bounds on p in L' fl L'v(R2) ; hlul2, pIui2 in L'(R2) ; u in D1"2(R2), Du in L2(R2). Finally, if we assume that (6.11) holds instead of (5.10), we write

I

Jul IfI dx <

Ilfll,,1I6IILmIIP<x>b IILr

1 + 1 = 1 and < x >= (1 + where m is arbitrarily large, b > ,2 y1 + m Therefore, we have in view of Appendix B of part I, if r < y

in

plul if I dx <- C(m) I IUl l

where a = S y

Ix12)1'2.

(6.25)

I IPI Ie., I IP < x >a I ILI'

f,1,2(R2)

r , A = ' 11r . We shall use the inequality when -y > 2: indeed,

in this case, q > 'y/(-y - 1) thus we can take m arbitrary large and thus r arbitrarily close to q/(q - 1) < -y. Notice that b and a go to 0 and that 6 goes to 'y / ((-y - 1)q) as m goes to +oo. When 1 < ^y < 2, we modify the preceding argument as follows: we write

fiui If I dx < C II p < x >6 II"2 Ii,/ Jul IIL,rlIuiIDI'2IlfII

Lq

(6.26)

r-1 + 2 r + 2 = 1 and r E (1, y) . 2 This is possible since q > 2/(y - 1): indeed, as n goes to oo (and b goes to 0), r goes to 1 + 2/q < -y. Next, for a > 0 small enough, (6.11) implies that h < x >aE L1(R2) and multiplying (6.1)-at least formally-and integrating by parts, we deduce if a < 1 where m is arbitrarily large, b >

p< x >a dx < C fR2

n,

(i+a[J

R2

Q+

p I ul

< x >a-

I dZ


IIPIIL2)

(1+allvullL2).

Inserting this bound in (5.21) and (or) (5.22), we deduce s(1-9)

fR2 pu - f dx

< C (1 +a II,"P- ulIL2)

where s = r/2 if -y < 2, s = 1 if -y > 2

IIPIIL,, C'

IIUIIL2

+ IIuIID1,2)

and r is arbitrarily close to q 41 < y if 'y > 2 while r is arbitrarily close to 1 + 9 < 'Y if < 2. We ,

6 = ryry l

rT 11

A priori estimates

61

may then conclude choosing a small enough provided 7

(1 + s(1- e)) < 2 or

,y(1 - s) > es(2 - ry), an inequality which is obvious if ry > 2 (and thus s = 1) and which holds if 1 < y < 2 since in this case s = a and e = yry l ''r-1 hence -y(1- s) > es(2 - ry) is equivalent to (ry -1)(2 - r) > (r -1)(2 -'y) or to r < In this case (St = R2 and (6.11) holds), we have shown a priori estimates for p in L' n Lt(R2) ; pIuI2, hIu12 in L1(1R2) ; u in D1'2(]R2) and p < x >a in L'(R2) for some small a E (0, 1). In fact, in view of (6.11), we also obtain an estimate for p < x >a in L1(1R2) for the same 6 as in (6.11): indeed, we argue as above and multiply (6.1) by < x >a using the L2 bound on f it we just proved. 0

Step 2: Proof of (6.12)-(6.13) in the case when 11 = 1RN(N > 3) and in the periodic case. We begin with the proof of (6.12) and then turn to the proof of (6.13), each time in the case when 11 = IRN, N > 3. Then, we shall briefly explain how to modify the proofs in the periodic case.

The idea of the proof of (6.12) is extremely simple: we observe that (6.2) yields

a V pry = F1 + div F2 - div(p u ®u) (6.27) where F1 = g - apu + pf is bounded in Indeed, p is bounded in L2N/(N+2).

L' n L with ry > N/2 and thus in LN12 while it is bounded in L2N/(N+2) . Next, F. = pVut + div it ei ((el, ..., eN) denotes the canonical orthonormal basis of RN) and thus F2 is bounded in L' . Similarly, f E L° where q + ,11 < 1 - ZN = 2N and thus p f is bounded in L2N/(N+2). As is well known, F1 can be written as div F3 where F3 is bounded in L' : for instance, solve 0q5 = F1 in RN, V E L2 (RN), ' E L2N/(N+2) (RN) and take F3 = V O. Therefore, we may rewrite (6.26) as

a V p' = div G - div(p u (9 iu) where G is bounded in L2 (RN), or equivalently in terms of Riesz transforms

ap' =

Ri R3(P uiu3

- Gi3).

Next, if N = 3, ry > 3, we deduce from (6.27) IIP''lIL2(R3) <,C < Ci

+ IIp IuI2 IIL2 (R3))

1+

11U112

IIPIILg(R3)

or equivalently IIPIIL27(R3)

< C Cl + IIPIIL6(R3))

and we obtain the desired bound on p in

L2-'(1R3).

(6.28)

Stationary problems

62

If N = 3, 3/2 < 'y < 3 or if N = 4, we recall that we have an estimate for pry in L' (RN) and we write for some r E (1, 2), to be determined later, pry

IRsRj(Puiuj)I

pry 1(aP,,<2jRijr'ijj) +

<_

,

a 1-6

where

I

IIPIILrry

< Ilpryll8l


a

I IRi Gig

IIL2

IIuhIL2N/(N-2) 11P11 Lq


where

1

2

e= r

if r < NN- 2

2

1+

e+

1 +N-2 q

N

=

1 r

-

Choosing q = ry, i.e. q = N 2 (y 1), r N 2 2-1 -notice that r < N 2 and that r < 2 if N > 4 or if N = 3, y < 3-we obtain the desired bound on p in Lq(RN). This completes the proof of (6.12) in the case when St = RN, N _> 3. We now turn to the proof of (6.13). We modify the above proof by writing

-IL Du -

V div u + a V pry = F - (pu - V) u

,

(6.29)

where F = F1 + (ap - h)u is bounded in L2N/(N+2) (RN) for the same reasons as above. Hence, in particular (it is in fact equivalent) we deduce pry }

p

[Du-V div u]_ IP-(pu.V)ul-fR[(R-P)-R.((pu.V)u)lI (6.30)

At this point, (6.13) follows easily provided we show that (pu 0)u is bounded in L2N/(N+2) + H'lq if y < N - 1 and in L2N1(N+2) + LQ if y > N - 1 where q = N_2)+ N i} - 1) : indeed, the Riesz transforms are bounded in 1-l'' for r > N+1 and q > N+1 if and only if y > 2 ! This claim on (pu V )u is shown as follows: first of all, we decompose pu =

v + w where div v = 0, curl w = 0. Since pu E L' n L''(RN) where r = (RN) for 1 < r, < r. N try ry+1 > 1, we can choose (uniquely) v, w in Next, we observe that we have (pu 0)u = (v 0)u + (w 0)u. In addition, Lr,,

div w = div pu = h - a p is bounded in

L" (RN ), therefore Dw is bounded LNry1(N_ry)(RN) if y < N, LO(RN) for all

in L1'(RN) and thus w is bounded in Q < oo if y = N, L°° (RN) if y > N. In particular, w is bounded in LN(RN)

in all cases and thus (w 0)u is bounded in L2N/(N+2)(RN). Finally, for all 1 <_ i < N, (v V)uz is the product of v E L' (RN) by Vu2 E L2 (RN), and div v - curl(Dui) - 0. Applying the results of R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes [110], we deduce that (v V)u is bounded in fq(RN)

A priori estimates

= 1 + 2, since q > N+i . 1=R',q N>3.

where

63

This concludes the proof of (6.13) when

The proof in the periodic case is almost exactly the same except that in (6.23) we just have to replace a p'Y by a (plf fn p'r dx), a modification that is easily handled since we already know that ho p7 is bounded. 0

-

Step 3: Proof of (6.12)-(6.13) in the case of Dirichlet boundary conditions (N > 2). We begin with the proof of (6.12). We may argue as in step 2 above to deduce

a V p'f = div G - div (p u ® u)

(6.31)

where G is bounded in L2(11). Notice indeed that in the case when N = 2, we already know that p is bounded in L7 and u is bounded in L' for all q < oo in view of the estimates shown in step 1 above. When N = 2, or if N = 3, -y > 3, (6.12) follows easily since we have

e -in p1dx

Lz

< C IIVP'IIH-1 < C IIGIILZ + C IIp IUI2 IIL,

whereq=6ifN=3, q>2 if N=2.

< C (1 + IIPIILq )

When N > 4 or N = 3, 2 < ry < 3, we take r = N 2 1 (recall that 1 < r < N 2 and r < 2). We then conclude as before once we observe that the

following inequality holds for any 1 < r < oo:

f

II(PIILr(n) < C IIGIILr(n) +

dx InI11r

n

if V = divG.

(6.32)

Indeed, by standard density arguments of {div e /q5 E Co (11) }, we deduce sup

LcouI dx /IIII Lr/(r-1)(0) < 1 in tb dx = o

< C IIGIILr(tl)

: 1, then III - f-n 0 dxllLr/(r-1)(n) < 2'

Next, if 101 = 1 and 11011

where s = T 21-the inequality is obvious for r = oo, r = 2, r = 1 and follows by interpolation-and thus

II(PIILr()) =SUP f cp n

dx

/II

< C IIGI ILr(n) + sup

< C IIGIILr(n) +

IILr/(r-1)(n)

fn

<1

4'J O dy dx

/I I Y' I I

Lr/(r-1) (fl) < 1

fwdx.

The general case follows by an obvious scaling argument. In fact, the preceding proof really shows the following inequality

Stationary problems

64

114"M < 2' inf IIW - cIIL,.(n) +

f V dx IQI1/*

for all cp E L*(SI).

n

Then, the inequality (6.32) follows from the fact that infCER IkO - cIIL,,(n) is bounded by C IIGIILr(.) (for 1 < r < oo) as is well known. We now turn to the proof of (6.13). We follow the argument of step 2 above and we find that [-pEu - V div u + aVp ] is bounded in 7{4 if N = 3, 2 < -y <

2, orifN>4, ry4,'Y>N-1. The main difficulty that occurs is the fact that we cannot directly deduce from this fact the regularity of D ((p + C) div u - aplf) and of D(curl u) because of the boundary of Sl and of the absence of boundary conditions for p, div u or curl u. Therefore, we have to use a standard cut-off argument in order to obtain the local regularity of these quantities. Indeed, we simply observe that we have, as in step

2 above, denoting d = dist(x, an) : - dV t (p + )div u - a pry} + pd curl u = H E 1-1q and thus

-v d {(p + )div u - ap"} + pd curl(curlu) E 7-(q since div u, curl u E L2 (1) and pry E Lt (SZ) with r = N 2 771 > q. We agree that 7-1" = Lq if q > 1 in order to simplify notation. Since d {(p + t;) div u - apry } vanishes on an, we may now easily complete the proof of (6.13).

Step 4: Proof of (6.12) in the case when Il = R2. First of all, we begin with the proof of (6.12) following the argument given in step 2. In particular, we may still write (6.27) where F2 E L2 (R2) and F1 = g - a pu + p f. Notice that pu is bounded in L' fl L2-,/('+1) (11 2) and that p f is bounded in L' fl L2''I ('Y+1) (R2) if (6.10) holds while pf is bounded in L1 fl L''(R2) with r = 9 + 1-y, < 1 if (6.11) holds. Next, we deduce from (6.9) that on the one hand g E LP (R3) for some p > 1 and on the other hand there exists v E D1"2(R2) such that v E L2(R2) and g = -Av + hv. Hence, g = div G1 +G2 where G1 E L2 and G2 E L1 nL2 In conclusion, we have

a V p" = div G + F - div(p u ®u)

(6.33)

where G is bounded in L2(1R2) and F is bounded in L' fl L''(R2) for some r > 1 (which we can always take to be in (1,2)). Next we observe that p u ® u is bounded in L1(R2) fl L q,,, (R2) for all q < 'y since u is bounded in D1'2 (R2). Furthermore, if (6.11) holds, we have already seen that & 16 is bounded in L' (R2). Hence, for e E (0, 2('y - 1)) and for any measurable set A fR2

lp

uiu,12

1A dx L}E

f

< (R2 lA

P2+,

< x >3 2+1 dx

(fR2

' (<x>°

dx

A priori estimates

65

where < x >= (1 + IxI2)1'2 and _a is arbitrary. We may then choose a = e and deduce from the bound on u in D11'2 (R2) that ei, (1R2 p < x >8 dx 1A P''dx 1A I Pu=ui l2 dx < C

2(1-e)

(j2

fR2

where e + ie = enough we find

e))-1

2+2E and 6 = (2(1-

f

I P uuI2 dx < C

A

e . In conclusion, choosing a small

(j1Ap2dx)°

(6.34)

as e goes to 0+. In particular, a/7 < 1 for e small enough. where A goes to 2 Next, we deduce from (6.28) that ,y

p = 71 +7r2 +

-a1 RjR3(puiuj)

r

where 7r1 is bounded in L2(R2) and 7r2 is bounded in L"(R2) for p E (2) 2) if r < 2, p E (2, +oo] if r > 2. Hence, we have, choosing A = {x E R2 /p" >- 2 I1r2I },

pY lA < 2 Iir1I +

IR%R.i(Pu;.uj)I

a and decomposing RR3(puiuj) into RjR3(1Apu=u1) + RjRR(lAcpusu3)

l2p2y1Adx

1+

f

21A

IPu,u;I2dx +

(1+(f1AP2-i) dx

J£2

i(py«Ii2I) p

2IuI4dx

airy (pry <2I' 2 t)

ply' dx

in view of (6.34), for any -y' E (-y, -y + 1) and where e' is defined with respect to 71 as A was with respect to y. Since fR2 dx < C fR2 I7r2I2y'/ydx,

we choose 'y' in (1, s) where s = r r 2 if r < 2, s = +oo if r > 2, and we conclude that fR2 p2' l A dx is bounded. On the other hand, fR2 ply 1(py «I -2 I) dx is clearly bounded. Therefore, in conclusion, we have shown that p is bounded in L'y (R2) + L9 (R2) for any q > 2-y. We conclude the proofs of the a priori bounds corresponding to Theorem 6.1 ,, by explaining, still in the case when SZ =R2, how to recover the L1 (R2) bound

on p when we no longer assume that (6.11) holds. We start from (6.33) and deduce for any cp E Co (R2), cp > 0

a V (cpp") = div(c'G) + cpF - div(cp pu (9 u) + [ap' - G] V w + pu(u V p).

Stationary problems

66

Next, we observe that (a pry - G) V + pu(u VV) E L' (R2) and thus we deduce V {acpp1' + R%Rj(co Gij - cppusuj)}

= cpF + [ap' - G] V + pu(u V W) E L1(R2). Therefore, we have 11W1'-,p11"2-y(R2)

=

< C+ a I IRRj (caGij

IIcPP'IIL2(R2)

- (apuiuj)II

L2(R2)

< C Cl + II(PPuiujIIL2(R2))

< C (1 +

II(P1/"PIIL2-,

and we conclude since u is bounded in D1'' (R2) and thus in L (R2) for all

q
-v [(µ + e)div u - ap' ] + 2 curl(curl u) = p f + g - p(u 0)u + hu.

(6.35)

If N = 2, curl(curl u) really means [- V 1(curl u)] . This allows us to deduce a certain regularity of (µ + C) div u - ap' and curl u separately from bounds on p, u (and Du). The second observation consists in writing (at least formally) for

allm>1

div(up') + (m - 1) div u pm + camp' = mh pii-1, hence we have

div(upm) + (m _ 1)

a

pm+,r

= mhpm-1 + (m +

+ ampm

)

pm

(ap'

-

(6.36)

(µ + )div u]

Integrating over 11, we deduce

r

I pm+r dx

C

(.L pm+ 1

dxl C

f I app - (µ + C)div uI sz

dx

A priori estimates

+C or

IPIIL-+ry

I

pm+ry dx

67

dx

hry

Jn

n

<- C [+ I Ihl

7ii(ry+1)]

(6.37)

(Notice that this constant C is independent of m > 1, provided (m - 1) is bounded from below.)

The idea of the proof is then to bootstrap the bounds on p, u, Du in the following way: given L9 bounds on pry and Du, we deduce from (6.35) L9' bounds

on (µ + C)div u - apry and curl u (where q' depends on q, N,'y) . These bounds, in view of (6.36) and (6.37), induce Lq' bounds on pry which, in turn, yield L4' bounds on div u, pry and curl u or equivalently on pry and Du. Before we implement the above strategy of proof, we wish to justify the manipulations that lead to (6.36) and (6.37). We thus assume that u E H1, p E L2 , h E Lgry(1+7) and (,u + )div u - apry E Lq for some q > 2 and we wish to show that p E L°ry and that (6.37) (and (6.36)) holds with m = (q -1)^y. In order to do so, we first prove that p E L9ry'°° by considering for all k > 0 the function (p - k)+. Since u E H1, p E L2, we deduce from the regularization argument used many times in the preceding chapter that we have

div {u(p - k)+} + div u k 1(p>k) + a p 1(p>k)

h 1(p>k) .

Hence, integrating over the period 11, we deduce a

I pryk 1(p>k)dx = fn

a+ prl

µ

+ fn divuk

- diva

k 1(p>k)dx

1(p>k)dx

< C k meas( > k

(9-1)ia

+

J

h1

dx

or

ki+ry meas(p > k) < C {ic meas(p >

k)(a-1)i4

+ meas(p > k)}

and finally

meas (p > k) < C In particular, p E LP for all p < (q

k-9-'

for all k>0.

Next, let m' E (1, m) (recall that m =

- 1)1). We then write for all R E (0, oo)

div

{upARm'} + (m' -1) divu (p A R)'' + am' (p A R)m'-1 1(pR) m'(P A

R)m'-1

Stationary problems

68

or equivalently div {u(p A R)m } + (m'

+ (m' - 1)

= m'h(P A

p7(p A R)m'-11(P
- 1) [div u - µ+t p7] (p A R) R)m'

+ am'(P A

+ [divu

+ µ+Cp7R 1(P>R)(P' A R)m

-

R)m

' -1

MP

1(P
+cp7] R 1(P>R)m'(P A

R)'n''-1

' -1.

We may now let R go to +oo since pm E L' , I div u - IA+C p7 j pm' E L1 , p7+m' E L1 , hp"`' E L', and we recover (6.36) with m replaced by any m' < m. From

(6.36), we deduce a bound on p E L7+" which is independent of m' E (1, m) exactly as we obtained (6.37). This implies that p E

L7+m, and thus (6.36)-(6.37)

hold (let m' go to m_). We may now easily prove Theorem 6.2, detailing the bootstrap argument sketched above. In order to simply the presentation and keep ideas clear, we begin with the case when s = oo (i.e. q, p > 2 , r = +oo). Let qo = 2: we know that p E L9°", Du E L90 and thus u E LP for all p < oo. In particular, Then, (6.35) implies that curlu, (p+ E L' for all r < pf 1'f' l;)div u - ap7 E W for all r < 7q+1 and thus to L' for all q < q1 (by Sobolev's Hence, by the above argument, embeddings) where e = q° + qo7 2 = p E L97 for all q < q1. Therefore, div u, p7 and curl u belong to Lq and thus Du belongs to Lq for all q < q1 = qoy. In particular, u E C and we may reiterate the preceding argument to find that 7q+i.

qu7.

curl u, (p + l;) div u - ap7 E

W1.9-fnti+1)

Du, p7 E L'

for all q < q1 i r < rl, where = a1 + qi7 - 2 if q1'(/ (-y + 1) < 2, r1 = +oo if Q1-y/ (ry + 1) > 2. Reiterating the above argument, we find in a finite number

of steps that curl u, (p + C) div u - a p7 E W 1'q for some q > 2 and thus (p + C)divu - ap7 E L. Then, letting m go to +oo in (6.37), we deduce that p E L°O and thus div u, curl u E L°°, hence Du E BMO, and (6.23) is shown in the case when s = oo. (6.24) then follows in view of (6.35) since pf E Lq, g E L", p(u V)u E L' for all r < oo and hu E L°°. The case when s < oo is treated by the same bootstrap argument that we do not wish to repeat. However, let us explain that the bootstrap argument will make the exponents grow indefinitely until they reach or exceed values due to p f , grh. For instance, if h E L'' (St), then the above argument shows that p E LQ7 where r = and thus Du, p7 E L'3 where s3 = Tryl r ; similarly, D(div u - µ+C p7 ), D curl u E L 13 where s = 3 + 337 = 33 771 = r if r < 00 (and t3 < oo if r = +oo). Similarly, if g E LP, D(div u - +t p7), D curl u E L`1 where t1 = p if p < oo, t1 < oo if p = +00. And Du, p7 E L'1 where s1 <

A priori estimates

69

if p < 2, si < oc if p = 2, si = +oo if p > 2. Finally, if f E L9, we find that L"2 and 82 = t2 - , = D(div u - µ+f p% D(curl u) E Lt' while p", Du E ryry l 2 z to ry and t2 < 2 is equivalent , t2 = q 1 + s if t2 < 2, i.e. S2 =

q<2wryhiles2
When q < 2, however, one needs a further argument in order to obtain the ryry i sharp values of the exponents s2 = , t2 = Q = 22y q . Indeed, choosing g and h E L°° to simplify, the above bootstrap argument only yields the fact that p E L , Du E L' , D (div u p'7) E L', D curl u E L for all 2-< s < s2i 1 <_ t < t2. In addition, we find or some constants C, independent of s E [2,S2)

a

D

y

+IIDcurluIILt
IIPIIL.

< C 1+ 11divu -

a

+t; P

< C(1+IIPIIL.1)

L, for all 2 < s < S2, where s = i - Z . In particular, I IpI (L,1 , II D(divu- +C P11 V and I I D curl u I I Lt

are bounded uniformly in s E (2, s2) . We then conclude letting

s go to S2--

Step 6: Proof of Theorem 6.2 in the case of Dirichlet conditions or when S = R2. In the two remaining cases, namely the case of Dirichlet boundary

conditions or the case when) = R2, the preceding proof has to be modified (and localized to be more specific) because we cannot extract from (6.35) global information on curl u or (div u - µ+ pry). In the case when f = R2, this is due to the fact that we have no global control upon u in LP and that p E L1 . In the case of Dirichlet boundary conditions, the Hodge-De Rham type decomposition (6.35) does not uniquely determine [µ curl u] and [(µ + ) div u - ap' ] since we have no information (or boundary condition) on any of those quantities on BSZ. We thus wish to localize the proof developed in step 5 by a simple cut-off argument. Let K be an arbitrary compact set contained in) and let cp E C0(St)

be such that cp - (0) 1 on K , 0 < cp < 1. Assuming that we know that p E

L (Q), Du E L

-v {cp [(µ + e)div u

where q > 2, we deduce from (6.35)

- ap'% - µp1 {cp curl u}

= cP [Pf + g - hu - P(u . 0)u] - [(µ + t;) div u - ap'y ] pcp - µ curl u01 co. (6.38)

First of all, we see that the term cp[p f + g hu] can be written (cpp) f + cog h(Vu) and thus is naturally localized. Next, the term cop(uV)u can be written (gyp) [(mu) ti0] u choosing co = V53 where E Co (fl). Therefore, the only difference with respect to (6.30) mainly lies in the terms [(µ + t)div u - app] Dcp and curl u V l cp which obviously belong to L4 (1) and thus are more regular than cpp(u V)u which belongs to L' (Q) where r = q + 97 if q > 2 and r < if

q=2.

Stationary problems

70

The proof made in the periodic case will thus clearly extend provided we can

localize (6.36) and (6.37), and also the regularity argument made in the case f E L9 with q < 2 (and g, h are bounded for instance). First, we write (6.36) with i > 1 and we deduce + (m - 1) + P+ arrtcp i` pm V

div (r

hp

= MW

+ p'"

P.

+

(1+!?!) co

V

[ap - (A + )div u]

u V V.

Hence, we deduce at least formally rn

IIcP''II m+y <- C IlVpryll

Li

hll LV

Li M

+ CIIcp'9IIL

IIc[apry-(p+C)divu]llLmTy

'r

EL

+

L

CIIccp1Ii'm+Y

IIu - VcIl

'Y

L

and thus I IcpryI IL

n


- (µ + )div u]

I IL,

(6.39)

+ II u IVcI IIL'

+ IIhIIL

The justification of this inequality is obtained in exactly the same way as in the periodic (global) case. The additional term Jul IDcI does not perturb the proof

made in the periodic case since it plays the same role, as far as regularity is concerned, as hu (for instance) and is more regular than hu; in fact hlulcp and Jul IVcI have exactly the same LP regularity when h E L°°(SZ). It only remains to adapt (and localize) the argument made at the end of the proof in the periodic case taking f E LQ (SZ) with q < 2, and g, h E L' (11) for instance. Then, we obtain that p E L" ' (Q), Du E (SZ), D(div u 14+c pry) E (Q), D(curl U) E (S2) for all s < s2 = 2q ryry- 1 and where t = +2 . Then,

-

the above arguments show that we have for some constants C independent of s E [2, S2) 11D

a

(div u

+ C

pry

C

L+ II D [V curl ul II Lt - C (1 + I I cPf I I LL ) Lt

11+

(1+cc(divu_

and we conclude easily since 0 < cp < 1. 0


PryL

C (1 +IIcpry9IL.Y)

A priori estimates

71

Step 7: Proof of Theorem 6.3 in the case when -y > 3. The proof is very similar to the proof of Theorem 6.2, using (6.35), (6.36) and (6.37) which are still valid in this case. The proof in the case when SZ = R3 is now entirely similar to the one in the periodic case since we now have some global control on u (u E L6(R3)). The modifications which are needed in the case of Dirichlet boundary conditions are exactly the same as the ones in the two-dimensional case, so we skip them.

Finally, in the periodic case when, for instance, f, g, h E L'(11), we follow the proof made in step 5 and start from the following informations: p E L°'Y , Du E L° where q > qo = 2. In particular, u E L3q/(3-q) if q < 3 , u E Lr for all

r < oo ifq = 3 , u E Lao ifq > 3; and thus, pf + g - p(u V)u + hu E L3 where s - q + q7 + r , i.e. s = q - 3 +-L if q < 3 , s > q + 47 if q + - if q > 3. This implies by the proof made in step 5 that p E L°1'', Du E L°' where q = s - 3 if s < 3, q' < oo if s > 3. Therefore, we find, ifs <3, q, = q-3-} 1 ifq<3, Q, > 1+973orq'<31if q=3,

q=3, q

while q' 3. For instance,

> q if -y > 3. More generally, one checks easily that ifq = qo = 2 , q' = 2 the map (q '- q') is increasing and that, after a finite number of iterations of

the bootstrap argument, we obtain a L°° bound on p while Du is bounded in BMO , D(div u - + p') E Lr , D curl u E Lr for all r < oo. The rest of the proof is then an easy adaptation of the proof of Theorem 6.2.

Step 8: The case ry = 3, N = 3. For the same reasons as in step 7, we only consider the periodic case when f, g, h E LOO. The other cases are indeed straightforward adaptations of this case along the lines of the arguments developed in steps 5 and 6. Next, the existence proof will be given in section 6.5 below and will show in particular that approximated solutions and the solution being constructed belong to a compact set of H1 and therefore of L6. This will be shown by a convenient use of energy-like identities. This allows us (with some uniformity which will be discussed in section 6.5 below) to decompose u for all e E (0, 1):

U = U1 + U2 ,

IIu1IIL6

< E , u2 E

L'

.

(6.40)

We are then going to obtain (formal) a priori estimates on p, u which a priori depend on bounds on IIu2IIL,,. (itself obviously depending upon e). These

estimates are obtained assuming, for instance, that p E LOO, Du E L° for all q < oo. Next, we remark that if -y = 3, the exponents obtained in step 7 reduce , 1 = 2 - ? + 1 = 7-22 and in particular q' > q if to 31 = 2q 31 + 13q = 7-9 3q q q 3 3q 3q q > 2 while q' = q if q = 2. This means that we only need to be able to start the bootstrap argument (and obtain that p E L34' , Du E L9' for some q' > 2) and then the proof is the same as in the case -f > 3-see step 7 above. We thus observe that p f + g - p(u V)u - hu = -p(ul - V)u + cp where cp E L3/s. Notice that II(PIIL3/3 C (1 + IIu2IIL.) = CE. Therefore, we find using

-

(6.37) and (6.35)

Stationary problems

72

IIDulILq

{1+Idivu_

+ Ilp3lU < C

p7

µa

-+- IlcurlullLq Lq


3

,

for all q E (2, 3).

L3+9

< 2 if q < 3. Therefore, we find for some positive Observe indeed that constant Co independent of e IIDulILq + IIpl13L3q < CE + CEIIPII L 3q

IIDuIILr

<

CE +

COEllPlIL3q IIDulILq

= s + 3Q and a + 12 e = r , and we find 0 = 3 .

where 1 + s + 3y = 11-2' i.e. In other words, we have

T 2/3

11P113

< CE + CoEI IPIIL3q IIDuil

IIDulILq +

This inequality obviously yields bounds on IIpIIL3q and IIDullLq if we choose e

small enough (e < ca

2-2/31.

Remark 6.11 We may now come back to the claim made in Remark 6.9 above. When 7 > 3, it is possible to use the proofs above to relax slightly the assump-

tions made upon f. We only explain this point for N = 3 and for N > 4, the two-dimensional case being treated in a similar way. Also, in order to simplify the presentation, we only consider the case of Dirichlet boundary conditions, the other cases being similar. We thus begin with the case N = 3, -y > 3 and then briefly explain how to modify the argument when N = 3, y = 3. Finally, we consider the case N = 4. Then, if -y > 3, N = 3, we assume that f E Ls7/(s7-3) and proceed to obtain bounds on u in Ho and on p in L2' . Of course, bounds on p in L' are automatic and (6.23) shows that we have 5/3

I1ul l2

HI


1+

Pe/s

Jn

If

Ie/s

(6.41)

d2

The argument given in step 3 above yields the following inequalities Ilp'hlL2

C

{IIphhIL1 + 1 + IIPfIILe/s + IIP Iu12

11L2

}

< C 111pile2ry + 1 + IIPfIILe,S + IlpfllLS,S IIPIlL2, }

using the Holder and Sobolev inequalities and (6.41), where e + 1 le = and thus 6 < 1. Notice indeed that 4-y' < 6 if and only if -y > 3. The above inequality ti then implies

A priori estimates IIPfiILe/'-1)

IIPIIL2,y < C I 1 +

I

73

< C {1 +

IIPIIL2y(-y-1)

}

since 5y; 3 + 2y = 6. Our claim is then shown since

y if (and only if)

y > 3. In the case when N = 3, y = 3, we assume that f E L 312 and decompose f = fi + f2 where 11f11L3 < e, f2 E Loo for any e > 0. We are going to obtain for e determined below (small bounds which depend upon bounds on I I f2 I I enough). Indeed, we obtain exactly as above

,

IIPf1IIL3

IIPIILe < c f 1 + II Pf IIL6,5 } < C {1 +

< C (1 + E3 IIPI13

e,S

+ IIf2IIL IIPIIL615 }

+ IIf211L00 IIPIIL15 IIPIILSS)

and we conclude choosing e small enough so that E3C < 1.

If N > 4 and -y > 3, we obtain a priori bounds on u in Ho and on p

in LN(y-1)/(N-2) assuming that f belongs to L9 where q = (2N(y - 1))

((N + 2)y - 3N + 2). These bounds are still valid if y = 3 but then depend on L°° bounds on f2 where f = fl + f2 , f2 E L°° , I1f11ILN12 < e (notice that q = when y = 3) : the proof in this case uses the same modification as the one 2 above when N = 3. Arguing as in the case when N = 3, we find the explained following string of inequalities H2

Hull

< C 1+ I I Pf II

IIPIIL(y_1) < C (1+ IIPf II L

<

C

07

+ 11 p Iu1211 L

+ IIuII2

1 + II Pf II L

IIPIIL

(,,,_1)

L

and thus, IIPII?

Lam(,-1)

< C 1 + IIPfhi

+ IIPfII' L7

IIPII

L-("-1)

We then deduce IIPII

x L

('r - I)

< c 1 + IIPfII'"

-

L

< - C 1 +IIPII'"(ry_1) IIfIIL

('-1)

L ALI

0

Stationary problems

74

We now conclude this section by explaining the statements of Theorems 6.1-

6.3 and of the proofs made above in the case when we replace the Dirichlet boundary conditions by other relevant boundary conditions. This matter was briefly mentioned in Remark 6.10 and we wish to present these extensions in some detail at this stage. The domain 11 is still a smooth, simply connected, bounded open set in RN (N > 2) and we denote by n the unit outward normal to M. In all the cases considered below, we use the following usual boundary condition on the normal velocity (which simply means that the fluid cannot penetrate the wall!)

un=0

(6.42)

on 1911.

The boundary condition is to be complemented by some boundary condition on the tangential part of u. We shall in fact consider various possibilities listed below and we briefly comment on them. In order to simplify the presentation, we only consider the cases N = 2 and N = 3. First of all, one may use for numerical purposes the following mathematical boundary condition

curl u = 0 on 811 if N = 2; curl u x n = 0 on 811 if N = 3.

(6.43)

Another boundary condition which corresponds to assuming that there is no tangential friction is given by: (d n) x n = 0 on on, where d = (Du + Dut) z (see P.G. is the deformation tensor. More generally, flow over a porous medium Saffman [458]) or over a perforated wall (see E. Arquis and P. Laplace [20], E. Arquis, P. Laplace and R. Basquet [21]) and general wall laws ("deduced" for instance from Boltzmann models) lead to boundary conditions of the following form

Au) x n = 0 on a fl,

(6.44)

where A is a non-negative matrix (not necessarily symmetric). In fact, for turbulence models, Au may even be replaced by non-linear quantities and a boundary condition closely related to (6.44) like i9U

Au 8n +

xn=0

on

811

(6.45)

is used sometimes (see 0. Pironneau [444] for example). We claim that, for each of these boundary conditions ((6.42)-(6.43) or (6.42)(6.44), or (6.42)-(6.45)), Theorems 6.1-6.3 (and all the preceding arguments, proofs and remarks) are stil valid globally in 11 (i.e. with K = 11) provided we assume that > - N (resp. N-2 µ) in the case when we impose (6.42)(6.45) (resp. (6.44)). We only have to detail two points, namely i) the estimates following from the energy identity (see step (1) and ii) how to obtain global (in 11) regularity information on D curl u and D(div u - µ+E p 'Y). We begin with the energy estimates. In the case of (6.42)-(6.43), we obtain in place of (6.22)

A priori estimates h- 22

+

,plu 22

+

(ae -

7 + plcurl u12 dx =

75

(u (6.46)

f pu f + u g dx. n

Next we claim that IdivulL, + IcurlulL2 + IuIL2 is an equivalent norm to the H1 norm on the space of functions u E H1(RN)N such that u n = 0 on 1911. This is rather well known and easy to check when N = 2. When N = 3, by localization

(and partition of unity), one just has to check this fact in a neighbourhood of a given point on 81 and thus we may assume that v is supported in Bi fl {x3 > o}, I ldiv v all L, < e IIDvl I L, , Ilcurl v - b1 1L2 < e II DvII L, where a, b E L' are fixed, v3 + civi + c2v2 = 0 at x3 = 0 where c1, c2 are Lipschitz , IIc2IILIQ < e, and e > 0 can be made as small as we wish (by and

-

IIci11L,,,

localization). Since A(v3 + civi + c2v2) = Ova + T = [V (div v) - curl curl v] 3 + < C(1 + T where IITIIH_1 < C(11vlIL2 + e llvvllL, ), we deduce that C IIVVIIL,) and thus Ilaivi+82v21IL2 , 11a2v1+a1v2II L, , I1a3VIIIL2 , II83v2IIL, <_ C(1+e IIVvIILZ ). Therefore, IIDvIIL3 < C(1+e 1lVvIIL2 ) and our claim is shown taking a small enough. Then, considering the following minimization problem IIv3IIH1

I = Min {fn (div u)2 + Icurl u12 dx /u E Hi (SZ)N

, u n. = 0

on aSt,

L Iul2dx =1 we deduce that I > 0. Indeed, if I = 0, the infimum is achieved at some u E Hi (St) such that u # 0 , u n = 0 on 8SZ and divu = curl u = 0 in 0. In particular, u = V where 0 E H2 (St) and AO = 0 in St , 94 = 0 on of Hence, 0 is constant and thus we reach a contradiction. In conclusion, we have shown the following inequality

in

(p + )(div u)2 + ,(curl u12dx > v

I Dull + Iu12dx

(6.47)

for some v > 0, independent of u E HI (11) satisfying (6.42). The combination of (6.46) and (6.47) allows us to repeat the arguments developed in step 1. In the case of (6.42)-(6.45), the argument is somewhat simpler since (6.46) is now replaced by

r h 122

Jn

+ ap l u2 + /'

+J

an

7

ry 1(apy

p(Au, u)dS =

- hp'r-1) + plDul2 + e(div u)2 dx Jn

(6.48)

pu f + u g dx.

Recalling that, by assumption, (Au, u) > 0, p > 0, we may then conclude easily at least when > - N . Indeed, in that case, we have for some 6 E (0, N )

(div u)2 > -6p(div u)2 > -N6plDul2 a.e. in St.

Stationary problems

76

If

= - N, we need to do a bit more work. Following the proof made below

(for the boundary condition (6.44)), we can show the following inequality for all u E H1(SZ)N such that u n = 0 on 8SZ

fn

1L(DuI2

- *(divu)2dx >

(6.49)

v fn (Du(2 + (u(2dx

for some v > 0 independent of u. Indeed, the argument given below shows that we only need to prove that if u E H' (fl) , u n = 0 on 8S1 and (Du (2 - (div U)2 = 0 N facts imply or equivalently Du N div u I then u 0. But, obviously, these

that u = ax+b for some a E R, b E RN and thus 0 =J div u dx = Na meas(SZ), therefore a = 0 and b = 0 since {n(x)/ x E 8S2} = S '. We next consider the case when (6.42) and (6.44) hold. First of all, writing Du = div(2d) - Vdiv u, we obtain n

h 122 + c p (22

+

7 1(ap'r 'Y

+ ( - µ)(div u)2 +

- hp7-1) + 2µ(d(2

Jan

2µ(Au, u)dS = J pu f + u g dx. n

(6.50)

We shall only detail the case when e = N-2),,, since the case when > N 2 µ is N in fact (clearly) simpler. We just have to show that there exists v > 0 (depending upon h, SZ and µ) such that the following inequality holds for all u E H1 (11) satisfying (6.42) /'

Jn

(22

+ 2µd12 - N (div u)2dx > v in (Du(2 + (u(2dx.

h

This will be achieved in two steps. First of all, we remark that (d(2 -

(d -

,

(6.51)

n

divu

I12

and that (dJ2 = 2 (Du(2 +.I 8iuj O ui, hence

n (d12dx = 2 f IDu12 + (divu)2dx + Since u.n = 0 on aQ

,

N

(div u)2 =

88ui dS. J s1 2 u3ni

u3ni ajui = -(Ku, u) > -C1u12 where K denotes the

curvature tensor. Hence, we have

f (d2 -

(div u)2dx > 2 I (Du2dx +

2

fn

(divu)2dx -

rn (Du(2dx

- C fn (u(2dx

an

1/2

1/2

2

C

fn (Du 12 + Iu12dx

4 J (Du(2dx - C [iui2cix ffn

n

where C denotes various constants independent of u. This inequality shows that (6.51) holds if h 1 since (d12 - N (div u 112 > 0 a.e.

A priori estimates

77

Next, we prove that (6.51) holds if h E L7 and h # 0. It is clearly enough to show that

I = Min

in

hlul2 + 1d12 - 1(div u)2dx /u E Hl(SZ),

0 on 811,

in

>0.

IuI2dx=1

In order to prove this claim, we argue by contradiction and assume that I - 0. Obviously, this infimum is achieved for some u E H' (1) such that u n = 0 on all , hu - d - N div uI - 0 a.e. in 52. In particular, 2

__

2

alai

1

N-1

j>_2

8z 8xi

u

2

N1

1

j>2

8x?9

u1 in D ' (52).

In other words, ui (and in fact U2, ..., UN) solves a uniformly elliptic linear secondorder equation with constant coefficients, namely 2

ax?j +

2

7- u1 = 0 1 1: ax3 i>2 -7

in

)

D'(cI), u1 E H1(S2).

In addition, ui vanishes on a set of positive measure since h # 0. This implies that ui - 0. We argue similarly for U2, ..., UN and reach a contradiction with the constraint f0 JuI2dx = 1. The contradiction proves our claim. We now turn to the second point we wish to explain, namely the regularity (in terms of LP spaces) of D(div u - 'Ua p7) and D curl u. First of all, we observe

that in all three cases, we may write the boundary conditions as (6.42) and

curl u x n = Lu x n on

(6.52)

acZ

where Lu is a linear multiplication operator with smooth coefficients. In particular, in two dimensions (6.52) reduces to curl u = Lu on 852. With these boundary conditions, the regularity issue is the following: assuming that u E W1,q(52) , p7 E Lq(SZ) where q > 2 (for instance!), we wish to extract some regularity on div u - µ+t p7 and curl u from the following equation

-v ((µ + e)div u - ap7) + µ Curl curl u = cp E Lr (52).

(6.53)

If N = 2, Curl = a and r > 97 + Q if q = 2 , r = -L + . ifq > 2, while, if N=3 Curl =curl andl1 ifq<3, 1> +13 ifq=3, r = 1q7 +1q N r- q71 +?q r 3ry 1

if q > 3. More precisely, we want to prove that curl u, div u - a p7 E W i,r (SZ) When N = 2, this is rather straightforward since curl u = L u E W 1-1/q,q (act)

on 8f and there exists 1i E W1,q(c2) such that curlu = V on 80. Therefore, we

Stationary problems

78

have µ Curl(curl u - ip) = cp + V ((µ + )div u - ap7) in 11, curl u - ip = 0 on 8SZ and thus taking the curl of this equality, we find

-µ 0 (curl u - i) = curl cp in 11, curl u - ip = 0 on all. Hence, curl u - lp E Wo ,r and curlu E W1,'' since q > r. Then, (6.53) shows that div u - µ+E p 7 E W1,r.

When N > 3, a more general argument is needed. Let us first explain the case when 11 is a half-space, say SZ = {x E RN /xN > 0}. Then, we have UN = 8u: {xN = 0} for 1 < i < N - 1, where 1pi is the trace on 8S2 of aXN - 0i, = 0 on 0 a function in W1,q(SZ). Introducing u E W2"q(11) such that ui = on {XN = 0} for 1 < i < N - 1 and uN - 0, and replacing u by u in (6.53), we see it is enough to consider the case whenz - 0. Then, if we decompose u = v + w, div v = 0, curl w = 0 in SZ, vN = 0 on {xN = 0}, we deduce from (6.53)

F = -µ0v + Vir E L'(St)

= 0 on {xN = 0} , div v = 0 in Q.

, VN = a2N

Next, we remark that we may extend VN and FN in an odd way and vi Ft in an even way to {x,, < 0} for 1 < i < N - 1, obtaining

, 7r

and

-µ 1v + V7r E L'(RN), div v = 0 in RN. Therefore, D2v E L''(RN) and thus D curlu = D curl v E L''(SZ) and we conclude.

It only remains to treat the case of a general domain. First of all, we decompose u as before: u = v + w, v, w E W1,q (SZ) div v = 0 , curl w = 0 in St

, v n = 0 on a Q. Then, (6.48) becomes -µ0v + Vir = F E L'(SZ), curl v = curl u,

v n = 0 on 852,

(6.54)

div v = 0 in Sl

and in particular, curl v x n = ip x n on 8SZ where 0 E W'(). Obviously, we only have to prove that v E W2"''(f ). This follows in fact from general regularity results on elliptic systems but it is possible to give an elementary direct proof. By standard localization and partition of unity arguments, it is enough to prove the regularity in an arbitrarily small neighbourhood of a given point of 8SZ which, by translation, we can take to be the origin, assuming that v (and ir, F) are supported in this neighbourhood. Furthermore, by a rotation, we may assume that, locally near 0, St = {xN > O(x')} where 0 is smooth, q5(0) = 0, VO(0) = 0 and we denote x' = (xl, ..., SN_1). We then change variables y' = x', YN = XN - l¢(x') and set v(y) = v(x), *(y) = ir(x), (y) = F(x) and .

we find

-p0v + Vfr = F + G in {xN > 0} , ON= h on {xN > 0}

(6.55)

A priori estimates

79

div v = D in {XN > 0} av=

aXN

= Ei on {xN = O}

where IGI < E [ID2vI + IDfrI] + G'

,

fo r

(6.56)

--

1
G' E L''

(6 . 57)

IlhJ1W2,r < E IIiIIW2,r E _ +A and IIAIIW1,r <- E IIDvIIWI,r and E > O can be IIDIIW1,r < E made arbitrarily small by restricting further the size of the neighbourhood of 0 in which v is supported. Next, we introduce cp E W 2.r ({xN > 0J) N such that Vi = = Fi on {xN = 0} and IIWIIWZ,r < C(IIhIIW2,r +IfEII",l,,.) < 0, WN = h, C (1 + EIIVIIW2,r) where C denotes various positive constants independent of E. ,

,

We then set 0 = v - cp and find that 0 satisfies

-µ0v + fir = F , div v = D in {xN > 01, avi vN=axN=0 on{xN=0} for 1
(6.58)

where we have I IFI ILr < C (1 + EI IVI I W1,r + EIIo*II Lr)

(6.59)

I IDI I W1,r <- C (1 + E I I v1I W2,r)

Next, we extend vn and FN in an odd way, vi , 7r , Fi , D (1 < i < N - 1) in an even way to RN, and we check easily that we have on RN

-µ0v +

F , div v = D in RN,

and thus in particular D curl v = D(-1L0)-'curl F E L'' and we deduce IID2vIILr + IIv IILr < C (1 +'61 IvIIW2.r + EIIV*IILr)

or (using once more the fact that q > r) IID2vIILr + IIV IILr <- C (1 + EIID2OIILr + EIIV*IILr)

-

Choosing e small enough, we conclude that v E W2,r and thus curl u and div u a

W1,r. P-1 E

+£These

-

bounds and regularity statements allow us to obtain the exact ana-

logues of Theorems 6.1-6.3 with the boundary conditions (6.42)-(6.43), or (6.42)(6.44), or (6.42)-(6.45). In addition, all regularity information is in these cases global in 11. Of course, we shall have to check in due course that the existence proofs given in the next sections in the case of Dirichlet boundary conditions also

adapt to these cases. We just wish to conclude this section by giving the weak formulation of (6.2) that corresponds to each of the three boundary conditions mentioned above. In the case of (6.42)-(6.43) we obtain for all v E C' (1) (for instance) satisfying (6.42)

Jt - puiuiaj vi + apu - v + (µ + t;)div u div v + µ curl u curl v

Stationary problems

80

-ap7divv-pf v-gvdx=0, while in the case of (6.42)-(6.44) we have

I-

pu=ujO,vi + apu.v + 2µdijaivj + (C

- µ)div u div v -

j

- apr'divv - pf v -

0.

n

Finally the corresponding weak formulation (6.42)-(6.45) is

in

- puz u3 88 v= + a pu v + A Du Dv + l; div u div v - a pry div v

- pf - v 6.4

Jan

0.

Compactness

In this section we are going to state and prove various compactness results which

are in some sense the stationary versions of the compactness results shown in chapter 5. They will play a crucial role in the existence proofs allowing us to pass to the limit in approximated problems and actually to construct the solutions.

We thus introduce two sequences of functions (p' )n>1 , (u'),,>, and we always assume at least that p" > 0 is bounded in L'nLq(11) for some q > 1, that un is bounded in H1(S2) N (except in the case when 12 = RN where we assume L2N/(N-2)(RN) if N > 3) that Dun is bounded in L2(RN) and u" is bounded in and that pn Iun I2 is bounded in L1(1l). Let us recall that, in the periodic case, we denote S2 = FIN 1(0, T1) and pn, un (and all other data) are defined on ][8N and periodic of period Ti > 0 in each xi (1 < i < N). Finally, if SZ is a bounded, smooth, open, connected set in RN, we always assume that un satisfies (6.42): we do not require un to vanish on 8S2 and we thus incorporate in our analysis the boundary conditions (6.43), (6.44) or (6.45) as well as the case of Dirichlet boundary conditions where un vanishes on 011. Without loss of generality, we may assume, extracting subsequences if necessary, that Pn converges weakly (in

L' n Lq) to some p > 0 E L' n Lq and that u" converges weakly in H'(1),

a.e. and strongly in LP(fl) for 2 < p < N2 N (except in the case when S2 = RN where the strong LP convergence holds on all balls of RN) to some u E H'(1) (with the same modifications as above for un in the case when S2 = RN) . Let us observe that plul2 E L1(fl): indeed, by the Egorov theorem (for example), we deduce that for all e > 0, there exists EE such that meas(EE) < c (if Q = RN, lungs meas(EE n B11E) < E) such that converges uniformly to lull on E. Hence, we have

C>

pnlunI2dx >

pnlunl2dx

JEe E and our claim follows upon letting c go to 0+. Jn

Plul2dx

as n goes to + oo,

Compactness

81

Next, we assume that we have

anpn + div(p"u") - EnApn = hn where en > 0

,

an > 0

,

an w a

,

fn

0

,

(6.60)

in 12

hn E L1(1)

,

hn > 0 and

hn n h in L'(12). Finally, we assume that the following assertion holds: div un - b(pn )^1

converges a.e. in 12

(6.61)

where b>0, 7>0. Theorem 6.4 If q > 2 and q > 7, then pn converges to p strongly in LP(Q) (resp. Li'C(RN) if 12 = RN) for all 1 < p < q as n goes to +oo.

Proof of Theorem 6.4. We follow the strategy of proof introduced in chapter 5 with a small technical modification due to hn and the possibility of zones of vacuum (where pn or p vanishes). We thus consider (E+pn)e where e > 0, e > 0: O has to be taken small enough in the argument below and will be determined later on. We claim that we have ean(e + pn)e + div {un(e + pn)e} - EnA(E + pn)e pn)e-1 + (1 - O)div > O [hn + E div un + anE] (E + un(E + pn)e

in Q. (6.62)

In order to justify this computation which is straightforward if pn (and 4.un) is smooth, we use our "standard" regularization proof which only requires e to be small enough (e _< 1 and O _< q/2 even if q were not assumed to be larger than 2). We simply observe that when cp is smooth and positive then e)We-2IV I2 -Ave = -OAV Ve-1 + e(1> -eocp ye-1. We next denote by ip the weak limit of a sequence cpn. With this notation, we see that divan - b(pn)7 converges in L''(SI) (Lr,, (RN) if 0 = RN) to divu bp 'for all r < 2, r < q/7. Using this information, we may pass to the limit in (6.62) (extracting subsequences if necessary) and deduce easily

-

p)e-1 Oa (E + p)e + div u (E + p)e > 9 [h + aE] (E + + 9 E div u(c + p)e-1 + (1 - O) div u (E + p)e

+ (1 - e)b

{p7(E +

(6.63)

p)e - 7 (E + p)e}

provided we choose O < q(1 - T ). We thus choose O > 0 in such a way that

O < min (1,q(1-

*)).

We then compute the inequality satisfied by ((E + least formally

1/e

and we find at

Stationary problems

82 p)e)1/e

a ((E +

+ div u ((E + p)e)1/e

s-1

J

> [h + aE] (E +

p)e-1

((E

S

p)e-1

+ E div ZL(E +

+

+ P)e)

ee) b {p(+p)e -

1

1

1

(E+P)eJ ((E+p)A)g (6.64)

However, the justification of this inequality is a bit delicate. Let us first observe that, denoting /R(t) = Rf (R) for R > 1 where 0 E C°°([0, oo)), 3' E Co ([0, oo)), 1 > /3' > 0, 3(t) = t if t E [0,11 , we have C OR ((E

+ p)A)

1/e

> (h + aE)(E +

1/e

+ div 1U/3R ((e

+ p)e) p)e-1 PR ((E + p)e) PR ((E

+Edivu(E+p)A-1 PR ((6

+ P)e) + p)8) PR ((6 + p)e)

+le - (div u) (e + p)e/3R ((E + p)A) PR ((E + p)el

1

-1

(E+p)A}/3R((E+P)e),QR((E+p)e

+(divu)/3R ((E + p)A)

1/9

-1

J

We then wish to recover (6.64) upon letting R go to +oo. First of all, we remark that we have (diva) {PR(( + p)9

< Cldiv uI

1e

-

+ (E

((6+p)e)

1 /A

P)e PR (E + P)e PR E + pe) 1

1

((E+p)e>R) < Cldiv ul(e + p) 1 (e+p)e>R)

and thus these terms go to 0 in L since divu E L2, p E L oC . Here and below C denotes various positive constants independent of f and R, and we have made use of the following inequality (E + p)9 _< (E + p)9 due to the concavity of (t H t9) on [0, oo) (recall that 0 < e < 1). Next, we estimate 16 div u(E +

p)9-1 PR ((E

+P ) 9PR

p)e-1 (E + p)e < C E Idiv u(E + < C E9 Idivul (E + p)1-9-

((E + p)9)

-l

This inequality shows that the left-hand side which obviously converges a.e. to E div u(E + p)9-1 (E + also converges in L as R goes to +oo and furthermore we have p)91/9-1

Compactness

E divu(E+p)e-1 (E+p)e in Lloc

1

83

< CEO Idiv ul (E +

P)1-0 __ 0

(6.65)

as E-->0+.

Then, we observe that we have

(h + a)(E + p)1f((E

+ p)e) OR (( + Pr)

g-1

1

> (h + aE)(E + p)e-1 (E + P)e

1((E+p)e 0

and

7 (E+P)eJ

QR

{p- + p)9 - 3

(E+p)ei-11

(E+P)e}

Here, Here, we have used the fact that (pY(f + p)e - p7 E + p)


> 0-

> 0 a.e. in view of 1/e

Lemma 5.2 of chapter 5. Then, the above inequality satisfied by /3R ((E + p) e)

shows that these quantities are bounded in

[h+ae]

while they converge a.e. to

and VI(f + p)e - pry (E+ Al I E-+-P-)

(E+p)e-1

The inequality (6.64) then follows from the above bounds and Fatou's lemma. Furthermore, we have shown that (h + aE) (E +

p)e-1

(E + p)9

are bounded in

L1

1 ,

p7(E + p)e - p1 (E + P?

loC uniformly in

(6.66)

E.

We next claim that (6.64) implies a

+ div {opal/e} > h+ 1 ee

b

pas-1. (6.67)

{v+e

Indeed, using (6.65) and (6.66) (and Fatou's lemma once more), we see that (6.67) follows from (6.64) upon letting E go to 0+ once we observe that we have

h (E + p)e-1 (E + p)es-1

>h

a.e.

Indeed, h > 0 a.e. and (e + t)e-1 = ry [(E + t)e] where -y(s) = s1-1/e is convex p)e1-1/e on (0, oo). Therefore, we have: (E + p)e-1 > -y ((E +-7M = (E + a.e.

Stationary problems

84

On the other hand, passing to the limit in (6.61), we obtain

a p + div {up} = h . Recalling that pe

(6.68)

1/e < p a.e. and subtracting (6.68) from (6.67) we deduce

1-6

b

E)

{ p7+e

l

_ P Pe } Pe -1 < div(ur)

where 0 < r < p a.e. In particular, r E L1 n L9(St) , rJu12 E L'(1). Integrating this inequality over SZ (and justifying it as in sections 5.3 and 5.4 of chapter 5), we obtain

(p+e_) fn

pe*1

dx < 0

while pry+e - pry pe > 0 a.e. Therefore, we have (p.r+e _ pry

Since 0+8 > Pry) ~ If Pry+e = Pry

pe )

pe

* -1

= 0 a.e. in St.

, p-'+" > (Pe) 2_ a.e., we deduce that -Y

Pry+e = (PI)

a.e. on {Pe > 0}

Hence, Pn converges strongly to p in Lry+e ({pe > 0). Finally, on the set {pe = 0}, clearly (pn)e converges strongly to 0 in L1(f n BR) (VR < oo). Therefore, we deduce that (pn)e converges to pe in L1() n BR) (VR < oo). Theorem 6.4 then follows easily. 0

6.5 Existence proofs We present in this section the proof of Theorem 6.1 and more precisely we prove the existence of solutions satisfying the bounds stated in Theorem 6.1 and obtained formally in the preceding section. We shall then conclude this section with the proof of the existence assertion in Theorem 6.3 in the case when N = 3, 'y = 3, thus completing the proof of Theorem 6.3. Step 1: Preliminaries. In order to establish the existence of solutions, we shall need several layers of approximations. First of all, we approximate (6.1)-(6.2) by

ap + div(pu) - EDp = h in S2 + 2pu.Du+a p 2 + Zdiv(pu®u) - pAu - V div u + aV Pry = P.f + g in c

h2

(6.69)

where f > 0. In the case of periodic boundary conditions, (6.69) really means that the equation holds on RN (and that p is periodic). In the case of Dirichlet

Existence proofs

85

boundary conditions, where 11 is a smooth bounded domain, we complement (6.69) with Neumann boundary conditions (for instance) on p

ap = 0 On

(6.70)

on 8SZ

where n denotes the unit outward normal to c9SZ. However, (6.69) is not the only approximation we shall need. In the case when N > 4, we shall have to approximate (6.69) further by

ap + div(pus) - EDp = h 2u

in SZ

(6.71)

+ 2 pub Du + a p 2 + z div(pu6 (9 u)

- ILDu - V div u + aVp1' = pf + g in ci

,

with the same conventions as above-in particular, we add the boundary condition (6.70) in the case of Dirichlet boundary conditions. In the equations (6.71) above, b E (0,1] and for each function cp we denote, in the periodic case, cPs = cp * rc6 where rc E Co (RN), 0 < r., < 1, Supp rc C B1i rc is even and res = 1 rc(6 ). In the case of Dirichlet boundary conditions, we denote u6 = (4u) * X6 and (Vp")6 = (6(rc6 * Vp"r) _ Cs(Vres * p'r) where C6 E CO '(0), Supp (6 C {x E 11 /dist(x, ac) > b} , Cs = 1 on {x E 0 /dist(x, 49S2) > 2S} and 0 < Cs <

1, [V((< C/8 where C > 0 depends only on 11. But before we explain how to solve (6.71) and then (6.69) and how to deduce Theorem 6.1, we wish to make some preliminary reduction. We claim that it is enough to prove the existence of a solution of (6.1)-(6.2) satisfying the bounds stated in Theorem 6.1 and obtained formally in section 6.3 in the case when f, g, h are smooth. Indeed, the existence then follows in the general case by approximating f, g, h by fn, gl, h" respectively in their respective integrability classes: in this way, we obtain a solution (pn, un) of (6.1)-(6.2) with f, g, h replaced by fn, gn, hn and (pn, un) satisfies, uniformly in n, -the bounds stated in Theorem 6.1. In particular, extracting a subsequence if necessary, we may assume without loss of generality that (6.62) holds. Furthermore, the restrictions on y in Theorem 6.1 ensure in particular that pn is bounded in Ll fl L4 (SZ) where q > 2, q > y. Therefore, we may apply Theorem 6.4 and we can pass to the limit proving in this way the existence of solutions in the general case.

Step 2: Existence for (6.69) (N = 2, 3 ; fl # RN). Therefore, we assume from now on that f, g, h are smooth (with compact support if fl = RN). We begin with the case when N = 2 or 3 and Q 34 RN. We first list a priori bounds satisfied by solutions (p, u) of (6.69) that we assume to satisfy: p, u E W2,q(fl)(dq < oo)in the periodic case, this really means that p, u E Wio, (RN), and that p, u are periodic. First of all, we perform the same "energy" computations as in step 1 of section 6.3 and we find

(ei p)1uj2 + (pu). 0I - µ0 + µI Du,2 - div(u div u) + '(div u)2 + div (aui_1 p'r) + -y' p"r-1(ap - e0p - h) = pu f + u - g hIuI2 +

z

Stationary problems

86

hence, integrating over 1, we deduce fh!.-

+

+ap1

1(p1- hp1)

+ p1 Du12 + e(div u)2dx <

eary

p2Vp2

fpu.i + u g dx.

And, exactly as in section 6.3, we obtain a priori estimates on u in H1, p in LI, hIu12, p1u12 in L' (which are in fact independent of e E (0,1]). Furthermore, we deduce E

f

dx < C

(6.72)

where C > 0 is a constant which is independent of e E (0,1) . In particular, u is bounded in Lq where q = 6 if N = 3, q is arbitrary in [1, oo) if N = 2. In both cases, q > N. We then deduce from elliptic regularity results that p is bounded in L°°. Of course, this bound depends one and from now on all the bounds we obtain depend upon e. And, in particular, pu is bounded in Lq while p1 is bounded in L°O. Therefore, using elliptic regularity results, u is bounded in LO° and then D p, Du are bounded in LP for all 1 < p < oo. Finally, this implies that D2 p, D2u are bounded in Lq for all 1 < q < oo and thus, in other words, p, u are bounded in W2,q for all 1 < q < oo. We may then prove the existence of solutions of (6.68) by a fixed point argument. Indeed, for t E [0,1], and for all cp, v E W""O°, we define Tt(cp, v) = (p, u) as the solution of (for instance)

cep + div(pv) - eAp = th in

11

, p E W2,q(`dq < oo)

v+2e1pv-µ0u-lVdivu+aVpl p(tf) + tg in n , u E

W2,q(Vq

(6.73)

< oo)

with periodic boundary conditions in the periodic case, or u = = 0 on O in the case of Dirichlet boundary conditions. Notice that (p, u) is well defined by (6.73) since we can solve first the equation for p and then solve the equation for u. Let us remark also that, by the maximum principle, p is non-negative. Finally, it is easy to check that Tt is a compact mapping on (Wi'O°)N+1 that depends continuously on t E [0,1], that To (cp, v) = (0, 0) and that any fixed point (p, u) of Tt solves

ap+div(pu)-etp = th, p>0 in 11 (1-t)hu+ div u + aV p1 = p(tf) + tg in Sl. Furthermore, the proof made above immediately yields uniform bounds in t r. [0,1] for such fixed points in W2,q for all 1 < q < oo. This is enough to apply standard fixed points results (such as Leray-Schauder topological degree results) and to deduce the existence of a solution (p, u) of (6.69)-(6.70).

Existence proofs

87

Step 3: Existence when N = 2,3; y > 2 and n # RN). We now pass to the limit when N = 2, 3; -y > 2 and 11 ,E RN-in fact, the proof below may be extended to the case when y > 1, N = 2 and S1 34 RN. We wish to let a go to 0+ in (6.69) recalling that pE > 0, uE is bounded in H1, pE is bounded in L'y (and PEIuEI2 , hluE12 are bounded in L') uniformly in e E (0,1]. Furthermore, the bound (6.72) holds. Before passing to the limit, we are going to show that pE is bounded in Lq

whereq=2yifN=2orN=3, y>3andq=3(y-1)ifN=3, 2<-y<3,

= .1 + s if N = 3notice that p > 2 unless y = 2, N = 3 in which case p = 2-and that these

while VpE is bounded in L" where 1 < p < 2y if N = 2,

.1

bounds hold uniformly in e E (0, 1[. Indeed, from the equation satisfied by pE we deduce that we have

Ile VPEIIL, < C

(6.74)

C

denoting here and below by C various positive constants independent of e E (0,1]. When N = 3, this inequality holds since ap, is bounded in L7 and -lr - s < a 3 = s < (if 2 < -y < 3). Next, we apply the same proofs as in section 6.3 writing p

-

aVpr,r

= pj + g + u div(VuE) + V(div uE) - div(pEuE (D uE) - c pEuE + 2 div

uE] -

(eVpE V)uE 2

and we obtain as in section 6.3 defining s by a - 1 = z if N = 3, s > 1 arbitrary

ifN=2 IIPEII7


1 + IIPEIuel2IILq/ + 11 lucI(eIVPEI)IIL9/, + IIEIVPEI IouEI II L.

< C (1+IIPEIILq +Ile IVPEI IILq)

Indeed, ifN = 2,

= 2, ! + s <

= 2, p > 2; while, if N = 3, y > 3,

andp+2<;;finally,ifN=3, 2<-y<3, Q=3 2-1, p+sq+39 and .1 + 1 = v + 3 = q + s = . Then, combining the preceding inequality with (6.74), the bounds on pE and VpE that we claimed above follow easily. Next, we want to show some bounds on V ((µ + e)divuc a(pE)'7). In the periodic case, these bounds take a simple form and we claim that .5

-

(p + )div u, - a(PE)7 +

2R`Rj

axj

j

is bounded in W1,1 where 1 < r < if N = 2, .1 = Q + (= a where s was defined above) if N = 3-in fact, if y = 2, N = 3, then r3= 1 and not only derivatives of the preceding quantity belong to L' but they in fact belong to xi

Stationary problems

88

using similar arguments to the ones introduced in section 6.3. Indeed, we apply the arguments of section 6.3 and we write

-v [(µ + )div uE - a(p,)"] +,0 curl (curl uE) +

2div (EVpE 0 uE)

PEf+9-(PEuE.O)uE+n(vPE-v)u,-hu,

We conclude observing that the bounds shown above imply that the right-hand side is bounded in L'. In the case of Dirichlet boundary conditions, we only have to localize as in section 6.3 the preceding argument and we obtain in this way that, for each cp E C00001), rcp {(µ + l;)divuE - a(pE)"} +1 RSRj is bounded in W1,'. In a similar fashion, we can show W1,1 bounds on µ curl uE + 2 R x [R (eVp 0 u)] with the same modification as above in the case of Dirichlet boundary conditions. The last bound or information we need in order to pass to the limit as e goes to 0+ consists in showing that we have

e VpE -- 0 in L2 as f - 0+

(6.75)

(and thus eVpE -> 0 in L'' for all 1 < p' < p). Indeed, we first observe that (6.72) yields for all 6 > 0 (recall that -y > 2) lie

IVP I2 1(Pt>_6)

IIL2


E1/2 b1-ry/2.

(6.76)

On the other hand, multiplying the equation satisfied by pE by pE A 6, we find E

I IVpEI2 1(Pe<6)dx < V 6 +

I pEuE ' VpE 1(Pe<6)dx

C b (1 + II IVPCI 1(Pc<6) IILZ)

hence

IIE IVp6l 1(Pc<6)

IIL2

:5 C (b + (Eb)1/2)

Combining this inequality with (6.76), we then` obtain IIE IVpEI IILZ

:5 C

(b

+ (E5)1/2 + E1/2 61-ry/21

and we complete the proof of (6.75) upon choosing 6 = Ea with cr E (0, 712 ).

We may now pass to the limit. First of all, we observe that (µ + )div uE a(pE)'t is relatively compact in L' for all 1 < r < q/-y since EIDp4Iu converges to 0 in L'. Then, we may apply Theorem 6.4 and we deduce that, extracting subsequences if necessary, pE converges to some p > 0 E Lq in Lq' for 1 < q' < q

Existence proofs

89

while uE converges weakly in Hl to some u and also converges strongly in L" for 1 < p < rr a Next, we pass to the limit in (6.69) writing 2 2

+21pu. Vu+

apu+ 1div2

2

(pu®u) = apu+div(pu®u) -

eOpu 2

Opu = -

div(EDp 0 u) + 1 e(Vp 0)u] converges to 0 2 a (in the sense of distributions for instance) in view of (6.75). Let us observe in addition that we can also show that Du, converges to Du in LP for all 1 p < 2 since div uE - µ+f (p6)'f and curl uE are relatively compact in Ll for all 1 r < q/y. We have thus built in this way a solution of (6.1)-(6.2) satisfying the properties listed in Theorem 6.1 including (6.13) at least in the case when N = 2,3 and y > 2. We have already explained in section 6.2 how to obtain when N = 2 or N = 3, -y > 3 a local energy identity (which, by the way, shows that Du, converges in L2 to Du). Let us observe that when N = 3, 2 < -y < 3, the local energy inequality (6.14) follows from the above construction. Indeed, we have

and observing that [-

shown above that (pc, uE) satisfies 2

h IuI2 + 2apIuI2 + div u [P-j-- + a71 p I

,y

+ 7a

1

[app

- hp7-1] -

y

2

AAI 2 +14IDuI2

-

ediv (u div u) + e(divu)2 -

+

Eary

(6.77)

Eary

ry-1 div(e-1 Vp) 0-2I Vpl2 = pu. f + U. g

and thus hIuI2

In

2

+ 2 apjuj2 + pI DuI2 + e(div u)2 +

- pu f - u.gdx = 0.

VpI2 (6.78)

We then pass to the limit in (6.77) as e goes to 0+ and we recover (6.14) observing that earyp7-2IVp 2 > 0 and that Ep7-1V p converges to 0 in L1 (say).

Step 4: Existence when N > 4, 7 > N - 1 and fl 54 RN. We want now to adapt the preceding proofs to the case when N > 4, -y > N - 1 and 11 34 RN. In this case, however, we first have to solve (6.71) and this is done exactly as in step 2 above once we observe that ub and (Vp")b are smooth and that the fundamental "energy" bounds still hold in this case (and in fact hold uniformly in 6 E (0, 11 and c E (0, 11). Let us detail this important point. We deduce from (6.71)

hu + pub Du + EDpu - pAu - V div u + a (Vp7)6 2

hence

= p f +g in St

Stationary problems

90

(22

hIuI2 + pub v + IeApIuI2 _ µ1 122 + µIDuI2 - l; div(u div u) + t;(divu)2 + a u (Vp7)6 = pu f + u g in )

,

therefore we also have

(o_.) - µ0 2 122

2hIuI2 + 2apIuI2 + div

2

+ µIDuI2 - ediv(u divu) + l;(divu)2 - a [u6 Vp7 - u (Vp7)b] ary -e div (p'V p) + (ap7 - hp7-1) + +div

(o1)

Earyp7-2IVpI2

+

= pu f + u g

ry-1

in 11.

(6.79)

Next, we observe that we have in the case of Dirichlet boundary conditions j'1

ub V pry

- u (Vp7)bdx = fJ

xn

dx dy [C5(y)u(y) - Vp"(x)ica(x - y)

-u(y) (s(y)V p7(x) rc6(y - 2:)] = 0 since rc6 is even. In the periodic case, this integral also vanishes for obvious reasons. Therefore, integrating (6.79) over n, we deduce easily (as in section 4.3)

IIIIHI + e IIP"2IIHI < C,

IIPIIL-Y < C

(6.80)

where, here and below, C denotes various positive constants independent of e and 6 E (0,1]; and, as in step 3 above, we also obtain

f IIVPIILZ < cc 1/7

(6.81)

.

Using the bound (6.80) as we did in step 2 above, it is then easy to construct solutions (u,,6 , pE,6) of (6.71) (or of (6.70) in the case of Dirichlet boundary

conditions) in W2' for all 1 < q < oo such that (6.80) (and (6.79)) hold. We may now pass to the limit as 6 goes to 0+, for e fixed in (0,11. Extracting subsequences if necessary, we may assume that uE,b

,

PE,b , (pE,6)7"2 converge

weakly in H1, a.e. and strongly in L" for 2 < p < N 2, respectively to some pE a.e. and in L4 for 1 < q < N try, LN/(N-2)7). In partticvular, (uE,6)6 = ((6 u6,6) * r.6 converges a.e. and and pE E in LP to uE for all 2 < p < Nt . In this way, we recover a solution (u, PE) of (6.69)-(6.70)) satisfying in addition (6.80) and (6.81). More precisely, we have in the sense of distributions (while (6.70) holds in a straightforward weak sense) both (6.69) and

uE, PE, PE/2 E H1 (so that pC,b converges to

hu + pu- Du+ 2div(Vp0 u) -2-VP - Vu - µ0u - tV dE u + aVp'r = pf + g apu + div(pu ® u) - 2div(Vp (D u) +

2(Vp

V)u

(6.82)

- µ0u-lVdivu+aVp7 = pf +g. The passage to the limit is straightforward except for the terms (p u6.V )u, Vp.Vu and (Vo7)b which require a few explanations: first of all, we observe that PE,6 is

Existence proofs

91

bounded in L9 where q = N 2 If > N while (uE,6)6 is bounded in H1 since, in the case of Dirichlet boundary conditions, D(uE,6)6 = D(6 uE,6 + (6DuE,6 and J D(61 luc,6I < 6 1(26>d>6) IuE,6I < C "d61, denoting d = dist(x, 8ft). Therefore, pE,6(uE,6)6 is bounded in LP for some p > 2 and thus converges to pEuE

in P. In addition, from the equation satisfied by pE,6i we deduce that V PE,6 is bounded in LP for some p > 2 ( p = a - N + 9 = 2 - N + N-2) Furthermore, EOPE,6 = apE,6 - h + (u6,,6) VPE,6 + pE,6 div(uE,6)6 is bounded in L' where = 1- 2 + q and thus V p,,6 converges strongly to VpE as S goes to 0+ in L2 (and in L" for all 2 < p' < p). We also deduce, by the way, that VpE E L' and Ape E L''. Before letting e go to 0+, we wish to make a few observations: first of all, we always have, by an easy passage to the limit, 2hIuI2

1

+ 2apIuI2+pIDuI2+ S(divu)2+ a"l (apy - hp ') 4ea

+

IVp7/2I2-

pu f

(6.83)

0.

Next, if N = 4 or N = 5, we claim that we also have if ry > 2 a7

(h + ape) IuE,2 + 2

+ div [u,

PEI 2

7-1 (apE - hpE -i)

+-

1 pE

1

+ pIDu6I2 +((divu.)2 - E + E a'Y pr,-2 I Vp I2

a

7-1

- Y1 2I2 - ( div(u6 div

6.84)

(pE-1VpE)

div g

PEuE ' f + uE

uE)

in 11.

The inequality is easily deduced from (6.79) using the preceding bounds and convergences for pE,6, V pE,6 and uE,6: for instance, uE,6IuE,6I2 converges to uE IuE I2

in La for a < 3 N 2 while p,,6 converges to pE in Lb for b < N2 and 3(N-2)

< 2N

3(N-2

+

= 1.

Similarly, pE 61VPE,6

= pe,62

Y-T

[v(p2)J

+

and

PE 12 while V (P'/2) b2) converges weakly in L2 pe'.2 converges strongly in L2 (say) to to V(pE/2) (in fact, it converges strongly). Two terms require some explanation, namely piDuE,6I2 + ((div UE,6)2 (since we only assume that p > 0 and p + ( > 0)

and uE,6 (Vp5)

6

- (uE,6)6 V pE6. The latter converges to 0 in the sense of

distributions since we have for all cp E Co (S2) fixed and for S small enough Jin

Rb6

st xcl

(Vp',5)5

-(

7

dx

dxdy u6,6(x) VPE,6(y) rc6(x - y) [W(X)

- P(y)I

Stationary problems

92

IIlX0

dxdy uE,b(x)pE,6(y) Vy {rcb(x

- y) [W(x) - V(y)J}

and we conclude easily since uE,b converges to uE in LP for p < NN2, e,,6 converges to pE for q < N2 and since we can write Dy {rc6(x

- y) I WW - W(Y)II

8

_ where I roI < C x

-

x

y b

VV(X) - (x

- y)

N rc (x_Y)v V + r,6

+, J -vrc() V ,(x)

10rc

+1

<

and we have for all

- rc(l;)V (x) d = 0.

Finally, assuming without loss of generality that p I DuE,b12 + (divuE,6)2 converges to some bounded measure m (in the sense of distributions or weakly in the sense of measures), then we claim that we have: m > ptDuEI2 +l;(divuE)2. This is straightforward since we know that fnplDvl2 + t;(div u)2dx is a positive quadratic form on Ho and thus is convex. Hence, we have for all cp E C0 li minf

J

pI D(cpuE,s)f 2 + (div uE,6)2 dx > fn J2 D(W%,)12 + (div(cpu6))2 dx

from which we deduce easily using the strong convergence in L2 of uE,b lim inf

Jn

1W2I DuE,6 I2 + Ecp2(div uE,b)2dx

=

Jn

W dm

fco2(izIDu2+(divu)2)

,

and we conclude the proof of (6.84). We may now let E go to 0+ and we follow the proof made in step 3. We prove exactly in the same way that pE and EDp, are bounded respectively in L9 and

L" where q = N 2 (-y - 1) and .1 = 4 + zN ; notice that all the computations LN/(N-2)-r. made in step 3 are easily justified since pE E Let us also observe that the convergence (6.75) is ensured by the inequality (6.81). The rest of the proof

is then exactly the same, and we obtain a solution of (6.1)-(6.2) satisfying all the properties listed in Theorem 6.1. Let us conclude this part of the proof by showing that (6.14) (the energy inequality) holds if N = 4 or 5 and -y > N . This fact follows from (6.84) upon letting a go to 0+. We only have to explain how one can pass to the limit in the

Existence proofs

93

terms pEt I uE I2 or uE pE and why c pE -' VpE converges to 0 in L' (say). The last point is easy once we observe that E

E P7-1 IVPEI = /2-11I

1

E

PC

IVPEI

PE/2

pE and OPE is bounded in L2 in view of (6.80) while p7/2 is bounded and a > 2 since 7 > . We could also observe that in La where a = NNZ E pE -1 VpE _ V (, pE) converges to 0 in the sense of distributions. Next, in order to pass to the limit in pEu( (u6 12 and uEPE , we recall that we already obtained (up to the extraction of subsequences) that uE converges to u in Lb for 1 < b < while pE and pE converge respectively to p and p7 in Lc for 1 < c < N2 of - 1) and in Ld for 1 < d < N 2 ryry 1. We then conclude easily since I

z

3

N-2 2N

N-2

1

N y-1

+

N-2 N-2 2N +

ry

N -y-1

N.

2N -21 (3ry-1) <1 if 7 >

6 At this stage it only remains to prove Theorem 6.1 in the cases when 1 < both ry<2, N=land 3 <-y<2, N=3,andwhen Z <'y
Step 5: Remaining exponents ry when D # RN. We are going to deduce the existence of solutions of (6.1)-(6.2) in the case when 1134 RN; 1 < 'y < 2, N =

2; 3 < yy < 2, N = 3 and a < ry < N -1, N > 4, from the existence results (and proofs) obtained above. In order to do so, we approximate (6.1)-(6.2) by

c p + div(pu) = h , apu + div(pu ® u)

pf+g in 11

(6.85)

with p > 0 and p given by

p = ap'r + Epm

(6.86)

whereEE(0,1], m=2ifN=2or3, m>N-1 ifN>4. Then it is not difficult to check that the preceding proofs yield a solution (PE, uE) of (6.85) (periodic in the periodic case, such that uE E Ho (SZ) in the case of Dirichlet boundary conditions) satisfying: PE E LN/(N-2)(m-1) if N > 3, pE E

L2m if N = 2, uE E H', D (div uE - 111 p, and D curl uE E L'(K) where r = 1- N + N-2 $m-1) and K = 12 in the periodic case while K is an arbitrary compact set included in fl in the case of Dirichlet boundary conditions, and we have

Stationary problems

94

n

2(h+ap6)IuEI2 +

a ,y

y1(apE -hpE-1) m 1(ap - hpm-1)

+

+m

t; (div uE )2 dx <

if N = 3

(h + ap6) ju6I2 +

m 1(a pE ,

+

- µ0 12I

-

,y

J in

pEuE

(6.87)

b + uE g dx,

a

ry 1(apE - hpE -1) 2

hpm-1)

+ div U.

PEI 2I

2

+

'y

ary 1 pE

+ m -1 pm

+ /IDu6I2 - div(u6 div uE)

+ (divu6)2 < pEuE f + uE g in 11. (6.88)

Next, we deduce from (6.87), exactly as in section 6.3, that uE is bounded in H1, pE is bounded in L1' and epE ` is bounded in L'. Next, we follow the argument made in section 6.3 (steps 2-3) and we obtain, denoting pE = a p7 + epE ` and letting C represent various positive constants independent of e, IIpEIIL* < C (1 + IIpE 1UE12 II Lr) < C (1 + IIp6IIL9)

where r = 2 if N = 2,

7-1

ifN>3.

Notice that pE E LI since pE E L9' with q' = 2m if N = 2, q' = V-_2 (rri - 1)

if N > 3 and thus q' > q, m > r. We then deduce easily from this inequality that pE is bounded in L4 and that pE" is bounded in L'. We now wish to obtain as in section 6.3 (steps 2-3) bounds on D(div uE 7+-,-p,) and on D(curl tion 6.3 and write

Therefore, we follow the argument introduced in sec-

-V [(p + e)div uE - pE] + curl (µ curl uE) = H1 - div(p6u6 0 uE)

where H1 is bounded in 7-(' where s < + if N = 2, s = 1 - N +

7 1 if

N

N > 3 and where we agree that W = L3 if s > 1; in fact, H, is even bounded in

La for some a > 1. Next, we claim that div(p6u6(&u6) is also bounded in 7-(s. This was shown (at least formally, that is assuming pE, uE have smooth) in section 6.3 and we give a short proof of it. Decomposing pEuE into pEuE = wE + zE where

div wE = 0, curl zE = 0 (and wE n = zE n = 0 on On in the case of Dirichlet boundary conditions, while wE and zE are periodic in the periodic case), we write div(p6u6 (&

uE) = div(w6u6) +

Then, div(w6u6) is bounded in fl3 in view of the results of R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes [110] while zE is bounded in W 1,q and thus div(z6u6) is even more regular. We then deduce as we did in section 6.3 (steps

Existence proofs

95

D(curl u,) are bounded in %,'(K) where K = SZ 2-3) that D(div uE - µ+E in the periodic case and K is an arbitrary relatively compact smooth open set such that K C 11 in the case of Dirichiet boundary conditions. In particular, we are now in a position to apply the compactness results shown

in section 6.4. Let us only observe that (6.56) holds (extracting a subsequence if necessary) because of the bound we just proved and since epE' converges to 0 in L' for all 1 < r' < r as e goes to 0+. Therefore, extracting subsequences if for all necessary, we may assume that pE converges to some p > 0 E L9 in 1 < q' < q , uE converges weakly in Hi and strongly in L" for all 2 < p < NN-2 to some u as e goes to 0+; and we easily recover a solution of (6.1)-(6.2) which satisfies all the properties listed in Theorem 6.1. In the case when 11 # RN, the only information left to check is the energy inequality (6.14) when N = 3, 3 < 7 < 2, and we easily deduce it from (6.88) upon letting a go to 0+ provided we show that euepE' converges to 0 in L'. Indeed, u, is bounded in L6 , epE' converges to 0 in La for all 1 < a < 3 yry 1 L9'

and 6+3y-1) =firy-1 <1sincei'>

3. O

Step 6: The case when SZ = RN. We now prove the existence part of Theorem 6.1 in the case when St = RN. In fact, we are going to deduce the existence of solutions in this case from the existence results already proven in the case of Dirichlet boundary conditions. Indeed, under the conditions of Theorem 6.1 (on f, g, h, ,y and N), we obtain the existence of the solution (pn, un) of (6.1)(6.2) in S = Bn such that pn > 0, pn E LQ(Bn) with q = 2ry if N = 2 or

N = 3, 7 > 3, q = N 2

ifN = 3, 3 < -y < 3orN > 4, u,, E

(^I

Ho (Bn), D {(O2 + e)div un - a(pn)ry} , D curl u,, E xloc(Bn) with r =

(recall that we agree that 71

f

(h+ apn)IunI2 + n

a 7

N

1-1)1

= LloC if r > 1) and

ry1(apn -hpn-1

)

+ pI Dun I2 + (div un)2 dx <

IBn Pnun . f + un . g dx) (6.89)

2 (h

+ apn) IUnl2 + ,?,a71(aPn u2 i

2

- hPn-1) µA 12u I

2

+ p l Dun 12 en -e div(un div un) + (div un)2 < pfzun f + un 9 in 11,

div un

[Pn l

+

7al(

(6.90)

if 2 < N < 5, 7y > N+2 6-N '

In addition, integrating (6.1)-see chapter 5 for similar justifications and arguments-over Bn, we deduce that a fBn pn dx = fBn h dx and thus pn is bounded in L'(B,,). From the inequality (6.89), we deduce exactly as we did in section 6.3 (step 1) that Dun is bounded in L2(Bn), hlunl2 and pnlunl2 are bounded in L1(Bn), pn

Stationary problems

96

L21V/(N-2)(Bn) if N > 3 and un is bounded in L' n L'Y(Bn), un is bounded in is bounded in LP(BR) for all 2 < p < oo and for each R E [1, oo) fixed. Finally, when N = 2, if (6.11) holds we also obtain a bound in L'(Bn) on pn < x >6 where < x >= (1 + 1x12)1/2 exactly as we did in section 6.3 (step 1). Furthermore, following the proofs made in section 6.3 (steps 2-4), we obtain bounds on pn in Lq (Bn) except in the case when N = 2 where we obtain bounds in L2,t(BR) for each fixed R E [1, oo) and in L27 + L7(Bn) for any q E (2, +oo] if (6.11) holds. The fact that these bounds are uniform in n is due to the fact that in (6.32) the measure of 12 = Bn appears with an exponent equal to 1 - 1 < 0.

In addition, we also obtain bounds in f-lt (Bn-1) for D(div un -+ pn) and D(curl un) if N > 3 and in fl'' (BR) for each fixed R E [1, oo) if N = 2. We may then pass to the limit as n goes to +oo using the compactness result shown in section 6.4. The details of this passage to the limit are very similar (and in fact somewhat simpler) than those made in the preceding step (step 5) and we thus leave them to the reader. At this point, we have completed the proof of Theorem 6.1. O The proofs of Theorems 6.1-3 are now almost complete: it only remains to prove the claim made in Theorem 6.3 about the case N = 3, y = 3. More precisely, we wish to prove there exists a solution of (6.1)-(6.2) satisfying (6.12)-

(6.22) and (6.23) when N = 3, -y = 3, f E Lq with q > 2 , g E LP with p > 6/5, h E L'' (n L' if SZ = RN) with r > 3. One possible argument consists in approximating the problem by considering 'yn > 3 and letting -fn go to 3. In view of the results already shown and their proofs, we obtain solutions (pn, un) corresponding to the exponent N satisfying the following bounds uniform in n: Pn > 0, pn is bounded in L''" (Q) (nL' if n = RN ); un is bounded in Hi (SZ) if SZ # RN; Dun is bounded in L2 and un is bounded in L6 if ft = RN; D(div un µ+£ pnn) and D curl un are bounded in L615(w) where w = 0 in the periodic case, w = RN if I = RN and w is an arbitrary smooth open set such that Co C S2 in the case of Dirichlet boundary conditions. In addition, we have

f

t

h lul l

+pn 2 + alyn (apn" 'yn - 1

hpnn-i)

1

(6.91)

+ pl Dunl2 + t; (div un)2 dx = fPflUfl.f+Ufl.9dX}

a7'n

'(h + apn)lunl2 -{"''n div fun Pn

2

Iu2 l

+

l (aPn7n - h

-fn

a - 1Pnn

7n

11

P7nn-i)

- A lug l

2

(6.92)

+ µl Dun l2

- S div(un div un)2 + (div un)2 = Pnun f + un g

in SZ.

The fact that these identities hold (instead of the mere inequalities proved above) has been shown in section 6.2.

The isothermal case in two dimensions

97

We may then pass to the limit and recover a solution (p, u) as n goes to +oo and yn goes to 3+ : the proof is very similar to proofs we have already made several times before. In order to prove the regularity statements contained in Theorem 6.3, we remark that, in view of the proof made in section 6.3 (step 8), we just have to show that un converges to u in L6. Notice that we only know a priori that un (and pn) converges to u (and p respectively) in LP for p < 6 (L oc

ifSZ=RN). One possible proof consists in using the result shown in Appendix C since IunI2 is bounded in L3, while undivun , pnunlun and unpnn are bounded in a-In (apnn hpn^-1) converges L3/2 and pnun . f + un g - z (h+ap)Iun12 - 7n-1

-

weakly in L' to pu f +u g - (h + ap) IuI2 - a (ap3 - hp2), and (6.92) allows us to apply Appendix C.

z

2

It is also possible to use the energy identities satisfied by un and by u to deduce the strong convergence of Du in L2 and we shall only sketch the argument: indeed, passing to the limit in (6.91), we deduce that we have

pIDunl2+e(div un)2 dx = n 12

lim

inpu- f

22 -ap

122

- 32 (ap3-hp2) dx.

On the other hand, it was shown in section 6.2 that we also have

fiIDuI2+(divu)2dx = fPtL.f+u.g_hi!- -apt

2

-

(ap3-hp2)dx.

Comparing these two equalities easily yields the strong convergence in L2 of Dun and thus the convergence in Ls of un.

6.6 The isothermal case in two dimensions We consider in this section the case when y = 1 which corresponds "physically" to an isothermal situation. In other words, we consider solutions of

ap + div(pu) = h , p > 0

(6.93)

apu+div(pu®u)-piu-CVdivu+aVp = pf+g

(6.94)

where a > 0; a > 0; h, f and g are given and h > 0, h

0; p > 0, p + > 0.

The only situation we can solve is the two-dimensional case and therefore we assume throughout this section that N = 2. In order to simplify the presentation, we begin with the cases when the equations (6.93) and (6.94) are set in a bounded domain, that is we restrict ourselves to the periodic case (in which case, the equations hold in R2 and all functions are required to be periodic as usual) or to the case of Dirichlet boundary conditions (in which case, the equations hold in a bounded open smooth domain 1 in R2 and we request that u vanishes on 81). Later on we shall treat the whole space (R2) case. Let us first make precise what we mean by solutions of (6.93)-(6.94): we look for (p, u) solving (6.93)-(6.94) in the sense of distributions and such

Stationary problems

98

that p E L', pLogp E L' and u E H1 (Ho in the case of Dirichlet boundary conditions, H1 ,(R2) in the periodic case). Let us observe that these conditions imply that pIuI2 E L1 since we have

pIuI2 < A p Log p + A exp

(2i)

for all A > 0,

using the straightforward inequaliy valid for all A > 0, a, b > 0

ab < A a Log a + A exp

b -1

(6.95)

.

Then, we see that pIuI2 E L' since edI"I2 < oo for some c > 0 if u E H1-in fact, the inequality is true for all c but without bounds-see J. Moser [400] for more details on such facts.

Let us also mention that if p E V (p > 0) then p Log p E L' if and only if p Log p 1(p>K) E L' for any K < oo since we have obviously for some positive constant C depending on K

I

plLogpl 1(P
Our main existence result uses the following assumptions on h, f and g

hEL',hLoghEL1 3 p E (0, 2aa2)

(6.96)

(6.97)

exp(If 12 /(3) E L1

9 E H-1

(6.98)

Let us remark that (6.98) holds as soon as g E L' and I9I ILogIgII1/2 E L1 (using once more the exponential integrability of H1 to functions and more precisely the existence of co > 0 such that fn eco I"1' dx is bounded if v E H1-or Ho in the case of Dirichlet boundary conditions-and IIvIIHI < 1).

Theorem 6.5 Under the assumptions (6.96)-(6.98), there exists a solution (p, u) of (6.93)-(6.94) satisfying in addition fo ap - h dx = 0, h 1(p=o) = 0 a.e. and if exp cl f I2) E L1 for all c > 0

inn

(h+ap)I uF +a(apLogp-hLogp)+pIDul2 2 +C(divu)2-pu f -

(6.99)

0.

The isothermal case in two dimensions

99

Remark 6.12 Let us make precise the meaning of the integral contained in the inequality (6.99). First of all, -h Log p > 0 if p < 1 hence this term makes sense since

h Logp 1(p>1) < h Logh 1(p>1) + e p 1(p>1) E V. In particular, (6.99) yields the following bound

h ILog PI E V. a9i where 90,

axe

i=1

2

Ugo + E

a

Of course, if g E L1 and glLoglgl

1 gi 1112

Finally, pu f E L1 since pIuI2 and

E L2 ,4.b e n

91, g2

2

9i =1

=1

and fn u . g dx = fn u - go - E

(6.100)

au aau t

az dx (an obviously meaningful integral).

E L1 then u g E V. pi f 12

E L1-the latter being true because

of the condition (6.97) on f . 0

Proof of Theorem 6.5. The proof is divided into several steps. First, we approximate (6.93)-(6.94) replacing aVp by aVp1' for y > 1 (and letting y go to 1+). Next, we collect a priori bounds and prove that p , plLog pl, p1 ul2 , hlul2 are bounded in L1 and that u is bounded in H1. Then, we pass to the limit as y goes to 1+ : this is the heart of the matter and in this case the only difficulty lies with the term pu ® u. The strategy of proof will be to use the concentrationcompactness method of P.-L. Lions [348] and to prove that pu ® u passes to the limit except for a "defect measure" which is purely atomic. We then show that such a defect measure cannot be present in the equation (6.94). The final step of our proof consists in deriving (6.99) from the corresponding energy inequality satisfied by approximated solutions.

Step 1: Approximation. As explained above, we consider an approximated system of equations, namely

aPn + div(pnun) = hn, P > 0, aPnun + div(pnun 0 un) - p1un - V div un + aV pn = pnln + gn where y = 1 + n for instance-we simply write y instead of yn in order to simplify notation. The data hn, fn, gn are convenient approximations of h, f, g which belong to L°° and are defined as follows: fn = f 1(ifk
Stationary problems

100

boundary conditions) such that pn E L°°, Dun E BMO, D(divun - t4a pin) and D curl u E LP for all 1 < p < oo. In addition, we have (see Theorem 6.1) 2

(hn + aPn) Iu 2 I + a

'Y

(aPn

1

7

- hnpn-1) + AIDunI2

(6.101)

pnun f n + un gn dx.

+ t; (div U") 2 dx = fo

Step 2: A priori bounds.

obviously have

in

apn dx =

Jn

hn dx.

(6.102)

Since hn is bounded in L' (Ihnl < Ihi a.e.), we deduce from (6.102) that pn is bounded in L'. Next, we deduce from (6.101) for any a' E (0, a)

fa

hn

2

un 2

+

a-a

1

Pn Iun I2 +

2

I

a'Y 1

'Y -

+ (div un)2 dx <

(Pn-1 -1) (aPn

- hn) + µI Dun l2

2

1

2a'

Pn) fnI dx + IIunIIN1 Ilgnl IH_1.

Observing that, in the periodic case, (f) 4. + i I DuI2 + l; (div u)2 dx) 1/2 is an equivalent norm on H' where h = hn0 (< hn if n > no), and no is large enough so that h # 0, we deduce from (6.95) and (6.97) a priori bounds uniform in n on un in H1 , pn and pn Log Pn in L1 provided we show that we have for all 6E (0, 1) 1)

(Pn-

1 1

J

a

dx > a9

h

Log

C

1

where C denotes here and below various positive constants which are independent

of n (but may depend upon 9). Indeed, we just need to observe that the following list of inequalities holds:

fn

(pn

- 1) (apn - hn)

> 9a

n Pn

/////

1(Pn >

max(a

Pn-1 -1 -1

-g',1))

1-

-Y

> Oa f Pn Log Pn 1(Pn>max(_ i2

fPnLoPn 1

1(1
r-1 n

,1))

dx < C

f

+1))

+1))

dx dx

dx,

h (ILogh) + 1) dx < C

- 1) (apn - hn) 1(1
,1))

dx

,

The isothermal case in two dimensions

> -C > -C

hn

fn ' -1

all - e)

and finally

-1

r

1(hn>a(1-A)) dx ry-1

hn Loga lhn

6

> -C fnh(ILoghI +1) 1

7-1

hn,

101

7-1

(a(1 - 6 Rn-1

1(hn>_a(1-A)) dx

dx > -C,

-1) (apn - hn)

fj

1)

dx > 0 ,

- 1) (aPn - hn) 1(h apn <1) d2

ry-1 f2 l

> - fn aPn ILogpnI 1(Pn<_1) dx > -C.

In conclusion, we have shown that un is bounded in H1, and that pn and pn Log pn are bounded in L1. Let us also observe that these bounds allow us to deduce from (6.101) hn

C(-y -1) > Jn pn -pn-1 d2 > fn en J

>pn (L)'

1 (Ln) ry

-,

- 'Y-1 Pn d2

dx.

Since hn < hn Rn ' , we see that On is bounded in V.

Step 3: Convergence and defect measures. Without loss of generality, extracting subsequences if necessary, we may assume that un, pn converge weakly respectively in H1, Ll to some u E Hl , p E L' such that p E L1 and p Log p E L' (and p, u are periodic in the periodic case while u E Ho in the case of Dirichlet boundary conditions). In addition, we may assume that un converges a.e. to u and in LP for all 1 < p < oo.

We want to pass to the limit in the equations satisfied by pn and un and thus we have to analyse the non-linear terms pnun, Pn, Pnfn and Pnun ®un Only the latter creates serious difficulties which we shall address later. We begin with pnun and pn fn: we claim that PnUn and pn fn converge weakly in Ll to p u

and p f respectively; indeed, we have by Egorov's theorem for all e E (0,1), a measurable set E such that meas (E) < e , u, f are bounded on E' and un, fn converge uniformly on EC respectively to u, f . Hence, we may write Pnun - Pu = (Pnun - Pu)1 E + l E pn (un - u) + (Pn - P) 1.-U Pnfn - Pf = (Pnfn - Pf)1E + lEcPn(fn - f) + (Pn - P) 1Edf

and we see that the last terms converge weakly in L1 to 0. The second terms converges in L' to 0 since we have

Stationary problems

102

fEC Pnlun - uI+pnlfn - fldx

< EP[Iun-ul+Ifn-fll

0

Finally, we claim that the first terms can be made arbitrarily small in L1, uniformly in n, by taking a small enough: indeed, pulE and pf 1E converge to 0 in Ll as a goes to 0+ and we have, in view of (6.95), for all A E (0,1)

e" + e I"dx

CA+A

f(PnIUnl+PnIfnI) 1E dx

E

r

< C (A +A e1/2

< C (,\

1/2

2 eI[..I

dx

n

e1/2

+

C(A))

and we conclude taking A > 0 small and then e > 0 small enough. Next, we claim that pn converges to p weakly (in the sense of distributions or in the sense of measures). Indeed, without loss of generality, we may assume that pn converges weakly in the sense of measures to a bounded non-negative measure ir and we have by the convexity of the function ([0, oo) t H V)

(Pn)' >- 1 +'y(Pn - 1) a.e. Hence, ij(pn)1dx > p. On the other hand, we have seen at the end of step 2 that we have <

-1) + 1 J hn dx -

an

1 rn h dx

as n - oo.

Indeed, hn converges in L' to h since hn converges a.e. to h and hn < h hn-1 = h Rn 1, while Rn 1 goes to 1 as n goes to +oo. Therefore, we deduce

r

a Jn h dx >_ meas(fl) limn

j

n

7

pn dx

=

! Jin h dx

in view of (6.102). Hence, fn dir _ fn h dx. Finally, (6.102) obviously yields the following equality (global conservation of mass)

In

p dx = 1 h dx. a in

Since ir > p, we conclude that 7r = p. In fact, it is even possible to show that pZ converges weakly in L' to p since we have for all K > 1

fPfll(>K)dx < Jn pn dx - if Pn1(pn
The isothermal case in two dimensions

<

<

fnPndx-

f

['YPn+(l

!)] I(on<_K)dx

f Pn dx - y r pn dx + (ry - 1) meas(c) n

n

< fn pn dx - 7

103

j Pnl (pn>K) dx

pn dx + (ry -1) mess (11) + C(Log K)

At this stage, we see that the only term left to investigate is the term PnUn ® un. This term is indeed delicate since concentrations can occur. An ex-

ample of the phenomenon is provided by the following explicit construction: take (logn)-1/2 log [max(IxI, for n > 2 so that 11 = B1, pn = i sn 1B11n, un = n)] Pn, Pn log Pn are bounded in L' while un is bounded in Ho (pn - 0 , un - 0, pn log pn 2i bo). Then, pnun = n2 1Bl1n does not converge weakly to 0 but does converge to irbo. The following lemma shows that such concentrations are precisely the only obstructions to the passage to the limit. It is clearly a variant of the general concentration-compactness lemma shown by P.-L. Lions [347], [348] with some specific difficulties in its proof that we postpone until the end of the proof of Theorem 6.5.

n

Lemma 6.1 Let pn > 0 converge weakly in L' to p, and let un, vn converge weakly in H1 to u, v respectively. We assume that pn log pn is bounded in L1. Extracting a subsequence if necessary, we may assume that Pnunvn converge weakly in the sense of measures (on f) to a bounded measure v. Then, there exists an at most countable set I (possibly empty), distinct points (xi)iE1 C Il, and constants (vi)iEI C R - {0} such that

v = puv+Evibyi

,

E IV,1112

< 00.

(6.103)

Step 4: Elimination of the Dirac masses. In view of the facts shown in step 3 and of Lemma 6.1, we deduce the existence of an at most countable set I (possibly empty), distinct points xi E l (i E I) and non-negative symmetric 2 x 2 matrices vi (vi # 0) such that we have

ap + div(pu) = h ,

-u-e0 div u+aVp = pf +g.

c pu+div iEI

Let us recall that p, pu, pu 0 u, h, pf E L', g E H-1, and Du E L2. We are going to deduce from (6.104) that vi = 0 or in other words that I = 0. Let us argue by contradiction and consider some point xio E SZ for some io E I. The matrix vio is non-negative and symmetric and thus if vio # 0 we know that (vi. )11 or (via )22 is strictly positive. Let us consider for instance the case

Stationary problems

104

when (vi. )11 is strictly positive, the other case being treated similarly. Making a translation, we can assume without loss of generality that xio = 0. Let b > 0 be such that [-b, b] 2 C 11 and let W E Co ((-S, b)) be such that: 0 < w < 1, and ]. We then multiply the equation satisfied by u1, contained in V = 1 on 2 the system (6.104), by cp(x2)xl cc (E) for C E (0,1) and we obtain easily

J (Pul + ap) V (El) W(X2) dx + E(vi)11 W

(()i) cp((xi)2) + (Vio)11

i

< f F cp(x) xico (e1 1 dx + +,/

G co(x2)

- E(Vi)11

[w (C1 /

(xi-) W1

+

J

-

(E)1

cp'(x2)x1

W, (e1 / J

W

(e1) dm2

dx - J (pU2 + ap) E W/ (E)ca(x2)dx

V ((xi)2)

iEI

where F E L1(fl), G E L2(SI), and m2 is a bounded measure. Since pui +ap > 0, cp > 0 and (vi) 11 > 0, we deduce from this inequality (Vi.)11 < Ce + Cf + J(pu2 + ap) 1(Ixil<<e6) 1(Ix21<6) dx

+CE(vi)11

1(V_
iEI

Since EiEI (vi)11 < oo and (vi)11 >- 0, we observe that the right-hand side of the preceding inequality goes to 0 as e goes to 0+. This shows that (vio)11 = 0 and thus vi. = 0, and we conclude that (6.104) reduces to the system (6.93)-(6.94). 11

Step 5: Properties of the limit solution. We have already seen that fn aphdx = 0. We next show that h1(p=o) = 0 a.e.. In order to prove this claim, we recall that, in fact, we have shown in step 2

in

In-1

- 11 Iapn - hnl dx < C(ry-1)

(6.105)

.

Indeed, we showed f,, (pn-1 - 1)(apn - hn) dx < C(y - 1) and 711(pn-1-1) (apn - hn) > -zn where zn is bounded in L'(S2). In particular, we deduce from (6.105) for all E E (0,1) hn l(pn<e) dx < Ce +

C

< Ce + I logEl C

hence, we have for all b E (0, 1) e-

fn hn

dx < f hn 1 (p,. <E) dx + in hn dx

a-6/6

The isothermal case in two dimensions

tI C

< Cc +

og e

+

105

e`16.

Without loss of generality, we may assume that e_Pn'6 converges weakly (in L°°(!n)-weak*) to some e6 such that (by convexity) e6 > e-p'6 a.e. Therefore, we obtain for all e, S E (0,1)

I

Jhe-P/6dx < Ce+

C

+e-E"6

log e

I

and choosing S = e2, we deduce letting a go to 0+ the equality fnh1(p=o)dx=0. It only remains to show (6.99) which we wish to deduce from (6.101) or equivalently from

in

-

2(hn+c pn)IunI2 +a ryI [a(Pn-Pn) hn(Pn-1-1)] + plDunl2 + (div un)2 - Pnunfn - un gn dx = 0.

(6.106)

We already know that we have ll nm fn pI Dun 12 + (div un)2 dx > j pIDul2 + e(div u)2 dx and

fun.gn-*ju.gdx

fpnIUnI2dX4jpIUI2dX.

,

In addition, by Fatou's lemma, we immediately obtain hnlunl2

lim n

n

dx > [hItLIdX c1

(in fact, it is possible to show that hnlunl2 converges in L' to hIuI2 as n goes to +oo)

We also claim that fa pnun fn dx converges to fn pu f dx; indeed, we prove exactly as we did for pnun that pnun fn 1(Ifnl <M) converges weakly in L' to pu f 1(I fkM) for all M E (0, oo). Then, our claim is shown provided we prove

that sup f I Pnun . Al 1(If..I>M) dx - 0 n

as M - +oo .

Indeed, we have for all e > 0 using (6.95) with A = 62

In

IPnun . fnI 1(Ifnl>_M) dx <

2

Jin

pnlunl2 dx

< CE + 2

+ . i Pnl,fnI21(If.

Pn log Pn 1(If,.I>_M) dx

M)

dx

Stationary problems

106

IfnI2

+2 f0 exp

-1

E2

1(Ifnl>M) dX 1/2

< Cc + C

r e2lfl2/e2 dX

meas(I f I ? M)

n

< CE + C(e)e-M j elfI dx st

and our claim is shown. At this stage, we see that (6.99) follows from (6.106) provided we show the following inequality linm ry 1 1

(6.107)

and « with hn and h re-

Notice that we have changed notation, replacing spectively.

We begin by proving (6.107) when h E LOO in which case hn = h for n large enough. Then, we observe that the function ([0, oo) E) t - ,x,11 [t"-t-h(t7-1-1)]) is convex and thus we have for any function cp E L°O (11) such that infess cp > 0

y11 [p;n'-Pn-h(Pn-1-1)] >

1

cpry-1

-

1

- hurry-2

-Y

'Y

(Pn - (p)

a.e.

}

We easily deduce from this inequality that the following holds lnm

>

1

7-1 f Pn-Pn-hn(Pn-1-1)dX Jn

V log W - h log V + 1 + log cp -

(p-cc) dx.

Next, for e E (0,1), M E (1, oo), we choose cp = min (max(p, e), M) and we see that the right-hand side is equal to

f(Pve) log(p v e) - h log(p V c) dx + J [M log M - p log p] 1(p>M) dx n

+

1(p<E)

(1+loge_ h

+ n 1(p>M)

(p-e) dx

(1+logM_)(P_M)dx

denoting a V b = max(a, b). Since p (1 + I log pi) E L1, it is easy to check that the

second and the fourth integral go to 0 as M goes to +oo and thus, letting M goes to +oo, we are left with

f(P V ) log(pve)-hlogpvedx+n J 1<E1+loge-

(p-e)dx.

107

The isothermal case in two dimensions

Then, we can bound the second integral by 1(p<E) [E(1 +I logEI) + h dx] ,

an expression that clearly converges to 0 as e goes to 0+ since h 1(p=o) = 0 a.e. Also, (p V e) log(p V e) converges in L1 to p log p while -h log(p V E) increases to -h log p as e decreases to 0+. Therefore, the first integral converges to fn p log p - h log p dx as c goes to 0+ and (6.107) follows in the case when h is bounded. Next, in the general case, we observe that we have for all M > 1 and for n large enough

fn 'Y >

1 1(Pn-11) (Pn - hn) dz

f

n 1(h<M)

-1 1

(P7n-1-1)(Pn-hn)d2

1

(Pn-1-1)(Pn-hn) + in 1(h>M) 1(1
> fn

in 1(h>M) ,

1 1(hn-1-1)hn dx

1(Pn-1-1)(Pn -hn) dx - f 1(h>M) (log hn)hn dx

In

(Pry,,-1-1)(Pn-hn)dX

> in

1(h>M)

logh hdx

Rn-1.

Using the above proof on the set SZ fl {h < M} (on which h is bounded!), we deduce lim n

1

St'Y-

1

(pn-1-1) (pn-hn)d2x

fn 1(h<M) [P log p - h log p] dx - fo 1(h>M) log h h dx = fn 1(h<M) P log P - i(h<M) h log p 1(p? 1) -h 109P 1(h<M) 1(p<1)

- fn 1(h>M) h log h dx .

We thus conclude easily letting M go to +oo since p log p, h log h, h log p 1(p>1)

belong to L' and [-h log p 1(h<M) 1(p<1)] increases to [-h log p 1(p<1)] as M increases to +oo. Proof of Lemma 6.1: We wish to adapt the proof of the (second) concentrationcompactness lemma in P.-L. Lions [347], [348]. First, we observe that we can assume, without loss of generality, that un = vn; indeed, we just have to write

Stationary problems

108

2

Pun vn = Pn (u 2 Vn ) - pn ("" 2 "") 2 . Next, we claim that it is enough to consider the case when u = 0. This claim is quite easy to show: we just need to write Pn(un-u)2 + 2pnunu

Pnt n =

- PnU2 .

Then, we check that pnu2 and pnunu converge weakly (in the sense of measures for instance) to pu2. We have already shown in step 3 that Pnun converges weakly in L' to pu and thus pnunu l(Iul<M), Pnu21(IuI<M) converge weakly (in Li in fact) to pu21(luI<M). In addition, we have in view of (6.95) for all e,6 E (0,1)

fn

PnlunH Jul 1(lul>M) d x <

2

rn pnlunI2 dx + 25 l. pnIul21(IuI>M) dx

J

< CS +

J

1

2S

n

PnIu121(lul>M) dx

and

Pn logPn dx + -C I e4 1(lul>M) dx en 1 Pn1U121(IuI>M) dX < E in J Cc + We(M)

where C denotes various positive constants independent of E, S E (0,1), and we (M) goes to 0 as M goes to +oo for each E E (0,1) fixed (recall indeed that e1 E L'(Q) for all e > 0 in view of the estimates obtained by J. Moser [400]). We then conclude letting M go to +oo, then c go to 0+ and finally S go to 0+.

We may now begin the proof of the lemma in the case when un - vn, u - 0. Also, we only treat the case when un E Ho (fI), the general case following by a simple extension argument for instance. First of all, extracting subsequences if necessary, we may assume that J Dun 12 andpn log+ pn converge weakly in the sense of measures to some bounded (on S1) non-negative measures µ, in respectively, where we denote log+ t = log(max(t, 1)) = max(log t, 0) (V t > 0). Next, we claim that we have for all Sc E C(?!), cp > 0

J

cp3 dv < 8 (fcc2 dµ

dm

(6.108)

.

Admitting temporarily this claim, we conclude easily. Indeed, we deduce easily from (6.108) that we have for all cp E C(SZ), cp > 0 nL

J cp3 dv <

3/2

1/2

8ir

cp2 d{max(µ, m)}

And, see P: L. Lions [347],[348], this inequality implies that v is atomic that is (6.103) holds with, however, the following bound: E Ef v?/3 Goo. At this stage,

The isothermal case in two dimensions

109

going back to (6.108), we deduce easily, using cp supported in B(x2, S),1 > cp > 0,

cp(xi) = 1 (for i fixed in I) and letting S go to 0

Vi < $1 µ({2i}) m({2i}) . Hence, we have for all i E I 1/2

1 8x

<

vi

1/2

1/2

µ({2i})1/2 m({ai})1/2 < 81r 1

(.21 Il({2i}) + 2 m({2i}))

therefore,

1: vi

1/2

1/2

< 81r

iE!

1

2 E EI[µ({2i}) + m({xi})] <

1/2

8

1 (p(?) + m(R))

and Lemma 6.1 is proven. It remains to prove (6.108) and we first observe that it is enough to establish

it for all cp E C1 (U), cp > 0 by density. Next, we remark that we have for all

a,b>0,e>0

ab < e a log+ a + min(b, E)

e(b-,)+/e

and we deduce for all e < 0

f

n

(P Pn (

n)2 dx < e fn SP pn log+ Pn d2

(6.109)

+ f- go min((goun)2, e) e((SPun)2-E)+/E dx .

Next, we observe that we have

f I0(cpan)12 dx = fn

21 Dun12

+ 2goVgo Vv,, un dx + IOcp12 un dx

J

(P2 dFi

since un converges a.e. and in L2 (for instance) to 0.

Therefore, if we choose e > co where co = 8 f V2 dµ, we deduce from the above convergence and J. Moser's sharp inequality [400] that e(Wun)2/e is bounded in La for some a > 1. Since un converges to 0 a.e., we deduce easily

that

1 go min((

n)2, E)

e((Wun)'-a)+/E

dx _4 0 . n

Hence, passing to the limit as n goes to +oo in (6.109) yields

f lp3 dv < e

r Vdm,

and (6.108) follows upon letting c goes to co. The proof of the lemma is complete.

0

Stationary problems

110

Another possible proof of Lemma 6.1 (still based on the results of P.-L. Lions [348]) is to apply the following lemma that we state, in order to simplify the presentation, in the case when un E Ha (SZ) converges weakly to 0 in Ho l. Let us recall that, in view of the preliminary reductions made in the beginning of the above proof of Lemma 6.1, this particular case is sufficient to obtain Lemma 6.1 in its full generality.

Lemma 6.2 Let SE be a bounded open set in R2, let un E H0 (1) converge weakly to 0 in Ho'. Then, there exists a set I C Il which is at most countable and possibly empty such that we have (u^)2 converges to 1 in L"(f) for all c > 0, p E [1, oo), (i) if I = 0, then a= ("^)2 converges (ii) if I is finite, i.e. I = {x1, ... , x,,,,} C SZ, then, for all b > 0, e < to 1 in LP (St - Um, B(xi, b)) for all e > 0, p E [1, oo), (iii) if I is infinite and I = (xi)i>I C SZ (where all the points xi are distinct),

then there exists, for each k > 1, ek > 0, ek goes to 0 as k goes to +oo such that, for all b > 0, eck (u")2 converges to 1 in LP (c - Ui==1 B(xi, S)) for all pE [1,00).

This lemma immediately implies that v is purely atomic since we have in view of (6.95)

P.(un), < ePn

logPn+ee<(u..)2-1

and thus v = 0 on 12 in case (i), v = 0 on St - Ix1, ... , x,,,,} in case (ii), while in case (iii) we have for all k > 1 k

U

- U B (xi, 6)

< C Ek

i=1

and thus, in all cases, we deduce that v is supported on I. Proof of Lemma 6.2. Without loss of generality, we may assume that J Vun 12 converges weakly in the sense of measures to some bounded non-negative measure

p on SZ. We denote by I the set of atoms of j. If I = 0, we use an observation shown in P.-L. Lions [348] to conclude. If I is finite, we take, for each S > 0 fixed, < E COO (RN), 0 < 1, ((X) = 1 if Jxi > b, ((x) = 0 if Ixi < b/2. Then, we consider un = 11m 1 C (x - xi)un and we observe that we have IDunI2 (fj; ` 1((x - xi)) u = µ weakly in the sense of bounded measures on S2.

n

Since µ has no atoms, we deduce as above that e+(u )2 converges to 1 in L1'(Sl) for all 1 < p < oo and (ii) is shown.

Finally, if I is infinite, we consider, for each k > 1 fixed and for each b > 0 fixed, un = (n1 ((x-xi)) un; and IVun12 converges weakly to µk = ((x-xi)) µ Obviously, itk < µ, the measure of the largest atom of µk is less than or equal to maxi>k+1 µ({xi}) = bk, and bk goes to 0+ as k goes to 11k

i=1

The isothermal case in two dimensions

111

+oo. We conclude easily using the fact, shown in P.-L. Lions [348], that ei Iunl2 is bounded in L1(11) if e > s . 0 We conclude this section by deducing from Theorem 6.5 an analogous result in the case when !Q = R2. In all the rest of this section, we therefore assume that 2 = R2 and we request that the following conditions hold

hEL1, hloghEL1, hlxl8EL1 for some 6E(0,1], h#0, h>0 VK>0

,

3 9 E (0, 2acY2)

,

(6.110) (6.111)

1(IfI>K) exp(If 12/9) E L'(R 2)

3C>0, VuECr (R2),

feudx

< C (L2 IVul 2+ hlul +2 1B1 I(I2 dx

(6.112)

1/2

The latter condition holds as soon as g = g1 +g2 where 9,(y)-1 is bounded on R2 and 92 satisfies for some m E (1, 2] JR2

1921' <x>2(m-1) [log(<x> +1)]0 dx < oo;

denoting <x>=

with

6 > 2m - 1

(1+Ix12)1/2. Indeed, (6.112) then follows from the results shown

in part I [355], Appendix B. Of course, we may even assume that g2 may be split into several pieces each of which satisfies the previous assumption for some m. Let us also observe that we may discard 1B1 Iu12 in (6.112) without loss of generality: indeed, let Ro E (1, oo) be such that h # 0 on B&, then (fBRO IVU12+hIuI2 dx)1/2 is an equivalent norm to the usual H1 norm on H1(Bp, ). This fact is easily shown since fBR hIuI2 dx is a continuous quadratic form on H1(B,) in view of (6.110). We may now state our main result

Theorem 6.6 Under the conditions (6.110)-(6.112), there exists a solution (p, u) of (6.93)-(6.94). 0

Proof. We approximate (6.93)-(6.94) by the same system of equations set in a ball BR (for R large enough so that h # 0 on BR) with Dirichlet boundary conditions. Using Theorem 6.5 we obtain a solution (PR, UR) satisfying the conditions listed in Theorem 6.5. In view of steps 3 and 4 of the proof of Theorem

6.5, we just have to obtain a priori estimates on PR, PR log PR, hIuRI2 in L' and DUR in L2. Notice that such bounds imply, as seen above, bounds on uR in D1,2(R2). Since we are going to deduce these bounds from (6.99), we also approximate f by fR = f 1(If1
fBR pR dx = « fBR h dx and

IBR

+ PRIURI2 + ac pR logpR - ah log PR 1(PR:51) + vIDURI2

Stationary problems

112

I

BPRURfR + uR9 + a IBR aPR(- log PR) 1(pR<1) + h(log PR)1(pR> 1) dx R

<

2PRIURI2 dX +

K

Tb-

J

PR dX + 2b

BR

r

+a /

IDuRI2 +hIuRI2dx

fB<

1pR>1dx

2 IDuRI2 + 6 PR log+ PR dx

r

f

PRIXIbdx+CJ

BR

BR

JBR

h log h + 1PR

PR(- logPR)1(1>PR>e_isi6) dx +

r PRIuRI2 + 4 J BR 2 P I

PR(-logPR)1(pR<e-1=I6) dx

JBR'BR

BR

<

PRIII21(IfI>K)dz

1/2

+CJ +a

JBR

PR 1(pR<e-i=i6)dX

2PRIURI2+4hIuRI2+2IDuRI2+ bPR1og+PRdx

+C

1+f

PR I x I6 dx R

for some v > 0 independent of R, where C denotes various positive constants independent of R, PR and uR, and where b E (20 , 1). In the above string of inequalities, we have used conditions (6.110)-(6.112) and the inequality (6.95). In particular, we obtain JBRPRIuRI2

+ hIuRI2 + IDuRI2 + PR I log PRI

-h log PR 1(pR<1) dx < C

(6.113)

1+1

PRI xI6 dx R

Next, we use (6.93) in order to obtain a bound in L' on PRIX 16 or equivalently PR < x >6. Indeed, we multiply (6.93) by < x >6 and we deduce

of

BR

PR

<x>6 dx = fB h<x>6 +PRUR R

I

< C+JBBR PRIURI dx

x

<x >2

dx

C 1+

since S < 1. Inserting this inequality into (6.113), we complete the proofs of the a priori estimates we claimed above and we conclude the proof of Theorem 6.6.

6.7 Stationary problems In this section, we study the stationary problems (6.3). We begin by explaining some of the difficulties associated with such problems; namely we show that there

are in general no solutions in the case when Q = 112N or in the periodic case.

Stationary problems

113

Then, we study the case of a bounded, smooth, open domain SZ with Dirichlet boundary conditions (or those introduced in section 6.3). As explained in section 6.1, (6.6), in this case, possesses or may possess infinitely many solutions. The set of solutions may be parametrized by the condition (6.6) which is a natural parametrization from a physical viewpoint since it corresponds to prescribing the total mass of the gas. We thus begin by studying (6.3)-(6.6), and, we next turn to the study of modified periodic problems, the modifications being in general necessary (as mentioned above) for the existence of solutions. The last issue we develop in this section concerns again the set of solutions but with a different parametrization than (6.6); in other words, we build solutions of (6.3) satisfying different constraints from (6.6) and we use constraints of the form (6.7). This will allow us to shed some light on some borderline cases such as ry = 3 , N = 3 and to consider arbitrary exponents ry.

Example 6.2 We thus begin with some non-existence results. First of all, we consider the case when SZ = RN, -y > 1. We then choose g - 0 and f = V c where -D is smooth, decays fast enough at infinity and 4P < 0 on RN, and for any c > 0 each connected component of {(P > -c} is unbounded (take for = -e_1z12). Finally, we assume that N > 3 in order to avoid the example usual specific difficulties associated with N = 2. We claim that there are no non-trivial solution (p, u) of (6.3) such that p > 0, p E L1(RN), Du E L2(RN), L2N/(N-2)(RN) uE and p E LO°(RN)-this last assumption is imposed in order to fix ideas: as will be seen from the proof below we need only p E L"(RN) and an energy inequality. Using the regularity proofs introduced in section 6.3, we readily check that Du E LP(RN) for all 2 < p < oo. Next, we claim that u - 0. Formally, this is a simple consequence of the energy identity which becomes for

the stationary problem (6.3), using the fact that f = 0c

JRNI)

2 d2;

= LN pu fdx = 0.

(6.114)

L2N/(N-2)(RN)). If (6.114) holds, then Du - 0 and thus u - 0 (since u E The fact that (6.114) holds is shown by multiplying the equation satisfied by u by

ucpR where WR = W(- k), R > 1, V E Co (RN), cp = 1 on Bl, Supp cp C B2 and 0 < cp < 1 on RN. Doing so, we obtain easily

f

N

Pf ucpRdx =

f

N

cPR {plDu12

uVWR LN{2P+_lP}

and we obtain (6.114) letting R go to +oo since pf u, IDuI2, (divu)2, ups', and upIu12 belong to Ll(RN) At this stage, we have shown that (6.3) reduces to u - 0 and a Vp1' = pV . In particular, p is continuous on RN (even C'1 ' on RN) and goes to 0 as (xj

Stationary problems

114

goes to +oo. Hence, on each non-empty connected component C of the open set as shown in Remark {p > 0}, we have for some c E R, p = (7 -f (0 + 6.8, and obviously C is a connected component of the open set {0 > -c}. Since

< 0 on RN, we see that c > 0. Therefore, by assumption, C is unbounded:

ry-1) there exists (x,,),,>1 C C such that (xtz) -,, +oo. Hence, cp(x,s) --n and we reach a contradiction. The contradiction shows that p - 0. ry-i Let us observe that the requirements on 0 are in general necessary for this (ry-Y1 4,)+ (7-1) non-existence assertion. Indeed, if {0 > 0} is non-empty, then p =

is a non-trivial solution, and even if 0 < 0 on RN and there exists one connected component C of {0 > -c} which is bounded for some c > 0, then p = 1c(

(4, + c))+ (ry-1) is a non-trivial solution. 11

Example 6.3 We wish to show that, in the periodic case, no non-trivial solution exists. Indeed, if we integrate (6.3) over the period n, we immediately obtain

L pfi+gidx = 0

1
for

and we reach a contradiction if, for instance, gt > 0, f i > 0 for all 1 _< i < N and (for instance) g 0 0. If g - 0 and fi > 0, f 0- 0, then we obtain the trivial solution p - 0, u = uo for any constant ua E RN. We now state our main existence result for the system (6.3) in the case of Dirichlet boundary conditions

Theorem 6.7 We assume that y > 1 if N = 2, -y > 3 if N = 3 and -y > 2 if N > 4; and that f, g E L°°(f ). Let M > 0. We request in addition if N = 3, y = 3 that M2/3II f II Lc is smaller than an absolute positive constant which depends only on a, µ, and 11. Then, there exists a solution (p, u) of (6.3) satisfying

I

p dx = M,

U E Ho (St) , p E LP (Q),

(6.115)

with p=2-yif y> 1,N=2orif-y>3,N=3,p= N 2(-y -1) otherwise. Furthermore, if y > 1, N = 2 or if -y > 3, N = 3, (p, u) satisfies PELOO

D div u

,

divuELOO

-+

curl u E Lf , D curl u E Lf0

Du E BMOto°

(6.116)

for al11 < t < oo .

(6.117)

Remark 6.13 In the case when N = 3, -y = 3 , it is possible to replace I I f I I L°° by IIf2IIL°° where f = fi + f2, fi = VC Indeed, we just have to observe in the proof below that we have

in

=J

and we argue exactly as in the proof below.

in

div(pu)dxn

Stationary problems

115

Remark 6.14 An interesting open question is the possible removal of the assumption on M in the case when N = 3, -y = 3. We shall come back on the set of solutions of (6.3) in this case.

Remark 6.15 The above result with the same proof still holds if we replace the Dirichlet boundary conditions on u (replacing of course Ho by H1!) by the boundary conditions studied in section 6.3. More precisely, we impose the condition (6.42) on the normal velocity and either (6.43), or (6.44) with C > N 2 µ or (6.45) with a >_ - N . In the case when we impose the boundary condition (6.44), the above result holds provided we assume the following condition

A>0orC>

11andforanyu0ERN,

Q antisymmetric N x N matrix (Qx+uo) n(x) does not vanish identically on all unless uo = 0 , Q = 0

.

(6.118)

The role of this condition is to restore the coercivity of the quadratic form [fn 2pIdI2 + (e-µ)(divu)2 dx + fan(Au, u) dS] on {u E H'(1) / 0 on 8111. When A > 0, the coercivity follows from the inequalities shown in section 6.3, upon observing that if u E Ho (11) N and (d - N div u) = 0 a.e. in 11, then u - 0 in SZ: indeed, as shown in section 6.3, each component of u, i.e. U1, U2, ... , UN, solves a linear second-order elliptic equation in 1. Similarly, when e > N 2 µ, if u E H1(SZ)N and div u = d = 0 a.e. in fl, then, as is classical in continuum mechanics, one can check easily that u = Qx + uo for some uo E RN and for some N x N antisymmetrical matrix Q. Therefore, u - 0 and we conclude.

Proof of Theorem 6.7. Step 1: A priori bounds. We are going to obtain the solution (p, u) that we seek by approximating (6.3) by the system (6.1)-(6.2)-with Dirichlet boundary conditions-with h = am meas (ci)-1 (in fact any h > 0 such that fn h dx = M could be chosen). We may now apply Theorems 6.1-6.3 and we obtain a solution (pa, ua) of (6.1)-(6.2) satisfying all the properties listed in Theorems 6.1-6.3 and

M Jc2ameas(SZ)

IuaI2 2

+apa IuaI2 2

a-ya I 7 _ pry-1 11'r 7-1 +-y - 1 meal 1l LPa

+µlDuo, l2+C(divua)2dx < J Paua f +ua gdx,

(6.119)

}

n

L pa dx = M

(6.120)

(since a fn pa dx = fn h dx = aM). We are going to obtain a priori estimates on (pa, ua), uniform in a E (0, 1]. First of all, we deduce from (6.119) and Sobolev embeddings VIItiaIIHo <_ CIIuaIIHI + CoIIfIILo II P«II L2NI(N+2) IIuaIIHH +

Ca

for some v > 0, where C denotes various positive constants independent of a and Co denotes various positive constants independent of a and of 11 f 11 Loo (n)

Stationary problems

116

and where L2N/(N+2) is to be replaced, here and in all that follows, when N = 2 by L" for an arbitrary p E (1, ry). The above inequality allows us to deduce IIuaIIH1

< C +CO IIfIIL°°

IIPaIIL2N/(N+2)

.

(6.121)

Next, we argue as in step 3 of section 6.3 and deduce from (6.2) IIPa)ILr s C+CIIPaIIL2N/(N+2) +COIIPaIua12IILr +C

IIP«IIL

C + C II Pa II L2N/(N+2) + COIIPaJIL-rr IIUaII i + CIIPaIIL7

if N > 4 or N = 3, y < 3, and r = 2 if N = 2 or while N = 3, 'y > 3. Let us observe indeed that -L + N 2 = r if r = N 2 where r =

yry 1

N-""i

3.

z-f+

Combining this inequality with (6.121), we deduce IIPatILr* <- C [1 + IIPatIL2N/(N+2) + IIPaIIL'Y- + IIPa{ILr] +C0!I fIIL°° IIPaIIL-r IIPaIIL2N/(N+z)

Next, w e observe that -yr > N+22 N and we estimate II Pa II i-, by II Pa I I L--1f, II Pa II r, i M(1-0)ry using (6.120) where 0 = 7* and we deduce easily

ii

IIP«IILry*

e)'

_

PCIIili-a)

IIP«IIL

< C + COIIf IIL°° IIPcIILV?a

= C+ CO IIfIIL°°

M2(1-a)

1

IIPaIIL

2a

where,. + 1- a = 2N . If N = 2 or N = 3, 'y > 3, r = 2 and 1 +2a < ry. Next, if <'y N=3, 3 <-y<3orN>4,-yr= N unless N = 3, y = 3 in which case a = 3 , 1 + 2a = -y = 3 and IIPcIIL2 < C+COIIf1IL°° M4IIP«IIL2

In conclusion, we have shown that pa is bounded in L"'' and thus, in view of

(6.121),uais bounded in H0 ifN=2and-y> 1, N=3and -y> 3 orN>4 and -y > E.. In addition, these bounds also hold when N = 3, -y = 3 provided Ilf IIL°O

M2/3 is small enough.

Step 2: Convergence and conclusion. We now wish to use Theorem 6.4. Clearly (6.54) holds with e = 0, and in view of the bounds we obtained in step 1, we see that we only need to check (6.55). In order to do so, we observe that the arguments introduced in section 6.3 show that we have

if N=2,-y> 1 orifN=3,'y>3 div ua, pa, curl ua are bounded in LOO (Q), (µ+C)div ua - a(pa)ry , curl u,,, are bounded in W , (SZ) for all 1 < q < 00 (6.122)

117

Stationary problems

D{(µ+e}div ua - a(pa)'Y} , D curl ua if N > 4 or N = 3 3 < < 3 are bounded in ?l

(6.

+ 2 (recall that q> N+1) and where I i+ q= 7 q>1,which is the case when ry>N-1.

where

10CV

123 )

L oc

if

These bounds immediately yield (6.55) with b = µ+C . We then deduce from Theorem 6.4 that pa (or appropriate subsequences) converges to some p > 0 in

L1()) for 1 < p < N (-y - 1) or 2ry depending whether N > 4, N = 3 and 3 < -f < 3 or N = 2 and y > 1, N = 3 and -f > 3. In addition, u,,, converges weakly in Ho ()) to some u and using (6.130), we deduce the existence of a solution of (6.3) satisfying the properties listed in Theorem 6.7. 0 Remark 6.16 Of course, when M = 0, p - 0 and we find a unique solution u of

, u E Ho (SZ) N It may then be tempting to guess that there exists a unique solution for M small but this is not the case in general. Indeed, let us consider the case when g - 0,

-tiAu - l; V div u = g

in SZ

f - V 4b, is smooth, 4P < 0 in Il except at m distinct points (71,. .. , z,,,,) in SZ where 1 vanishes. Then, in general for M small, we can build (2"t - 1) distinct

2 Ix - X,12 near Yt for i = 1, 2 solutions. For instance, if m = 2 and '' with ai > 0, then we find for M small enough three solutions given by u = 0, near Yi (i = 1,2) and p3 = [(4 +.1)+ ti-1J1/(i'-1) Pi = [( + Ai)+ near 21 and Z2 Where A,, A2, A > 0 are small and such that fn Pi dx = fn P2 dx = '27

fnp2dx=M.0

We have seen in Example 6.3 that (6.3) has in general no solution in the periodic case. The result which follows (and the methods of proof) shows that the only real obstruction to existence is the lack of coercivity (or control) on "constant flows", i.e. u = constant. We treat in fact two cases, namely the case when f - 0, fn g dx = 0, and the case when we have a dissipative force and more precisely we consider the example of an electromagnetic force with a non-trivial magnetic field. We thus consider in the periodic case the system div (pu) = 0

,

(6.124)

div(pu ® u) -pOu-eV div u+aV'p'' = pf +g+ (u x B) x B

where p, u, f, g, B are periodic and the meaning of (u x B) x B is

-

(uiB2 when N = 2 and the usual exterior products when N = 3. We make the following assumption on (B, f, g) : B, f, g E L°° and we have u2Bi)(B2)

either B is not parallel to u a.e., for all u E RN, u 54 0 ; 1 or B = b(x)u for some b E L°°, periodic, b # 0, u E RN, r

i orB-O, f -OandJgdx=0. u#0, f -0and In

n

I

(6.125)

Stationary problems

118

We then have Theorem 6.8 Under assumption (6.125), the same result for the system (6.124) as Theorem 6.7 for the system (6.3) holds.

Proof. We only have to show that (6.121) (or some appropriate modification) holds under condition (6.125). The rest of the proof is essentially the same. In fact, the only modification is really the existence of approximated solutions that is not granted by the results obtained above. However, showing the existence of solutions of the following approximated system of equations

div (pu) + ap = aMmeas (c)-1

,

.

pf +g+ (u x B) x B

apu+div(pu(&

(6.126)

is a simple adaptation of what we did above in the preceding sections (in the case when B = 0), and we thus skip this part of the proof which presents no new difficulty.

Therefore, it just remains to prove (6.121). The only new term in the energy identity is

j[(uxB)xBJ.udx = -JnIuxB12dx where u x B = u1B2 - u2B1 when N = 2, and we obtain easily the following inequality

j

vJn IVuI2dx+

r

Iu x BI2dx < C

(i+j pf

(6.127)

In the case when B is not parallel to a non-trivial constant vector, one shows easily that (fn vIDuI2 + Iu x BI2 dx)112 is a norm on (Hp.,)N which is equivalent to the usual norm since fn Iu x B 12 dx does not vanish when u is constant (and

u 0- 0). This allows us to copy the argument made in the case of Dirichlet boundary conditions.

In the case when B - 0, then, by assumption, fo g dx = 0 and f = 0 where ' is periodic; therefore, fn gu dx = fn g (u f-n u) dx < CII VUIi L2 (n) and we

-

obtain a bound on Du in L2 (12). In addition, integrating over St the momentum equation in (6.126), we find that fn p u dx = 0. Then, we have

M

Lu

Lu

[uJu] dx

< CI DuIL2 IIPIIL,

< C IIPIIL,

Therefore, we have obtained the following modification of (6.121)

IIuIIH1 < C (1 + IIPIIL,)

(6.128)

where C may depend upon M. Next, we follow the argument in the proof of Theorem 6.7 observing that div [pu 0 u] = div [pu ®

(u - 1 u) I +

(j(o )/

div pu

Stationary problems

=div

pu® u-

f

n

u

+

119

If u n

a

M meal(1)

-ap

We thus deduce the following bound IIPIIi7r
+ IIPIIL-yr IIPIIL,

and we conclude easily in all cases of dimensions and exponents ry. Finally, if B b(x)u for some u E RN, u # 0, b E L°°, periodic and b # 0,

we decompose u = v + w where v = u - (u, u)IuI-2 i and w = (u, u) U. We then observe that we have for some ,c > 0

fziIDuI2+ JIB x ui2 dx >

Jin

IDvI2 + I DwI2 + IvI2 dx

Indeed, if this inequality were not true, we would find sequences un, vn, wn such

that Dun and B x un converge in L2 to 0 while fn I vn I2 dx = 1; hence, un converges in H1 to some constant u and thus vn converges to u - (u, u) I uI -2 u whose L2 norm is 1, while B x ii = 0 a.e. and a must be proportionnal to U. The contradiction proves our claim. We then go back to (6.127) and write

J

=C 1+

f

=C 1+ J

n

Therefore, v is bounded in H1 and Dw is bounded in L2; in particular Du is bounded in L2 and (u f-u is bounded in H1. Furthermore, multiplying the

-

momentum equation by u/IuI and integrating over 11, we find that fn pw dx = 0

and thus

M

fwH

JP[WJWJdX

From these bounds, we deduce once again (6.128) and we may follow the argument above.

Remark 6.17 Let us observe that we have shown in fact the existence of a solution which satisfies fn pu dx = 0 if B - f = 0 and fn g dx = 0, or (fn pu dx, 1) =

OifB-b(x)u,uERN,u54 0,bEL°Operiodic, b00, f -0and

The rest of this section is devoted to the study of the same stationary problem (6.3) with a different parametrization from the prescription of the total mass. Namely, we shall use the constraint (6.7) in place of (6.6) for some convenient p. Typically, p will be chosen large enough but it need not be so. In fact, we shall

Stationary problems

120

use this idea in two types of situations: first of all, when y = 3 , N = 3 in which case we choose p E (1, 2); and when y > 0 is arbitrary (for all N > 2) in which case we choose p large enough (p > po(y, N) to be determined later on) and we shall consider "modified Dirichlet boundary conditions". Before we state precisely such existence results, we wish to make a general comment and draw some conclusions. First of all, we are going to prove the existence of a continuum C of solutions containing the trivial solution p = 0,

u = uo and such that for all M E [0, oo) there exists an element (p, u) in C satisfying fn pP dx = M, where uo denotes the solution of

-pAuo - V div uo = g

in St

,

uo = 0 on 00

(6.129)

(or different boundary conditions as we shall see later on). By continuum we mean a closed connected set in an appropriate function

space (typically p E Lq for some q > p, u E H1 for instance). In fact, the proofs below also show that, in Theorem 6.7 (or Theorem 6.8), we also have the existence of a continuum C of solutions, containing the trivial solution (0, uo),

in LP x(W1,q)N(for all p3,N=3,y
N

Z

(y - 1), 1 < q < 2 if 3 < -y < 3, N = 3 or N > 4) for instance, such that

CCLPx(H1)Nwithp= N z (y-1)if <-y<3,N=3orN>4andforall

M E [0, oo) there exists an element (p, u)3in C satisfying fn pdx = M. Next, we observe that I fn p dx / (p, u) E C} is necessarily an interval of the form [0, Mo] or [0, Mo) where 0 < MO < +oo since C is connected, p > 1, (0, uo) E C and C does not reduce to (0, uo). In other words, the results which follow show the existence of some Mo E (0, +oo] such that, for all M E (0, Mo), there exists a solution (p, u) of (6.3) satisfying (6.6). It is a very interesting open question to decide whether MO < oo in some situations with y > 1. However, if y E (0, 1), it is possible to conclude as we now show.

Remark 6.18 If y E (0,1) and N > 2, one can check that, in general, there exists a critical mass beyond which no bounded solution exists. Indeed, let us take g 0, f - VD where is Lipschitz (or even C°°) on 1, (D < 0, max.- ( _ D(xo) = 0 for some xo E S1 and (P behaves like -jx - xo1m near xo (m = 1 or 2 for instance). As we have seen several times above, in this case for any solution (p, u) we have u - 0 and aVp"T = pV , pry (and thus p) E W1,°°(S2) if p E L. It is then easy to check that we have pti-1 = (.1)(A -') or p = b(\ - -1>)-1/(1-7) where A > 0 (and b Next, as A goes

to 0_, fn pdx converges to Mo = fn a'f dx < oo if lmf < N. If m = 1, our claim is shown provided y < 1 - N and, if m = 2, we need N > 3 and y < 1 - N . In fact, it is possible to show that no solution other than the ones above (with A < 0) exist with u - 0 and p E V: indeed, we have then pP E W1,1 and a E+P V p 1 = EP V15, hence, letting e go to 0+, we deduce easily

that

Opt'-1 = -l(ogo) 0' a.e. in fl. In particular, pry-1 is Lipschitz, thus

Stationary problems

121

pry- 1 + 4P is constant on Q. Obviously, this constant denoted by A is non-negative and we conclude.

P > 0 a.e. on St and

We may now state our main results.

Theorem 6.9 Let 'y = 3, N = 3 and p E (1, 2). Then, there exists a continuum C of solutions of (6.3) in LP x W1,P for all 1 <_ p < 2 and such that C n {(p, u, M) / 0 < M < R) } is bounded in L2 x Ho for all R > 0, containing (0, uo) and such that, for all M E [0, oo), there exists (p, u) E C satisfying (6.7).

Remark 6.19 Of course, one can obtain a similar result in the periodic case with the same assumptions as those made in Theorem 6.8. The second result we now wish to state is of a slightly different nature. First of all, we consider in this case arbitrary 'y > 0 in arbitrary dimensions (N >_ 2) but we modify the boundary conditions imposing (6.42) (the normal velocity vanishes) and either (6.43), or (6.44) with the conditions > N 2 µ and (6.118) -A.. Those conditions are imposed in order to or (6.45) with the condition enforce "enough coercivity" as explained in section 6.3 and in Remark 6.15. We shall not recall below these assumptions and conditions.

Theorem 6.10 Let N = 2 or N = 3, y > 0. Under the above conditions and if p is large enough (depending only upon N and -y), there exists a continuum of solutions in Lq x W 1,q (V q < oo), containing (0, uo) and such that, for all M E [0, oo), there exists (p, u) E C satisfying (6.7). Remark 6.20 As we shall see, we have in -fact for (p, u) E C p E L°° ,

curl u E W 1'q (V q< oo)

,

divu-

µ

a

p l E W 11q (d q< oo) .

This follows from the existence proof but it can also be deduced from the regularity analysis (results and proofs) performed in the preceding sections.

Remark 6.21 We are not able to prove a similar result in the case of "pure" Dirichlet boundary conditions for lack of compactness of p in LP "up to 892".

Remark 6.22 The analogue of Remark 6.18 also holds here.

Remark 6.23 Analogous results to Theorem 6.10 can be shown when N >_ 4. However, the form of constraints to be used has to be modified slightly (or at least it seems so!). The modifications being too technical, we prefer to skip such extensions to higher dimensional situations which, anyway, are not physically meaningful.

Remark 6.24 In Theorem 6.10, the trivial solution uo obviously corresponds to the equation -pAuo - V div uo = g in S2, with the same boundary conditions as those mentioned before Theorem 6.10. In particular, uo is not the same in Theorems 6.9 and 6.10.

Stationary problems

122

The proofs of Theorem 6.9 and 6.10 follow the same line of arguments with, however, important technical differences. Therefore, we wish to begin by explaining the common strategy of proof. The main (and simple) idea is to approximate (6.3) by

div (pu) + aPP = a

M , meas(g) apPu+div (pu(gu) - pAu-t;0 div u+aVpr' = pf +g inSl,

(6.130)

in the case of Theorem 6.9, and in the case of Theorem 6.10 by

div (pu) = 0 div(pu (9 u)

in 12,

J in

pP dx = M,

(6.131)

div u+V {app'+apP} = g f +g in St

where a E (0,1) and p > 3 is large enough. Of course, p is required to be nonnegative in SZ and u to satisfy the same boundary conditions as in Theorem 6.9 or 6.10. Then, essentially copying the existence proofs for the system (6.1)-(6.2), we shall obtain the existence of solutions of (6.130) (or (6.131)). In fact, we shall obtain a continuum Ca of solutions (p, u, M) in L2 x H1 x [0, oo) or Lq X W l,q x [0, oo) (p < q < oo) such that, for all M > 0, there exists (p, u) satisfying (p, u, M) E Ca. The existence of such a continuum is a rather direct application of the existence proofs and of topological degree arguments (together with a simple compactness observation described below). More precisely, we prove that

Ca is unbounded in L2 x H1 x [0, oo) (resp. Lq x W1,'? x [0, oo)), closed and connected for the weak topology (resp. in Lq X W i,q x [0, oo) and the strong topology) and Ca fl {(p, u, M) / 0 < M < R} is bounded in L2 x H1 x R (resp. Lq X W i,q x R) for all R E (0, oo). We then set

C = { (P, u, M) / 3 an

Mn n M ,

n 0, 3 (pn, un, Mn) E Cc,,,, (Pn, Un) n (PA t) }

the convergence being the weak L2 x H1 convergence in the first case, strong Lq x W' ,q convergence in the second case. We shall then prove, using compactness

results, that (p, u) solves (6.3), pn converges in L" (in particular) strongly to p

and thus fn pP dx = M. Then, as shown in Appendix D, C is a continuum, contains (0, uo, 0) and we shall prove that C n {(p, u, M) / 0 < M < R} is bounded for all R E (0, co) and C n {(p, u, M) / M = R} is non-empty for all R E [0, oo). The conclusion follows easily setting C = {(p, u) / 3 M E (p, u, M) E C} and observing that C = { (p, u, fn pP dx) / (p, u) E C} . We may now begin the proofs of Theorems 6.9 and 6.10.

[0, oo)

,

Proof of Theorem 6.9. We recall that throughout this proof -y = 3 and N = 3.

Stationary problems

123

Step 1: Bounds for solutions of (6.130). We begin with formal a priori estimates for solutions of (6.130). First of all, integrating over SZ the equation satisfied by p, we immediately obtain fo pP dx = M. Next, exactly as in section 6.3, we obtain the following energy identity (writM ing h = meal O ) 2

Jcthi!_ + app 12 + a'y (P7 - hp-1) + µI DuI2 +t;(divu)2-pu f-g-udx =0

(6.132)

from which we deduce as in the proof of Theorem 6.7 (see (6.121) and the following inequalities) IIUIIHI

s IIPIIL <- C (1 + IIPIIL6/5)

<- C (1 + IIPIIL6/5),

where C denotes various positive constants (independent of a, p, u,). Next, if p E (1, 5) , we use Holder's inequality and deduce 3

IIPIIL < C (1 + IIPIILP

11P112(1-8)

)

> 3 hence 2(1 - 9) < 3 . If p > 5 , we simply where 2 + 121 = 6 , i.e. 9 2 use the above LP bound. Therefore, we deduce in all cases bounds in L2 for p and bounds in H1 for u (which depend only on bounds on f, g, the coefficients a, µ, t; and the constant M).

Step 2: Approximated problems and continua. We follow the existence proofs in section 6.5 approximating (6.129) by (for e, 5 E (0, 11)

am

app + div (pu) - -A p = m am u 1 pu Vu + app meas(St) 2 + 2 -

in SZ

lop

0 in M

( )+ 1 div (pu ® u) - µ0u 2

(6.133)

2

pf +ginfl, u=0on8Q,

and we obtain, as in section 6.5 with the same modifications as above, a priori bounds (depending upon a and e) on p and u in W 2,q (St) for all q < oo. We then define a family of compact mappings parametrized by M >_ 0 as follows: for (W, v) E W1"°°, we solve

app + div(pv) 8p

8n

-ep=

aM meas(1)

in S2

= 0 on 8SZ, p E W2,q (V q < oo)

Stationary problems

124

eAp v - p Du - CO div u + aOp^'+ 6V p2 em ) + pv Ov + z m( =pf+g in fl, u=0onO), uEW2iq(Vq
continuous on [0, oo) x W""°° X (W'°)' and that T (M, , ) is compact for all M > 0. We may then apply a classical and fundamental result due to J. Leray and J. Schauder (on T(M, , + uo) - (0, uo) ...) to deduce the existence of an unbounded continuum Ca'6 of fixed points of T in (W1,°° X (W 1,°O)N) x [0, 00)

containing the trivial solution p - 0, u - uo, M = 0. In view of the above a priori estimates, we also know that for each M E [0, oo), there exists a solution (p, u) E W2,q X (W21q)N (V q < oo) of (6.133) which belongs to C. It is worth emphasizing that Ca 6 is connected and closed in (W1,00 x (W 1,00 )N) x [0, oo) and thus (using the a priori bounds) in L2 X (HI )N x [0, oo) (or even L4 x (H1)N x [0, oo) for all p < q < 2). Finally, we observe that if (p, u) solves (6.133) then fn pP dx = M and the following analogue of (6.132) holds:

f

In

2 hIuI2 + 2ap'I

ul2

+ AJDuI2 + (divU)2

+earye-2IVpl2 + 2eblopl2 +

+ 2Sa(pP+1 - hp) dx =

aa1 ,1

()9ry+P-1

- hp-r-1)

(6.134)

fu.f+u.9dxand

combining the arguments of step 1 above, of step 3 in section 6.5 and of step 2 in section 6.6, we deduce L3 bounds on p and H1 bounds on u for M bounded, uniformly in e E (0, 1], for a, b fixed in (0, 1]-we just need to observe that fn pP dx = M.

Step 3: Limits. We first let a go to 0+ for a, S E (0, 1] fixed. In order to do so, we apply the results of Appendix D choosing E = Lql X (W l,Q2 )N where 2 < qi < 3 and p < Q2 < 2. Obviously, (D.1) holds since we have fn pP dx = M and (D.2) holds with xo = (0, uo). We thus have to check (D.3) or equivalently (D.5). In order to do so, we consider a sequence of solutions (for a, S E (0, 1] fixed) en > 0, En --n 0, ptz > 0, Mn > 0, Mn --n M, un of (6.133). As shown above, pn,un are smooth (W2'q(St) for all q < oo), pn is bounded in L3 and un is bounded in Ho l. We wish to show there exists a subsequence of (pn, un) which converges (strongly) in Lql X W',qz and in order to do so, we just have to check that we can apply the results of section 6.4. First of all, we extract a subsequence still denoted by (Pn, un) such that pn

and un converge weakly to some p > 0, u E Ho in L3, Ho respectively. In addition, we may assume that un converges a.e. to u and strongly in LP for

2
Stationary problems

125

Next, we may copy the argument made in step 2 in section 6.6 to deduce the (relative) compactness of div un pn/3 $ pn in L8(11) for 1 < s < Z +f and of curl un in L'(Sl) for 1 < r < 2. Therefore, without loss of generality, we may assume that (6.55) holds with bpn replaced by pn/3 + A+C pn. Then, we 6 +{ simply follow the proof of Theorem 6.4 (observing that the main modification

-

concerns the term apn which is easily handled once we observe, with the notation pP+e-1 _< (pp)1+(e-1)/P if 0 E (0,1)), and we of the proof of Theorem 6.4, that obtain the compactness of pn in Lg1(11) for 1 < qi < 3, from which we deduce the compactness of div un, curl un and thus Dun in Lq2 (11) for 1 _< q2 < 2. Thus (D.5) holds. Therefore, we may apply the results of Appendix D and we find an unbounded continuum Ca in L1q1 X (W1'g2)N x [0, oo) (1 < q1 < 3 , 1 < Q2 < 2) containing (0, uo, 0) of solutions of-

apP + div (pu) = a

M

,

meas (SZ)

p>0

in .11

(6.135)

ap"u+div (pu 0 u) -pzu-CV div u+aVp5/3+bVp2

=pf+g in 11.

Furthermore, the proof presented in Appendix D immediately yields the following additional information: CC C L3 X (Ho) N, fn pP dx = M for all (p, u, M) E Ca and

I

hIU12

2

+

+ 2 ap"l uI2 + i IDuI2 + (div u)2

aory

7-1 <-

(pP+'r-hpry-1)

(6.136)

+2Sa(p9+2-hp)

pfor all (p, u, M) E Ca fu.'f+u.9dx,

.

In addition, Ca fl {(p, u, M) / M E [0, R]} is bounded in L3 x (Ho )N x R for all RE (0, oo). We next let b go to 0+ for a E (0, 1] fixed. Once more, we use the results of Appendix D. We now choose E = Lq X (W l ,q) N for p < q < 2, and we

first observe that, in view of (6.135)-(6.136), the arguments of step 1 above combined with step 3 in section 6.5 show that Ca fl {(p, u, M) / M E [0, R]} is bounded in L2 X (Ho )N x R, uniformly in b E (0, 11, a E (0, 1], for all R E (0, oo]. Since fo pP dx = M, Ca is unbounded in E x R and obviously (0, uo, 0) E C.

Once more, we just have to check that (D.5) holds, and once more, we use the compactness results of section 6.4 and the proofs in section 6.5 to deduce the compactness of div u- . p5/3 in L3 for 1 < s < 5, curl u in L' for 1 < s < 2, p in L' for 1 < s < 2 and thus of Du in L' for 1 < s < 2 (whenever (p, u, M) E Ca,

b E [0,1], M E [0, R] for all a E (0,1], R E (0, oo)). Thanks to Appendix D, we obtain an unbounded continuum Ca in Lq X (W1'q)N x R, contained in Lq x (Ho )N x R where q = max (2, p + ), of solutions of (6.130). In addition, 3

Stationary problems

126

C. n { (p, u, M) / M E [0, R] ? is bounded in L4 x (Ho) N x R uniformly in a E (0,1] for all R E (0, oo) and we have for all (p, u, M) E Ca: fn pp dx = M and

ff 2

hlul2

+

pplul2

2

+ p!Dul2 + l;(div u)2

(6.137)

+ ac i (pp+7 - h py-1) dx < fn pu f + u g dx.

We finally let a go to 0+ using once more Appendix D in the same setting as the preceding limiting procedure, the details being exactly the same: we just have to observe that (6.137) yields a bound on app+ r-1 in L1 and thus app L(p+'r-1Vp, and the crucial compactness follows from an easy converges to 0 in adaptation of the proof of Theorem 6.4. This allows us to complete the proof of Theorem 6.9.

Proof of Theorem 6.10. As explained above, the proof of Theorem 6.10 is very similar to the proof of Theorem 6.9, so we shall only sketch it. We begin by observing that the proof of Theorem 6.9 (and that of Theorem 6.7) yields the existence of an unbounded continuum Ca in Lq x (W 1'q) N x R for any q > p contained in LOO x (W i,r)N x [0, oo) for all r < 00 of solutions of (6.131) such that (0, uo, 0) E Ca and we have for all (p, u, M): (6.138)

L

Next, we claim that if we choose R E (0, oo) then, for all (p, u, M) E Ca with M E [0, R], p is bounded in L°O (St), u is bounded in W 1,q (11), div u + pry - µ+F pp is bounded in W 1'q (1) and curl u is bounded in W 1'q ( 1) for

all 1 < q < oo, uniformly in a E (0,1]. First of all, p is bounded in L"(n) and we recall that we assume that p is large (to be determined later) and in any case greater than 3. Then, (6.138) immediately implies that u is bounded in H1 since our assumptions on ti, and the boundary conditions ensure the coercivity in H1. From now on, we only consider the case when N = 3 since the two-dimensional case is similar and simpler. In particular, p(u V)u is bounded

in La' (1) where al = 1 + s + z and ai can be made arbitrarily close to 2 by choosing p large enough (notice that p > 3 implies that ai > 1). The usual (by now) regularity analysis then implies, if ai > 5, i.e. p > 6, that apb + app is bounded in L2(11). In addition, the argument detailed in section 6.3 shows that (p + l;) div u - (a p'" + app) and curl u are bounded in W1,al (1) and thus in LQ1(Q) where /3i = 33a1 can be made arbitrarily close to 3 by choosing p large enough. Then, the lemma stated and shown immediately after this proof implies that app' + app and thus div u, curl u and Du are bounded in Lpt (11). Hence, p(u 0)u is bounded in Lag (11)where-L=1+ pl + pl - 3 = p + « -1 and a2 can be made arbitrarily close to 3 by taking p large enough. Iterating the above 32 argument, we obtain a bound on a p7 + app, and Du in L02 where 32 = 3-a2

«

Stationary problems

127

and thus a bound on p(u V )u in La(S1) provided we choose p large enough so that 82 > 3 and thus u is bounded in LOO, and p + 102 = 1 < Then, exactly Ct as above, we deduce that ae + app is bounded in L°° and thus p is bounded in LOO, and that div u and curl u are bounded in L°O and therefore u is bounded in W1,11(52) for all 1 < q < oo. The rest of our claim then follows easily. We may then complete the proof of Theorem 6.10 by sending a to 0+, using the result of Appendix D. Once more, the compactness requested by the condition (D.5) is a straightforward consequence of the above bounds on p and u and of the compactness analysis performed in section 6.4. O 3.

Lemma 6.3 Let U E W1,9(52) for all 1 < q < oo satisfy u n = 0 on 852; and let p E L'(0) > 0 satisfy div (pu) = 0 in 52. Let cp be a continuous function on [0, oo) and let us assume that div u - V(p) E L' for some r E [1, +oo]. Then we have (6.139)

IIco(P)IILr s Ildivu-go(p)IILr.

Remark 6.25 This lemma and the estimate (6.139) is very much related to the estimate (6.32) shown in section 6.3 above. It is in fact possible to combine the argument made there and the above setting to reduce the regularity assumption on p and assume that p E L", U E W 1,q where 1 < p, q < -oo and 1 + 1 < 1. We do not give the details of this technical extension since we do notpneeA it. 0

Proof of Lemma 6.3. We begin by showing that we have for any continuous function 0 on [0, oo) :

fu.Vb(P)dx = 0 =

fdiv(u)b(p)dx.

(6.140)

Indeed, we observe that we have for any function 3, div [u/3(p)] = (div u) [,8(p)

- Q'(P)P]

in

S2 .

Therefore, (6.140) holds if we can choose 8 in such a way that Q'(t)t - /3(t) _ fi(t) on [0, oo). Then, we remark that we may assume without loss of generality

that ti(0) _ &'(0) = 0 and that b is smooth (once (6.140) is shown for such smooth functions i, the general case follows by density). Thus, we may choose ,3(t) = t fo s ° ds and (6.140) follows. At this stage it only remains to use (6.140) with = Icpl''-2cp (at least if

r < oo, the case r = +oo follows upon using (6.139) with r < oo and letting r go to +oo). Indeed, we find f I w(P) I r dx =

f{() - div u} I'(p)

<- II'(P)

(6.139) is established. 0

-

divullJr II,(P)IILr1and

Ir-2 w(p) dx

Stationary problems

128

Remark 6.26 As can be seen from the proofs of Theorems 6.7, 6.9 and 6.10, one can use many variants for the approximation of the stationary problems. In particular, the approximations detailed in the proofs of Theorem 6.9 and 6.10 are only possible ones (that work!); there are many other possible choices some of which might even be slightly simpler!

Remark 6.27 We wish to come back to a consequence of Theorems 6.9-6.10 mentioned before Theorem 6.9, namely the existence of some MO E (0, oo] such

that for all M E (0, Mo) there exists a solution (p, u) of (6.3) (in L°° (St) x W1,Q(SZ)N for all 1 _< q < oo) satisfying (6.6). We wish to point out that Remark

6.16 above shows that we may have for all M > 0 small multiple solutions (p, 0) which converge to (0, 0) as M goes to 0+. This example shows that such

existence results for small M cannot be obtained by the use of the implicit functions theorem (or variants of it that yield a unique branch of solutions).

6.8 Exterior problems and related questions We wish to consider in this section stationary problems like (6.3) set in unbounded domains of RN (N > 2). More precisely, we study the problem (6.3) (with p > 0!) and the domain S may be either i) SZ = RN, or ii) St = w° where w is a bounded smooth connected domain in RN, or iii) St = IR x w and w is a bounded smooth connected domain in RN' i . In the two last cases, we shall always assume that u vanishes on 8SZ (Dirichlet boundary conditions); even if we could allow for different boundary conditions as in the preceding sections we do not wish to consider such extensions or variants here. It remains to prescribe some conditions at infinity for p and u or, on other words, to prescribe the behaviour of p and u as lxi goes to +oo (cases i), ii)) or as xl goes to foo (case iii)). We have seen in the preceding section that, in the first case, i.e. when 11 =RN,

there may not exist non-trivial solutions if we request that p and u vanish at infinity. The construction of such non-existence examples is easily adap*ed to the

other cases considered in this section and we may thus restrict our attention to the case when p or u does not vanish at infinity. In fact, we only want to consider the natural situations (from a physical viewpoint) where the flow is constant at infinity, i.e. in cases i) and ii) p -+ p°°

,

u --+ u°O

as

lxi -+ +oo,

(6.141)

and in case iii)

p -+pf

as

xl --++oo

,

u-+0

as

xl -±oo.

(6.142)

Notice indeed that in the "tube" case iii), the velocities at x1 = ±oo have to vanish since they vanish on R x 8w. This section is only concerned with existence results when u°° = 0, p°° > 0

in (6.141) and p+ = p°O = p°° > 0 in (6.142). We entirely leave out the case when u°° 34 0 since this case already raises the delicate issue about problems set

129

Exterior problems and related questions

in bounded domains with non-homogeneous Dirichlet boundary conditions for u

(and thus Dirichlet boundary conditions for p on some part of the boundarythe so-called "inflow" part). This issue is investigated in the next section. Also, in case iii), we do not consider the case when p+ # p_° in (6.142) since we can construct non-existence situations in this case as we now show. Let (p, u) be a solution in that case; we assume (for instance) that p E L°O(11), Du E L2(11), u E L2(fl) fl L°°(0) (notice that we can use Poincare's inequality since w is

bounded), g - 0 and f - 0 (or f - V with 4P smooth, vanishing at xi = ±oo fast enough). Multiplying the momentum equation by u, and integrating over (-R, +R) x w, we find easily (a.e. R (=- (0, oo))

µa2 (R, x') - u(R, x') - div u(R, x') ui (R, x')

dx' w

+u(R,x') a P(R,x')1u(R,x')12 +a72 1 p(R,x')7

+

i

(-R, x') u(-R, x') + t; div u(-R, x') ul(--R, x')

- ui(-R, x') 11p(-R,x')ju(-R,X1)j2+a -Y

R

+J

r

IR

dxi

Jw

'

1

p(-R, x')7

2

dx' µ1Du12 + (div u)2 = 0.

We next observe we can find Rn going to +oo such that fw dx'{IDu(Rn, x')12+ 1Du(-Rn,x')12+1u(Rn,x')12+1u(-Rn,x')12} goes to 0 as n goes to +00. Letting

R = Rn and n go to +oo in the preceding equality yields the fact that u - 0 and thus Vp7 = 0, i.e. p is constant and our claim is shown. We thus begin with our analysis of the case when u°O = 0 and pO° > 0. In order to keep ideas clear, we consider the case when fl = RN and then explain how to modify the results or the arguments in the exterior case (case ii)) or in the tube case (case iii)). We thus wish to solve div (pu) = 0 in RN,

p > 0 in RN

pf +g inRN,

(6.143)

with the following conditions at infinity (to be interpreted mathematically appropriately)

p(x) - p°° > 0,

--+

as

1x1 -- +oo.

(6.144)

The data f, g are always supposed to satisfy f, g E Li fl L°° (RN). Less restrictive conditions are possible but we skip such easy (and technical) extensions.

Furthermore, we take N > 3 for rather clear reasons (due to the decay of Green's function for second-order elliptic operators): let us mention in passing that if N = 1 and p, u are "constant at infinity", integrating the momentum equation immediately yields fR p f + g dx = 0. Thus, if f > 0 and fR g dx > 0, there are no solutions, even if g =- 0, p - 0 if f > 0 in R and thus u is constant.

Stationary problems

130

Next, if N > 3, we shall always look for solutions (p, u) satisfying at least: L2N/(N-2)(RN), p E p°O + LP(RN) + Lq(RN) (when 1 < Du E L2(RN), u E p, q < oo are to be determined). These conditions are the precise mathematical (weak) translation of the boundary condition at infinity (6.144). We begin our analysis with the case -y > 3 and afterwards we shall look into more general exponents. Theorem 6.11 Let N > 3 and -y > max (3, 2) . Then, there exists a solution (p, u) of (6.3) such that Vu E L2(RN), U E L2NI(N-2)(RN), p - poo E L3(RN) n L°° (RN) if N = 3, (p - p°°) E L2 (RN) n Lq (RN) with q = N 2 (-y -1) if N > 4.

Remark 6.28 Of course, when N = 3 and -y > 3, we may apply the regularity results shown in the previous sections and we deduce that p E L°° (RN), curl u and div u - ' E W1,q (RN) l- {c2 E W1l' (RN) SuPyERN fy+Bl I,I q + IDcolq dx < oo}) for all 1 < q < oo and Du E BMO(RN). Remark 6.29 The behaviour at infinity of the solutions (p, u) is not clear. It is easily seen however that the best (that is the fastest) possible decay at infinity C.Indeed, if we take -r, lp(x) - p°°I < 777:7 is: Iu(x)I < Ix 2, IDu(x)I < first p = p°°, f = 0 and div g = 0, we thus find a solution u of

-µ 1u - eV div u = g in RN,

L2N/(N-2)(RN),

uE

Du E L2(RN)

and, in general, u decays at most like 1/IxIN-2 while Du decays at most like

1/IxIN-1 in general if g E Co (RN). Next, if u = g = 0 and f = V LNI(N-1)(RN))

W E L1 n LOO(RN), 4 E

(i.e.

then we may choose p to be defined

by

p7-1

= (P°O)ry-1 +

7

- 1 ID

a-y

and we see easily that the decay stated above is in general optimal. In view of these examples, it is natural to conjecture that u E L7I 0o(RN), Du E LN/(N-1),oo(RN) and p - p°° E LN/(N-1),oo(RN). O Proof of Theorem 6.11. Once more, the proof is divided into several steps. The first one consists in obtaining formal a priori estimates. Next, we build conveniently approximated problems (through a "double-layer" approximation) and we justify the a priori estimates derived in step 1. Finally, we conclude the proof in a third step passing to the limit.

Step 1: Formal a priori estimates. We first recall the "usual" local energy identity with a small modification, namely

div u

P

122

+a

1 Y

(pry-(P°°)ry-1P) 11 _µ0I 22 +pIDul2

(divu)2 = pu- f

inR'.

( 6.145 )

Indeed, we have incorporated the extra term div {u(-a(p°O)'1-'p)} which clearly vanishes since (6.3) holds.

Exterior problems and related questions

131

Next, we use the boundary conditions at infinity (6.144) and we deduce at least formally JRNII()= JRN

Using the Sobolev inequalities, and the assumptions on f and g, we deduce easily the following bound Hull L,

(RN) + II Dull L2(RN) < C (l + IIPIILO0+LQ(RN)) .

(6.146)

Here and below, C denotes various positive constants independent of u, p. Next, taking the divergence of (6.143) (the momentum equation) and using (6.144), we deduce easily that we have (at least formally) ap'y = a(p°O )"y + (µ+1;) div u + RjRj (pu=u j )

div (pf) - (-0)-1 divg

-(-O)-1

in RN

(6.147)

(recall that R, denotes the Riesz transform 88(-0)-1/2). In particular, we deduce IIP'IILo+Lq/ry(RN) < C 1 + IIDuIIL2(RN) + II PII L°O+L9(RN) IIUI12

(RN)

+ IIPIILo+L9(RN)

}

Notice indeed that when N = 3, q = 2, N Z = 6 and thus Z7 + .1 < 2 since N 7-1 < 2 and N-2 + 1 = N-2 C1 + 1 = z > 3, and when N > 4,

-

N-2 ry N q N Combining this inequality with (6.146), we finally obtain y

ry-1 J

q

IIPIII,o+L9(RN) <- C (1 + IIPIIL-+L9(RN) + IIPIILOO+L9(RN))

Since y > 3, this inequality shows that we have a priori bounds on p in L2N/(N-2)(RN) L°° + Lq(RN), on u in and on Du in L2(RN). In addition, we may now go back to (6.147) in order to deduce when N = 3 that p'7 - (p°O )'y

is bounded in (L3/2,o° n L°°) + L2 + L3 + Lr wherer 1 =q1 + 31 < ?; indeed, 3 p f and g belong to L' n Lq for some a > 3, puiu j belongs to L3 + Lr and

-

div u E L2. Therefore, p1' (p0O) f is bounded in L312'°° +L 3 . Notice also that by applying the bounds obtained in the previous sections, we also obtain a bound on p in L°°. In conclusion, p't - (p°°)'y is bounded in (L 3/2,o0 + L3(R3)) n L°O(R3).

-

Therefore, (p - p°°)1(jp-p°O jp°° /2) E L1(R3) (and in fact are bounded in those spaces). Finally, we have shown that p - p°O is bounded in (L1 + L3(]R3)) n L°O(R3) Similarly, when N > 4, we deduce that p1'-(pOO)1' is bounded in LN/(N-1),°0+ L2(RN) if y > N - 1, while if 3 < y < N -1 (and thus N > 5), p'7 - (p°°)" is LtN/(N-2)] 117' + L2(RN) since N2 Y = N-2 + N-2 1 < N-1 bounded in ry-1 N N ry-1 - N

Stationary problems

132

if and only if 7 > N - 1. Then, exactly as above, we deduce in both cases that p - p°° is bounded in (L' + L2 (R3)) n (Lq + LOO (R3)) with q = N (-f 1) . We then deduce bounds on p - p°° in L3(RN) n L°O(RN), L2(RN) n Lq(RN) when N = 3, N = 4 respectively using the following lemma (and its proof).

-

i) Let l < a < b < oo and let c E [1, oo]. If c < a or c > b, then

Lemma 6.4

(La+Lb)nLc=LdnLc where d= aifcb.IfcE [a,b],

then (La + Lb) n Lc = Lc. ii) Let 1 _< a < b < co, 1 < c < d < +oo. Then (La +Lb) n (L° +Ld) = Lb n Lc

ifb
Proof. We first observe that if f E La + Lb, there exist 91, 92 such that

IfI=91+92

,

g1ELa

92ELb

,

,

91,92>0 a.e.

(6.148)

Indeed, if f = fl + f2, fl E La, f2 E Lb, then If I = min(If I, if, 1) + (If I - Ifi I)+ and min(If I, Ifil) E La, (if I If1I)+ E Lb since 0 < min(I f I, Ifil) <- If1I E La, 0<(IfI-If1l)+ If2IELb. Then, if f E (La + Lb) n Lc, we see that 91 and g2 belong to L° since

-

-

191 I,1921 < If I

a.e. If c < a, then gi E La n L° and 92 E Lb n L` and thus

92 E La n Lc. Similarly if c > b, then gl E La n Lc and thus 91 E Lb n Lc while 92 E Lb n Lc. Finally, if c E [a, b], we just observe that Lc C La + V.

Next, if f E (La+Lb)n(L-+Ld), we see that 0 < gi < If I E Lc+Ld .(i = 1,2) and thus 91, 92 E Lc + Ld. Therefore, gi E La n (Lc + Ld) and 92 E Lb n (Lc + Ld) .

Using part i) of the lemma (already proven above) when b < c, we deduce that 91 E La n Lc while 92 E Lb n L°. Hence, f E (La + Lb) n Lc = Lb n Lc in view of part i). Finally, if a < c < d < b, we simply observe that Lc + Ld C La + Lb. 0

Step 2: Approximated problems and a priori bounds. Let a E (0,1] and let R > 1. We consider the following ("stationary") problem in BR:

div (pu) + ap = ap°O

in BR

div(pu(9 u)+apu-µ0u-l;Vdivu+aVpy = pf + g in BR,

(6.149)

u E Ho (BR) .

Applying the existence and regularity results of section 6.2, we obtain the existence of (p, u) solving (6.149) and satisfying in addition: p E LC* (BR) n L2y (BR), u E Woe (BR) for all 1 < p < oo if N = 3, p E Lq(BR) where q = N 2 (-y 1) if

-

N > 4 and

fBit

f

P°°Iu12

B R

2

a 71

+ ,y

pdx = P°°IBRI

(6.150)

+ 2 PIuI2 + µI DuI2 + t;(div u)2

(py - py-

i p°°)

dx <

(6.151)

pu . f + u g dx

B R

(with equality if N = 3).

Exterior problems and related questions

133

Our strategy of proof is to first let R go to +oo, keeping a > 0 fixed and then let a go to 0+. Therefore, we first look for a priori bounds uniform in R (but possibly depending upon a E (0,1]). To this end, we first observe that we have in view of (6.150) p7 _ poo ps-1 dx _

fBR

pti _ poop?-1 + (P°° )7 - P(P°° )7-1 dx f1BR

f

(P°°)''-1) (P - Poo) dx > 0

BR

In addiction, we may even deduce from this inequality the following lower bound

-1dx

JBR

(6.152)

J

p11(p>2p-) dX

IP-P°OI2 1(p:52p-) d2 + J BR

BR

for some v > 0. Then, we write JB R

pu f + u.g dx < II ulI L,. II9IIL,

+

where a + y + 2N

+ 2P°° IIuII L,3

IIP1(p>2p°°)IIL, IIuIIL,

IIf II L,a

IIfIILa

i.e.. + .1 = 2 + N (recall that

> 3). Therefore, we

have dx

<_ CIIDuIIL2 (1 +

IIP1(p-2p°O)IIL-,)

for some C independent of R. Inserting this bound in the equality (6.151), taking into account the inequality (6.152), we deduce that u is bounded in Ho (BR), PIuI2 and (p'r-1 - (p0O)7-1)(p - pOO) are bounded in L1(BR) uniformly in R > 0. In particular, in view of (6.152), (p p°O)2 1(p<2pc) and p" 1(p>2pm) are bounded in L' (BR) while p is bounded in LOO + Llf (BR) uniformly in R > 0. We next need to obtain further a priori bounds (essentially on p) in order to be able to pass to the limit as R goes to +oo (and then as a goes to 0+). In order to do so, we use systematically the observation made in Appendix E and deduce bounds on p'Y - f p' in various sums of LP spaces from the equation (6.149) and the bounds already shown. Before we do so, we first wish to estimate carefully f p" . Obviously, we have f pry > (f p) ry = (pOD) lf. Next, we remark that we also have (in view of the bounds shown above)

-

-

fB p'' dx < R

(LR

P-'-' dx P°° +

< (L, p" dx

('r-1) /'r

CR-N

poo +

CR-N

Stationary problems

134

<

1

'Y

'

p' dx + ry (P°O)ry + CR-N , R

hence

py dx < (p°O)y + CR-N , JBR

where C denotes various positive constants independent of R. In conclusion, we have shown the following inequality

fB Py dx - (p')' <

CR-N

.

(6.153)

R

We may now use the observation made in Appendix E to deduce further bounds (independent of R) on p using the bounds already obtained and the momentum equation in the system (6.149). We begin with the case N = 3: we then recall that Du is bounded in L2, p f and g are bounded in L6/5, and thus can be written as derivatives of functions bounded in L2. In addition, u 0 u is

bounded in L3 nL', pu®u is bounded in Li n (L31'/(1+3) +L3) = L' nL3y/(ry+3) in view of Lemma 6.4, while pu is bounded in (L2y/(1'+1) + L2) n (L6"/(y+6) + L6) L2 n L6,'/(y+6) using once more Lemma 6.4. Therefore, we find

py -

f

py dx R

< C (1 + IIPu ®ul 1L2)
Using (6.146), we deduce IIPII ioo+L2

C (1 + II PII L°°+L2ry)

and we have shown that p is bounded in LOO + L2y. We finally observe in this case that we have

I

IP

P°O12dx < f 1(p_<2p°°)IP-P°°12dx+

f

BR BR

R

R

B<


I

1(p>2p00)p2dx 1(p>2 per) py dx

BR

while

J I p- p°Ol2y dx < C R

1+1 R 1(p>2p0e) I P-P°O12y dx

C1+

I

BR

1(p>2p°°) p27 dx

< C.

In conclusion, we have shown that (p - p°°) is bounded in L2 n L2y. At this stage, we may apply the local bounds shown in section 6.3 and we deduce bounds on p

Exterior problems and related questions

135

in L0 (BM) and Du in LP(BM) (for all 2 < p < oo) for each fixed M E (1, oo), uniform in R >_ M + 1 and in M > 1. Next, when N > 4, we obtain in a similar way using Appendix E that p is

bounded in L' + Lq where q =

(7 - 1) and that p - p°° is bounded in

L2 n Lq. Let us finally observe that the arguments made in section 6.3 show that D(µ + ) div u - apry is bounded in L'(BM) or lf(BM) (if r < 1) for all

r < oo if N=3, and r = .1+1-N-L ifN>4andthus(p+l;)divu-apry is compact in LP(BM) for all p < oo if N = 3, p < N 2 ry-i if N > 4, for each fixed M E (1, oo), uniform in R > M + 1.

Step 3: Passages to the limit. We first let R go to +oo, denoting by PR, UR the solution of (6.149). Without loss of generality, extracting subsequences if necessary, we may assume that UR converges weakly in H1 to some u and UR converges in L oC to u for 1 < p < NN2 while PR converges weakly in Ll to some

p>0(q=27if N=3,q= N 2(7-1) ifN>4),andwehave pIul2ELi(RN), P - p°° E L2 n D. In addition, p E L°° (RN) , Du E LP (RN) for all 2 < p < 00 if N = 3. Finally, we may assume that div UR - µ+C (PR)" converges a.e. and in L PC for all 1 < p < q/7 in view of the bounds shown in the preceding step. Next, we wish to apply the results and methods of section 6.4 in order to conclude that PR is converging to p in Li for all 1 < p < q. However, we cannot just apply Theorem 6.4 because of the behaviour at infinity and we need to look into the proof. In fact, following the proof of Theorem 6.4, we only need to check

that

LN

u VW,,r dx - 0

as n --> +oo

where r = p - (_p8)'19 (recall that pe is the weak limit of pR, 9 E (0, 1) is chosen small enough) and Wn = Sp (n), cp E Co (RN), cp(x) = 1 if Ixi < 1. Obviously, PO - (pop)e E L2 n Lq/e since the set {pR < p°O/2 or pR > 2p°°} has a finite

measure bounded uniformly in R and the function (x - xe) is Lipschitz on [p°O/2, 2p°0]. Then, we deduce that (pe)'/e - p°° E L2 by a similar argument (it also belongs to L°° + Lq and thus to L' in view of Lemma 6.4). Therefore, r E L2 n Lq(RN) and we may apply (or follow) the rest of the proof of Theorem 6.4.

We may now pass to the limit in the equations (6.149) thus satisfied by (p, u). In addition, we deduce from (6.150)-(6.151) and Fatou's lemma the following:

f

N

2

#, IuI2 + 2 pIuI2 + µiDuI2 + (div u)2

+

a7

7

1

(pry-i

- (p0O)ry-i) (p-p00) dx < fit

(6.154) N

pu f +

dx .

We next need to pass to the limit as a goes to 0+, and we first have to obtain a priori bounds on (p, u) independent of a. In order to do so, we use the argument

Stationary problems

136

developed in step 1 above, the only difference being the terms involving a in (6.149).

Obviously, the inequality (6.146) is still valid since the terms involving a in (6.154) are non-negative. Then, going back to (6.154) and using (6.146), we obtain

a

r

1/2

N

IuI2 + pIuI2 dx <

J1z

N

pI uI2 dx

1/2

f I2 dx

JR + IIuIIL2(RN) IIgIIL2(RN)

hence 1/2

a

JAN

Iui2 + PIUI2 dx

< C (1 + IIPIILo+LQ(RN)) .

(6.155)

Next, since we know that p - p°O belongs to L2 n Lq(RN), we obtain instead of (6.147) apti = a(P°°)ry + (µ+6) div u + R;R1(Pu{uj)

-(-0)-1divg -a(-0)-1div(pu),

(6.156)

and, exactly as in step 1, we deduce the following inequality, II PII Lo+L9(RN) < C (1

+

IIPIILoo+L9(RN))

Notice indeed that aII PuIIL2+L'(RN) < C (aII,/TuIIL2(RN)) IIPIILa+L1q(1tN) where

-

2ry-1 N-2 if N > 4, 1 = 1 +- i and .1--L < z since 1 _ 1 N-2 < q2--L= -y-1 2N 2q N- 2N 2 N q 2q s 2

-

while a +-L- - 3 < 1 when N = 3. Therefore, we have obtained a priori bounds on p in L°° + Lq(RN), u in L2N/(N-2)(RN), Du in L2(RN), au in L2(RN) and apu in L2(RN) uniformly in a E (0, 1]. Furthermore, going back to (6.154), we deduce easily that _V/_a_pu

and /u are bounded in L2(RN) uniformly in a E (0, 11. In addition, repeating the argument made in step 1 above, we find that p - p°O = F + Vra- G where

F is bounded in L' + L''(RN) and r = 3 if N = 3, r = 2 if N > 4 while G L2N/(N-2) (RN) (uniformly in a E (0,1]). Finally, if N = 3, the is bounded in analysis performed in section 6.3 shows that p is bounded in LOO (R3 ).

We may now pass to the limit letting a go to 0+ and writing p = pa, u = ua. Extracting subsequences if necessary, we may assume without loss of generality that, as a goes to 0+, pa converges weakly in Li C to some p E L' + Lq (RN ) such that p - p°O E L' (RN) n (LOO + L3 (RN)) (s = oo if N = 3, s = q if N > 4), L2N/(N-2)(RN) and strongly in L O0 for 1 < p < NN2 to ua converges weakly in L2N/(N-2)(RN) satisfying Du E L2(RN). Finally, as in step 2 abovesome u E and in the preceding sections-we may assume that div ua - +C (pa)d' converges a.e. and strongly in L O0 for 1 < p < q. We conclude using Theorem 6.4 as we did

Exterior problems and related questions

137

above when we passed to the limit in R: indeed, r = p - (pe )110 E L3 n Lm(R3) if N = 3, E L2 n (L°° + Lq(RN)) if N > 4. Then, if N > 4, fRN Iu - OVnIrdx n f(n
dx)1/2

and thus fRN Iu Ocpn Ir dx goes to 0 as n goes to +oo. When N = 3, the argument is a bit more elaborate. First of all, we recall from the proof of Theorem 6.4 that we have

div (ur) > c (7)

77)

1

?0

in

V/(RN)

>1 where c =h+£ a 1-e e > 0. Thus, multiplying by Wn, we deduce for all n -

(-p7Pe) dx < C

(p)1-1

n J(n<jxj<2n)

Ln

Iulydx 113/2

<

n3/2 < Cn1/2.

(R3) n Next, we observe that we may choose 9 = 1/2 (using some arguments develIIUIIL6(FtN)

oped in Chapter 5 when studying the damping of oscillations for time-dependent

problems, we may in fact choose arbitrary 9 E (0,1)) since pa is bounded in L°° (R3) uniformly in a E (0,1] . Then, a simple use of Holder's inequality yields the following inequalities p = ;5:5 (p7+1/2)

1/2ry (_.j_)(27_l)/2

ry

1 7 < (p+1/2)(21"2'

hence pp7 < pry+1/2 p1/2 a.e. and (p1/2) (plf+1/2

7(___)1/27

- Pry pl/2) > p7 r a.e.

We thus find that f Ben pry r dx < CJ n1/2. Next, we recall that L3 (R3 ). Therefore, we deduce for all n > 1

f

n<jxj<2n)

rdx < C n1/2 +

r

(n
Ip7

a.e.

(poo)7 E

- (jow)-Y I rdx

< C n1/2 + IIP7 -- (P°°)9IIL3(n
J

<2n)

hence f(n
in a E (0,1] by uc, in view of the proofs in section 6.3), we may then conclude observing that we have

J(n
I ul r dx < C 1 r dx < Cn-1/2 . n (n<jx1<2n) J

0

Stationary problems

138

Remark 6.30 Let us point out that combining some of the arguments made in steps 2 and 3 of the above proof, one may obtain directly some a priori estimates on (p, u) uniformly in R E (0, oo), a E (0, 11-and more precisely the a priori estimates shown in RN for (pa, u,). We next mention without proof the analogues of Theorem 6.11 in the cases of an exterior domain or a tube-like domain. We skip the proofs since they are straightforward adaptations of the preceding proof, and we begin with the case of an exterior domain.

Theorem 6.12 Let w be a bounded smooth connected domain in RN, let .Q = RN - w and let N > 3. We assume that y > max (3, 2) . Then, there exists a L2N/(N-2)(SZ), u=0 solution (p,u) of (6.3) in SZ such that Vu E L2(SZ), u E p°° E L3(Q) n L27(SZ) n L°°(SZ6) for all 6 > 0 with SZb = {x E on 8 , p St / dist (x, i) > 6} when N = 3 and p-p°O E L2(Sl)nLq(SZ) with q = N 2 ('y-1)

ifN>4.

In the tube-like domain case, i.e. when fl = R x w and w is a bounded smooth

connected domain in RN-1 (N > 2), the results are quite similar with a slight modification due to the fact that Poincare's inequality is valid for such domains which are bounded in at least one direction.

Theorem 6.13 Let N > 2. We assume that y > max (3, 2) . Then, there exists

n a solution (p, u) of (6.3) in 11 such that u E Ho (St), p - p°O E L2(SZ) n L°O(SZb) for all b > 0 with SZb = R x w6i w6 = {x E w / dist (x, 8w) > a} when

N=2or3andp-pOOEL2(11)nLq(SZ)with qN 2(y-1)ifN>4.

We next explain how we may obtain another type of existence result for the same range of exponents y as in the case of a bounded domain. This will require the force term to be small (up to a gradient). More precisely, we write

f = VV + f, f, f E L1(RN) n L°O(RN)

(6.157)

and we shall assume that f is small. We only consider here the case when fl = RN, the other cases, namely the exterior domain case, or the tube-like domain case, being treated similarly.

Theorem 6.14 Let N > 3 and let y > 3 if N = 3, y > 2 if N >_ 4. If f is small enough in L1(RN) n L°°(RN), then there exists a solution (p, u) of (6.3) L2N/(N-2)(RN), satisfying u E Du E L2(RN), p - poo E Lr(RN) n Lq(RN) with

r=3,q=min (3('y-1),2y)if y>2,N=3and r=2,q= N 2(y-1) if N > 4, and p - p°° E L3 (RN) + L3(ti-1) (RN) if

< -y < 2, N = 3. 3

Remark 6.31 As can be seen from the proof below, one only needs f to be small enough in L2N/(N+2)(RN) n La(RN) wherea 1 = 2N N+2 - q'i

Remark 6.32 If y = 3, N = 3, exactly as in the preceding sections, there exists a solution (p, u) such that p e LO°(R3), Du E LP(R3) for 2 < p < oo.

Exterior problems and related questions

139

Sketch of proof of Theorem 6.14. We only present a sketch of the proof of Theorem 6.14 since it follows essentially the scheme of the proof of Theorem 6.11 with a few major modifications concerning in particular the a priori bounds. Observing that we have (at least formally) Du12 JRN pl

+ (dlv u)2 dx < fR N pu f + u g dx ,

we deduce the following variant of (6.146), namely II uIIL2N1(N-2(RN) + II DuIIL2(RN) <- C (EOIIPII L-+L9(RN) + 1)

(6.158)

where eo = IIf II L1nL- (RN) . We then follow the argument made in step 1 of the

proof of Theorem 6.11 and obtain IIPIILo+L9(RN) -< C (1 + IIPIIL-+L9(RN) + EOII PII LOO+L9(RN)) .

(6.159)

Of course, we may restrict our attention to the case when y > 3 (and thus in particular 3 _< N _< 5); indeed, the case -y > 3 is contained in Theorem 6.11 and if y = 3, we just take co small enough and deduce from (6.159) a bound on p in L°° + Lq(RN) and the rest of the proof is the same as in Theorem 6.11. Next, when -y > 3, we deduce from (6.159) that we have for co small enough IIPIILo+L9(RN) < Cl(EO)

or

IIPIILOO+L9(RN) > C2(E0)

(6.160)

where Ci(eo) < C2(Eo) are respectively the smallest and the largest positive roots of [xy = C(1 + x + Eox3)]. Let us remark that Ci(eo) converges to the positive root of [xl' = C(1 + x)] while C2 (eo) (Cc 2)1/(3-y) converges to 1 and in particular C2(eo) converges to +oo as co goes to 0+. The strategy of the proof is to show by a continuity argument that a solution satisfying the first (upper) bound exists for co small enough. In order to do so,

we consider a solution (p, u) of (6.149) with f replaced by ft = VV + ti where t E [0,1]. From the arguments and methods introduced in section (6.7) we may even assume that we have a continuum C of solutions (p, u, t) in L4 X Ho x [0,1]. Note there is here a minor technicality due to the fact that the continuum is included in Lq x Ho x [0,1] but the topological properties (closed and connected) hold for the strong topology of L" x W01 'r x R for p < q, r < 2. This difficulty is circumvented by observing that the alternative (6.160) is easily shown to be valid with Lq replaced by LP where p < q is close to q. This is why we shall ignore this problem in this sketch. Next, we claim that the alternative (6.160) holds for any such -(p, u) and for all R > 1 (large enough) and a E (0,1]. This claim really follows from the proof of Theorem 6.11 and the Remark 6.29. We just need to observe that the term

LR pVV udx = LR aV(p-p°°) dx

Stationary problems

140

can be easily bounded by a fB,, (p7-1 - (p°° )7-1) (p - p°°) dx. Indeed, without loss of generality, we may assume that V is bounded and V E LN/(N-1) (RN). Hence, we have o

f

BR

<

I V I IP-P°°I 1(IP-P°OI
2

J

IP-P0I21(IP-PI
IlZ

aB(pry-1

<

4

- (POO)ry-1) (p-p°°) dx + Ca

R

provided we fix b > 0 small enough, and where C denotes various positive constants independent of a, R, t, p, u. In addition, we have for all b > 0, denoting y = min ('y, N), a

f

BR

IVI

IP-P°°I 1(IP-P°°I>poo/2) dx

P_b

ab

JB R 4

f

I

P°OI71(IP-Pool>Poo12) dx +

(pry-1

Ca S1/(7-1)

- (P°O)ry-1) (p- p') dx + Ca

R

choosing once more b > 0 small enough.

Not only does this show that (6.160) holds but also it shows that, when t = 0, then IIPIIL-+L4 < C1. But this implies that for all (p, u, t) in the continuum C we must have II PI I L°° +LQ < Ci; indeed, (6.160) imply that the sets {(p, u, t) E C / IIPIILo+LQ < C' 2C2 }

and {(p, u, t) E C / IIPIIL-+L9 > C'

C2 }

make a disjoint open covering of the continuum C, and since the first one is non-empty, the second one has to be empty. Then, (6.160) implies our claim. At this point, we have shown the a priori bound we needed in order to be able to reproduce the proof of Theorem 6.11. 0 We now conclude this section with a few additional results that we state only

in the case when 0 = RN, the extensions and adaptations to the cases of an exterior domain or of a tube-like domain being straightforward. Once more we look at the situation when u "goes to 0" as IxI goes to +oo while p "goes to a constant". But, instead of prescribing the "value of p" at infinity, namely the constant p°O above, we are going to use different parametrizations in a slightly similar way to what we did for stationary problems on bounded domains in section 6.7. More precisely, we introduce a positive function on RN denoted by w such that: w E L1 n L°°, f'RN w dx = 1, inf essBR w > 0 for all R E (0, oo), I f I < Cw a.e. on RN for some C > 0. Such a function clearly exists: take for e-IXI2 instance w = (If I + e-Ix12)(fRN If I + dx)-1. The type of parametrization we shall use is described by the following constraint

Exterior problems and related questions

fR N

141

pw dx = M

(6.161)

where M > 0. Our main result is then

Theorem 6.15 Let N > 3, let -y > 3 if N = 3, ry > 2 if N > 4. Then, for each M > 0, there exists a solution (p, u) of (6.3) satisfying (6.161) such that L2N/(N-2)(RN), Du E L2(RN), p E L°O(RN)+Lq(RN) with q= N 2 (y-1) 4L E

ifN>4or3

E

Lr (RN) n Lq (RN) for some p°O>0with r=3if N=3,ry>2,r=2if N=4,

p - p°O E L3(-'-1)(RN) + L3(RN) for some pO° > 0 if N = 3, 3 < -t < 2.

Corollary .6.1 Let N _> 3, let ry > 3 ifN = 3, y > Z ifN > 4. Then, there exists p E (0, +oo] such that, for each p°O E [0, p), there exists a solution

(p,u) of (6.3) such that u E Lr(RN) n Lq(RN) (or L3(RN) +

L2N/(N-2) (RN),

L30y-1)(RN)).

Du E L2(RN), and p - p°O E

Remark 6.33 It is an interesting open question to decide whether we can take p = +oo in the preceding corollary. Of course, Theorems 6.11 and 6.14 provide examples of situations where indeed we know that p = +oo. Remark 6.34 It is possible to show that Corollary 6.1 still holds for a < 7 < 3 if N = 3. The proof of this claim follows closely the proofs of Theorem 6.15 and Corollary 6.1 above: one builds continua of (approximated) solutions satisfying fR, ppw dx = M for some large p. This constraint allows us to obtain a priori bounds on u in , Du in L2, p and u in L , Du in L o° for all 1 < q < L2N/(N-2)

oo, and p - p°O in L3 + L3(,y-1) for some p°O > 0. As seen from the proofs below, we only have to check that f R, pn w dx converges to fR3 pew dx as n goes to +oo

if pn satisfies the preceding bounds uniformly and if pn converges to p in LP10 for all 1 < p < oo. This is the case since fBR pnw dx converges to fBR p'yw dx as n goes to +oo for all R E (0, oo) while we have (1-9)'y

0-f/P

pn w dx <

pnw dx

J(Ixl> R)

(L,>RPnwdx)

where B + 1 - 6 = 1 P 'Y

r

J3

e7/P

w dx

,

_ IIwIIL3/2f1L3(y-1)/(3ti-4)(lxl>R)

and the right-hand side converges to 0 as R goes to +oo uniformly in N > 1. 0

Sketch of proof of Theorem 6.15 and Corollary 6.1. We first explain how we can obtain a priori bounds. We only have to modify a little bit the proof made in step 1 of Theorem 6.11. Indeed, we now estimate the integral

f

RN

pu f dx < CII DuII L2(IItN) IIPwII L2N/(N+2)(RN),

Stationary problems

142

and we deduce as in step 1 of the proof of Theorem 6.11 the following bound II DuII L2 +

< C (1 + II PW II L2N/(N+2))

IIuIIL2N/(N-2)

Next, we use (6.147) which holds for some p°O > 0 which is also an unknown and we write (6.162) e = (p°")" + F1 + F2

where F1 E L2, F2 E L" with p = N 2 2171 (or p = 2 if N = 3, 'y > 3) and IIF1IIL2 <-C(1 + II PW II L2N/(N+2))

,

(6.163)

IIF2IILP <
IIPWIIL2N/(N+Z))

In particular, we have II

PII

(1 + (P°O)" + II PW II L2N/(N+2) + IIPIIL°°+L9 11pW112 L2N/(N+2) ) II PII L-+L9

and 9

1IIPWIIL98

IIPWIIL2N/(N+2) < IIPWIIL1

with 0+ 1 q-9

N+2 2N

< <' IIPIIL00+L4

using the fact that f RN pw dx = M is fixed, and we thus deduce IIPIILx+LQ < C(1+(p°O)Y+ IIPIIL0+La)

where a = 1 + 2(1-0) < 7 thanks to the restrictions on 7-this fact was already used in order to obtain bounds on solutions of stationary problems in bounded domains. In particular, this inequality implies that we have in view of (6.163) IIPIIL°D+L9 < C(1+p°O),

IIF11IL2 < C(1+p°O)3,

IIF2IILP < C(1+p°°)°` (6.164)

where 0 = 1 - 0 E (0, 1). In order to obtain a bound on p°O, we are going to use once more the constraint (6.161) and we write using (6.162) P

,

I r, (P°O),

I F'2I (P°°),

11'r

+

Then, in view of (6.164), we have

meal

IF1I1 (P

)

+

(pI F2I)7

>2

< meas {IF1I >

(p°O)1 } + mess {IF2I > - (p°O)ry} 4

Exterior problems and related questions


{(1+poo)(3(p°°)-7

143

+ (l+p°O)a(p°°)-ry}

1, otherwise the bound on p°° is proven. Therefore, if p°° > 1, we have

M = I pw dx > p°O .N

JAcnBR

w dx 2-1l y

for some measurable set A whose measure is bounded by a constant C and where R is chosen in such a way that meas (BR) = 2C. Hence, fAcnBR w dx > (inf essBR w) C > 1/C and we have shown that p°O is bounded and thus, in view of (6.144), p is bounded in L°O + Lq, Du is bounded in L2, u is bounded in L2N/(N-2). From these bounds, we also deduce the fact that p - p°O is bounded in IT n Lq (or L3(7-1) n L3 if N = 3, 3 < ry < 2). The rest of the existence proof (Theorem 6.15) is pretty much the same then

as the proof of Theorem 6.11 and there is no need to provide further detail at this stage. Corollary 6.1 follows upon the observation that one can build, as in the preceding section, continua of solutions (or approximated solutions). Then, the key observation is to remark that p°O is continuous for the topologies with respect to which we obtain a continuum. This is done by observing that we obtain sequences of solutions (or sets of solutions) which are bounded in 2N (LOO +Lq) x L2N/(N-2) and compact in LP c x LO for all 1 < p < q, 1 < r < N-2 Then, we go back to (6.147) and we obtain the following identity

ap. = j ap'1wdx N +

rN(-0)-1

j (p+C)divuwdx - j RR?(puju,)wdx N

N

div(pf)w +

f(_L1 (divg)wdx.

Our claim then follows easily by arguments similar to the proof made in Remark 6.33 observing for instance that JRNR1R,(pujuj)wdx = f N

puju,(RRjw)dx

and that R,RRw E LP(RN) for all 1 < p < oo. 0

Remark 6.35 We wish to mention here some open questions concerning another stationary problem which, even though it is set in a bounded open smooth domain SZ C RN (N > 2), presents some similarities with exterior problems. We consider the stationary problem (6.3) in Q with the following boundary condi-

tions: u = uo on BSZ, p = po on {x E 9) / uo n(x) < 0} where n denotes the unit outward normal to acz, uo and po are given functions on &Q. The existence of stationary solutions in such situations is essentially an open problem.

Stationary problems

144

6.9 Regularity of solutions We begin with an example that shows that, even though all the data are smooth, we cannot expect the density p to be continuous or in W1,P for any p > 1 and the velocity field to be C' or in W2,P for any p > 1. As we shall see later on, this is related to the possible presence of vacuum since we shall prove that, in the absence of vacuum, the density p is (Holder) continuous and the velocity field is at least of class C'. Example 6.4 Let SZ be the unit ball centred at 0 (= B,) in R2 (for instance). We are going to build a solution (p, u) such that p - 1 Bi z and u E CO0(B1/2 UBi/2), u V C'(B,). Indeed, we choose p(x) = 1 if Ixi > 2= 0 if Ixi < .1 and we first take an arbitrary divergence-free vector field u E C°O (B1) or even Co (Bi) such that x Vii(x) = 0 on 8B112. Then, we set u(x) = u(x) if 2 < Ixi < 1. Next, we take = 0 in order to simplify the presentation even though the construction is easily modified to all cases where + p > 0, it > 0. Then, we choose u1 E C°°(B1/2)

such that ui(x) = u(x) if lxi = 2 and (2 V)ui = -µ j if Ixi = 2, and we set u(x) = ui(x) if Ixl < 1. Notice that u is Lipschitz on B1, but u ' Ci(B1), u ¢ W 1.1c (Bi) and p ¢ W11-.1 ,1(B1). Next, we define g to be an extension to B, of

-pLu1 such that g E Co (B1). Finally, we define f to be an extension to B, of -µou+ (u V)u - g E C°°(Bi/2) such that f E C°°(B1) and thus f E CO' (B1). Since the flux pu FXT vanishes as Ixl - 2 and as Ixi - a -, we see that we have

div (pu) = 0

in

Bi

while by definition of u, p, f, g we have both on B1/2 and on BI1/2

pf +g Finally, we conclude by observing that this equation holds on B1 since the jumps on 8Bi/2 cancel

(.v) ui(x) + a[1- 0]

-µ 1(1XX1 O u(x) - (1XX1

X

=0

1

by construction. 0 Example 6.5 We want to show here that the above construction can be adapted to the full compressible model, namely

div(pu) = 0 (pu

V)T - kLT + (y-1)(divu)pT = 1(auj +a;uti)2 - (divu)2.

(6.165)

Indeed, we choose once more p, u on B1 and u on B112 as in Example 6.4. Next, we wish to build u on B1/2. Writing x = (r, 0) where (r, 0) are the polar

Regularity of solutions

145

coordinates, we choose cp E C°° ([0, 2]) such that V vanishes for r small, w (2) 0 and cp' = 1 for r close to 2. We are looking for ul defined on B112 by

_

ui(r,6) = r T (2,0) co(r) Next, we wish to solve the temperature equation with the appropriate boundary conditions, say, for example, T = 1 on 8B1. We thus look for a solution T of

-kiT + (pu . 0)T = 2 82uj +ajui)2 - (divu)2 in Bi

,

T=1 on 8B1 i

where u is defined by ui in B1/2 and by u on Bi12 so that u is Lipchitz on B1. In

_ O(r) E Co ({ < lxi < 1})2 extended by 0 to B1-and look for T = T (r), then u = * T (2) W(r) for r < a fact, if we further choose u =

(-82") awhere

and (azuj + ajut)2 - (div u)2 is a radial function while (pu 0)T _= 0. In this case,2we simply solve

-kLT = [2 (auj + 8jui)2] + T (2)2 a(r)1(r<4) in BR ,

T=1 on aBi

where a(r) = [2 (8uj + ajU,)2 - (div u)2] 1(r<) and u = r (p(r), i.e. a(r) _ 2

[(`°r r )'J r2. This non-linear equation can be solved at least if k is large enough (for u, cp fixed) by a simple fixed point (or implicit functions theorem) argument.

We find T = T(r) > 1 on B1 such that T E W2'°O(B1) and T E C°O(B1/2 U Bi12). Furthermore, ui E COO(B112) and we define g to be an extension to Bi of

-Dul such that g E Co (B1). Finally, we define f to be an extension to Bi of -Du + (u - V )u + VT - g E COO(B112) such that f E C°°(B1). We then conclude checking as in the previous example that the equations (6.165) hold in B1.

We now turn to regularity results and we begin with the simpler cases when there are no boundary conditions, namely the periodic case or in the case of the whole space. We then consider a solution (p, u) of (6.3) such that p E L' (R'), u E Wuni (RN ), div u - µ+t p" E Wi; (RN ), and curl u E Wi; f (RN) for all 1 < p < oo. We denote Wurif = {u E Woop(RN), U E Lunif(RN), Du E Lunif(RN)} and Lunif (RN) _ {u E L OC(RN), SupyERN (fy+Bi JuI" dx)i/P < oo}. In order to simplify the presentation, we assume that f and g are smooth (Cb (RN) say). Finally, we assume that there is no vacuum, i.e. in

inf ess p > 0.

(6.166)

Let us point out at this stage that we simply assume that y > 0 and that we have shown in the preceding sections existence results of solutions in two and three dimensions having the regularity assumed above at least for some range of exponents y. It remains to decide when (6.166) holds, and we postpone the

Stationary problems

146

discussion of (6.166) until the end of this section where we present some situations

where solutions are known to satisfy (6.166). Of course, Example (6.4) above shows that p may not be continuous if (6.166) does not hold.

Theorem 6.16 Under the above conditions, p and Du are bounded and (uniformly) continuous. Furthermore, if we denote 8 = IA+f (inf ess #f (IDull Loo -1 , k = [8] (the integer part of 0) and a = 8 - k E [0,1), then we have

pECb'a,

UECb+l,aifa

pEC6-1,Q

0;

(6.167)

u

-µ+

curl u E Cb'Rfor all (3 E (0, 1) if a = 0

Remark 6.36 In the case when a = 0, the proof below shows that p satisfies IDk-lp(x)

- Dk-lp(y)I

<- Clx-yl I log Ix-yIl

if Ix-yI < 2.

Remark 6.37 For a general pressure law, i.e. if we replace ap"r by then the above result holds (with essentially the same proof) provided we assume that p is of class C1 and non-decreasing on [0, oo) and that v = inf {p'(t)t / inf ess p < t < sup ess p} > 0. We then need to replace in the definition of 0 the quantity a-y(inf ess p)'y by v. We only have to make the following observation in order to adapt the proof of Theorem 6.16 below: let q E C' ([a, b]) where a < b E R and assume that 0 < v < q' < L + v < oo on [a, b] for some positive constants v, L. Then, we have for all x E [a, b] q(x)

inf {q(s) - vs + L(x-s)+} + vx

sE [a,b]

inf

sE[a,b]

o
{q(s)

- (v+z)s + (v+z)x} . 0

Remark 6.38 Let us mention that Theorem 6.16 above also -holds for timediscretized problems (6.1)-(6.2). The proof below is easily adapted to that case assuming of course that h > 0 is smooth and replacing 0 by the following quantity

a

µ

II Du Ii L min

,ty + h (X)

x E RN

,

inf ess p < t < sup ess p

0

Proof of Theorem 6.16. The proof is rather delicate and we split it into several steps.

Step 1: log p is the unique bounded solution of a semilinear Hamilton-JacobiBellman equation.

Regularity of solutions

147

We set f = log p. Since p is bounded and satisfies (6.166), then f is bounded. Next, at least formally, we have log

p = (u.Vp)p-1 = -divu =

µ

+ p7 - divu - +

Thus, we find

+be"f = cp

D'

in

(6.168)

where b = +{+{ > 0, cp = div u - bp'y E C6'3 for all 0 < P < 1 by assumption. The justification of the equation (6.168) is straightforward by a standard regularization (a very simple particular case of regularizations such as those done several times in the preceding sections and chapters) in view of the regularity of u (E Wun f for all p < oo) and of the boundedness of f. Let us next observe that u E Cb(RN), divu E LOO (RN) (since p E L°°(RN)),

curlu E L°°(RN) and thus, by standard results on Riesz transforms, Du E BMO(RN). Indeed,

-Dui = 83 Fi3

where Fib E L°O(RN)

for all

1
and thus 8kui = RkR,,(FiJ) E BMO(RN) for all 1 < i, k < N. Next, we consider the equation (6.168) as a non-linear equation for f where cp is given. The non-linearity acts only on f and is strictly increasing and convex and thus can be seen as a semilinear Hamilton-Jacobi-Bellman equation at least formally associated to a particular deterministic control problem described below in step 2. Finally, we prove that (6.168) has at most one bounded solution. More precisely, we are going to show that if fl, f2 E L°° (RN) satisfy in D'

(6.169)

then fi > f2 a.e. on RN. Indeed, we first observe that we have in D'

u V(f2-fi) +V(f2-fi) <_ 0

where v = by exp [-y(inf ess f i)] . Using once more a simple regularization argument, we deduce 0

inD', for all 1
Then, in the periodic case, we simple integrate this inequality over the period fl and we find

pv / (f2-fi)P. dx < II(divu)+IIL°O

fn

in

P (f2-fi)+dx.

Hence, choosing p large enough, we have shown that f2 < fi a.e. In the whole space case, we slightly modify the preceding argument and argue by contradiction. We thus assume that sup essRN (f2-fl) > 0. Therefore, for 6 small enough, we have, denoting by <x>= (1 + (x12)1/2,

Stationary problems

148

sup ess

(f2-fl-6 <x>)

RN

2 supessRN (f2-fl) > 0.

Then, we observe that we have

V{f2-fl-6 <x>} + v{f2-fi-6 <x>} < S [IIuIILc - V <x>]

u

<- Co 6 1(Ixi<_xa)

for some fixed (independent of S) positive constants Co and Ro. In addition, since f, and f2 are bounded, (f2 - f, - 6 < x >) + is bounded and has a compact support. We may then make the same proof as in the periodic case and we find

I

(vp- If(divu)+IIL°°)

(f2-fl-S <x>)+dx N

1(1x1)+ ldx

<- CoSp J'N

< CoSp

f (f2-fl-S <x>)+dx

(P- l ) / P M1/P

JRN

for some positive constant M (= IB1I R4N). Therefore, we deduce

v-

f

/'

1

II(divu)+IIL°O

1/p

N(f2-fl-S <x>)P. dx

<

C0SMI/P.

6, and we reach the desired contradiction with the assumed lower bound on the supremum. Letting p go to +oo, we obtain sup essRN (f2 - f j - S < x >) _<

Step 2: Construction of a continuous value function. We now wish to build the value function of the control problem associated, at least formally, to the equation (6.168)-see for instance P.-L. Lions [3461 for more details on the relationships between value functions of deterministic control problems and solutions of Hamilton-Jacobi-Bellman equations. First of all, we recall that u is bounded and Du E BMO. Therefore, as is well known, u satisfies

Iu(x)-u(y)I < CiIx-yf I log Ix-yII for all Ix-yi < 1

(6.170)

for some positive constant C1. This continuity allows us to consider the solution of the following ordinary differential equation Xx = -u(XX)

for t > 0,

Xx(0) = x E RN

(6.171)

,

and this solution is unique. In addition, one deduces easily from (6.170) the following bound:

IXx(t)_X (t)I < exp {(log Ix-yf )e-c1t}

if0 < t < To,

for all X, y E RN such that I x - y I < 2 , where To is defined by log log 1

Ix

(6.172)

- y I e-c1 To =

Regularity of solutions

149

Next, we introduce the following value function defined on RN by

{J00 .f (x) = inf

[co(xr)+b['ye(t)_1]e0(t)J exp [_jtf.ye0(8)ds]

(6.173)

e meas. from [0, oo) into [ao,13o]}

where ao = inf ess f (= log (inf ess p)) and Qo = 7 log IkaIILLet us immediately point out that, if we know that u were Lipchitz, then-see for instance P.-L. Lions [346]-f would be uniformly continuous and would be the unique (viscosity) solution of the associated Bellman or Hamilton-JacobiBellman equation, namely

uvf+

sup a0 <9
[i'° + rye-ye (.f -9)]

= W.

(6.174)

Let us also observe that choosing 0 = (3o, we deduce from the definition (6.173) of f that we have I (x) < Ao for all x E RN. In addition, we shall see below that

f > f a.e. on RN and thus f > a on RN (since we are also going to show that f is continuous). Combining these two inequalities, we shall deduce that the above supremum in equation (6.174) reduces in fact to elf thus explaining the equality

between f and f that we shall address later on. We finally show that f is (uniformly) continuous on RN. In order to prove this claim, we follow the method of proof introduced in P: L. Lions [346] and write for all x, y E RN and for all T E (0, oo) 00

If(x)-f(y)I <sup f I (p (Xt) - (p (Xf )I exp (-fot 'iebel'lds ds/0 -- [ao,/3o]

r

<

0

I '(Xt) -V(Xy)I exp (-'yea°t) dt

f Iv(Xt) - w(X') I e-"tdt+Ce-vT where C denotes here and below various positive constants independent of x and y E RN. In particular, we deduce from (6.173) and from the regularity of p that we have for any fixed 0 E (0,1) and for all T E [0, To]

f(x)-f(y)C

JT

exp {/(log Ix-yl)e-clt-vt} dt+Ce-"T

< T exp {f3(log Ix-yI) e-c1T} +

Ce-vT

Next, we choose T = K log log x 17 with K > C1 and we observe that T < To if we choose K large enough and Ix - yI < -11. Hence, we deduce for all x, y E RN

with Ix-yI <4

Stationary problems

150

1f(x)-f(y)1 < C log

(6.175)

1x 1

y

for some 6 > 0 which we may always assume to be in (0,1) (in fact, 6 = K Let us also observe that f is clearly bounded since cp is bounded and 0 is bounded by definition. 0

Step 3: f = f and thus p is continuous. We are first going to show that (6.174) holds in the sense of distributions. One possible way to prove this claim

consists in regularizing u by convolution uE = u * tc6 (for e E (0,1)) and approximating f by fe defined in the same way as f replacing u by uE. Of course, we denote as usual s;E = - ri. () with n E Co' (RN), x > 0, Supp rc C B1, fRN , dx = 1. We thus introduce the solution XE = Xx'E of

exp

Xe = -uE(Xe) for

t > 0,

X6(0) = x E RN

and the value function fe defined, for all x E RN, as follows

ff(x) = inf

-Jo t'Yeh'9ds

0 meas. from [0, oo) into [ao, Ao]}

.

Then, exactly as in [346], fe is bounded, uniformly continuous and the unique viscosity solution of 'ue

V1,6

+

sup a0:50<'00

[e"° +

7e-Y19

(fE - 0), = V

in RN .

In particular, fe is also a viscosity solution of ue

Vir

= cP -

sup

ao<9<00

[eye +'ye'le(fE-0)]

in RN ;

and, as is well known in viscosity solutions theory (see the original proof in M.G. Crandall and P.-L. Lions [3461), this equation also holds in the sense of distributions.

We next check that fE converges uniformly on RN to 1. This convergence implies easily that (6.174) holds in the sense of distributions. We first observe

that X' converges to X uniformly in x E RN, t E [0, T] for all T E (0,00). Indeed, we have for 0 < E < e-1 : 1ue(x)-u(x)1 =

J[u(y)_u(x)] cE(x-y) dx _< C1E log 1 ,

Therefore, we deduce for all x E RN, t > 0, e E (0,

a-11

for all x E RN

.

Regularity of solutions

IXe-XI < C,

IX£-XI Ilog IXE-X II +E

151

loge

IXE-XI(s) < 2

if

for all s E [0, t]. This inequality easily yields for all x E RN, E E (0, a-1] if t E [0, Tel

I XE -X I < me

where me is the solution of mE = Cl [mc log

me(0) = 0

,

1 me

+ e log 1

me(t) E (o, a]

for t E [0, T e]

for t E [0,TE]

,

mc(TE) = 2

A simple argument allows us to check that TE goes to +oo while me goes to 0 uniformly on [0, T] for all T E (0, oo) as e goes to 0+ We then deduce the uniform convergence on RN of P to 1. Indeed, we have for all x E RN, by a similar argument to that in step 2 above T

Ife(x) -.f(x)I : CJ
IXE-XIQdt+Ce-vT

0

rT

I0

mc(t)O dt + C e-"T

where ,Q is fixed in (0,1), T is arbitrary in (0, oo), v is defined in step 2 above and

C denotes various positive constants independent of c and x. The convergence of f E follows upon letting first E go to 0+ and then T go to +oo. At this point we have shown that f E BUC(RN) satisfies (6.174) in the sense of distributions and thus satisfies in particular

+be-'f > W

in V.

This allows us to apply the comparison result shown in step 1 above and we deduce in particular that !':2: f a.e. in RN. As explained in step 2 above, this in turn shows that f also solves (6.168) and thus, using step 1 once more, we finally

deduce that f - f . In particular, p - of is continuous and since f is bounded we deduce from (6.175) the following estimate P(x) -P(y) I :5 C (log min

(

- yI 12)1)

x1

for all x, y E RN.

(6.176)

0

Step 4: Proof of Theorem 6.16 in the case when 9 E (0,1). We recall first that curl u and div u - bp7 E Cb (RN) for all f E (0,1). In particular, u = Vir +v where ir E C6'Q(RN), V E C610 (RN)N for all 8 E (0,1) and divv = 0. Therefore,

we have A7r = F where F - bpi E C. (RN) for all 0 E (0,1); and, neglecting

Stationary problems

152

smooth contributions due to infinity (or the easy adaptation to the periodic case) we find for some constant CN which only depends upon N

u = cN JN I

Y

F(y) X(x-y) dy + uo

in RN

(6.177)

uoECb(RN) for all /3E(0,1), where X E Co (RN), X is spherically symmetric, X 0 1 near 0 say on a ball B1. We shall then use several times the following lemma whose proof we postpone until the end of step 4.

Lemma 6.5 Let F satisfy (6.176). Then, Du is uniformly continuous on RN if 6 > 1 and if 6 E (0,1] we have for some positive constant C I u(x) -U(y) l

x-ylll-a

CI x-yI Ilog l

if 6 E (0,1)

(6.178)

Iu(x)-u(y)I < C'Ix-yl log(Iloglx-yll) if6 =1 for all x,y E RN such that Ix-yI < 1. 0

We are next going to prove that Du E BUC(RN). In order to do so, we deduce from step 3 (estimate (6.176)) that F also satisfies (6.176) since p is bounded. Therefore, assuming without loss of generality that 6 E (0, 1), the first estimate in (6.178) holds, thus improving the continuity estimate (6.170). From this improved estimate, we deduce easily the following improvement of (6.172), namely for all x, y E RN such that Ix - y I < 1

I Xz(t)-X1(t)I < exp

-

if 0 < t < To

log Ix yI

where To is defined by To= [-(log 2)b + (log lx 1 v )61. Then, we deduce as in c all x, y E RN such that Ix - yI 2 and for all step 2 above that we have for

TE(0,To) 1/b

b

I f (x)-f (y)I < CT exp -Q

- CT

log ix yi

+

Ce-vT

,

for some f3 E (0, 1) .

In particular, for each ic E (0, oo) fixed, we may choose for Ix-yI small enough

T = n log (log

x

1 y) and we find e

If(x)-f(y)I <- C'Ix-Y1 +C (log

I'-Y1

In other words, f and thus p, p1f and F satisfy (6.175) or (6.176) for all 6 > 0.

Regularity of solutions

153

Therefore, using once more Lemma 6.5, we deduce that Du E BUC(RN). In particular, u is Lipschitz. Then, we deduce from the results and methods of proof of P.-L. Lions [346] that f E Cb (RN) for all 0 < a < 1, a < 0. Therefore, p, pry, F and Du belong to Cb for all such a. This regularity combined with the regularity arguments developed in sections 6.3 (and 6.2) easily yield the fact that cp = div u - bp'' and curl u E C"' for all a E (0, min(1, 0)). In particular, cp is Lipschitz and applying the results of [346], we deduce that f and thus p, p'r, F and Du belong to Cb if 0 E (0,1), Cb for all 0 < a < 1 if 0 = 1. The rest of the claim in (6.167) then follows repeating the argument above. 0 Proof of Lemma 6.5. Without loss of generality we may assume that uO 0. Next, let C E Co (RN), is spherically symmetric, C(x) = 1 if Ixi < 1, e(x) = 0 if IxI > 2, and we denote E(x) = C (M) for all x E RN, for all C E (0,1]. We then write for all e E (0, 1) and for all x E RN

2 y F(y)

u(x) = CN +CN

(x --y) dy

x-y

]RN

JN Ix-ylr'

and we denote b uE (x) the first integral and u2 (x) theeecond integral in t1ie

right-hand side. Next, we have by symmetry for all x E R and for all e E (0, 4 )

N (F(Y)-F(x)) (x-y) dx

IuE(x)I = LIRN Ix

_< Cl log

yl eI-a

1(1x-yl<2e) dy _CCI log EI

N-1 JRN Ix-1 YI

where, here and below, C denotes various positive constants independent of c and x. In addition, uE is obviously Lipschitz continuous (in fact even Cb) and we have for all xERN and for all E E (0, ID uE(x)I

=

JAN

F(Y) D

(1-eE)(x-y) X(x-Y) dy

Ix yIYN x-

1N(F(Y)-F(x)) D

YI

N

X(x-y)

I

r


J(E

(log [mmn

1

(lx I Y1

2

Ix

Iloglx-yll-6Ix-YI-Ndy

(E
= C+CJ

I

Ilog rl-6r-'dr

E


if6E (0,1),

< C if6> 1,

-1YIN dy

dy

Stationary problems

154

< C log I log EI

if b = 1.

Therefore, if 5 > 1, we deduce easily that Du E BUC(RN). If b E (0, 11, we obtain for all x, y E RN and for all e E (0, 4 ) Iu(x)-u(y)I <- Cel log el-a + Clx-Y1 I log EI1-6 ell-8

by log I log EI when b = 1. Then, choosing e = I x - yl , we deduce (6.178) for all x, y E RN such that I x - y l < 1, an inequality which is obviously equivalent to the conclusion of Lemma 6.5. 0 replacing I log

Step 5: Conclusion. In fact, we are going to explain how to prove Theorem 6.16 when 0 E (1, 2). Indeed, this proof will make clear how we can go above 0 = 1 and repeating the argument we develop below allows us to treat the general case. The proof already made shows that f, p, pr, Du <'OE C6 (RN) for all 0 < 8 < 1 and < 1. In fact, if 0 > 1, using the thus div u - bprr, curl u E C"' (RN) for all 0 estimate shown in [346], we see that f , p and p7 are even Lipschitz continuous. Therefore, D2u E BMO since curl u E Cb'Q and divu E Wb'°O. Next, we differentiate (6.168) with respect to Xk for each k E {1, ... , N} and we find

(u.V)8kf +c8kf +(8ku- V)f = 8kw

in D'

where c = 6ye'r1. Notice that c E W 1,°°(RN) and that c > v = II DuII

(6.179) (inf ess p)"7 _

L'0.

Then, exactly as in steps 1-3 above, we show that ek f is the unique bounded solution of (6.179) and coincides with the following continuous explicit expression

akf(x) =

rt exp (- / c(Xx(s))ds dt

1000

(6.180)

o

where

k is the solution of N

E jaju(Xx)

for

t>0

(6.181)

j=1

satisfying (0) = ek (the k-th vector of the canonical orthonormal basis). In addition, 8k f E BUC(RN) and satisfies (6.176) for some b > 0. A simple way to convince ourselves that (6.180) does provide a natural candidate for the solution of (6.179) is to use a method due to N.V. Krylov [3131. We define an extended function on RN x RN, f, (x, ) = V f (x) and we deduce from (6.179) that f, satisfies the following equation N

eiaiu =1

.

f = C - DAP,

(6.182)

Regularity of solutions

155

and, we should of course expect f, to be given by 00

rt

f i (x, i) = fo V (X x (t)) - C(t) exp -

J0

c(X X) ds dt

for all x, i E RN

where C solves (6.181) with (0) = rl. Of course, (6.180) corresponds to the particular case where 71 = ek.

We then argue as in steps 4-5 using Lemma 6.5 twice to obtain that D2u E BUC(RN) and then that V E Cb'a for some a > 0. In particular, Dcp is Lipschitz continuous. In order to conclude, we adapt the argument in [346] and write for all x, y E RN and for all T E (0, oo)

IDf(x)

- Df(y)I <- C

f

T

0

00

+C

a-"t dt

Ix-yl + JT

[I

(t)I + 117(011

a-vt dt

where , i are the solutions of (6.181) corresponding to X1 and X Y respectively. Next, E we show easily by Gronwall's lemma that we have for all t > 0 and for all

x,y

-

RN

Ie(t)I <- eIID+LIIcet

Irl(t)I < ellDull.t ellDt4 t

IXx(t)-X"(t)I < e1IDull0tlx-yi (e014iot_i)

IlDuiloo

IID2uIIoIx-yI

Therefore, we obtain for all x, y E RN and for all T E (0, oo)

IDf(x)-Df(y)I <- C {c(2IIDuuoo-v)TIx-yI +e(IlDull.-v)TJ Recall that IIDuIIo < v < 2IIDufl, since we assumed that 0 E (1,2). Then, if Ix - y I < 1, we choose T =

u

log l and we deduce

IDf(x) - Df(y)I <- CIx-yla where a = IlD u ll. - 1 = 0 - 1. The rest of the assertion contained in Theorem 6.16 follow as in step 5 above. 0

In the case of a problem set in a bounded (smooth, open, connected) domain 11 of Rhr, we have to circumvent the difficulties already encountered in the preceding sections. More precisely, in the case of Dirichlet boundary conditions, we do not know how to deduce regularity informations up to the boundary on div u - " p' IA+C

and curl u from -pAu - V div u + aVp7. In particular, we do not kn ow if

[-µ0u - CV div u + aV pi' E CA (St)] implies that [div u - µ+£ e, curl u E CQ (SZ)]

for any /3 E (0, 1). If it were the case, the result above (Theorem 6.26) would hold mutatis mutandis. Of course, in the case of the boundary conditions (6.37)(6.38) or (6.37)-(6.39), or (6.37)-(6.40)-we do not need here the matrix A to be non-negative-arguing as in section 6.3 we see that the above difficulty is solved. So we state, without any further detail on the proof, the following result.

Stationary problems

156

Theorem 6.17 In the case of Dirichlet boundary conditions, we assume that div u - +E pry and curl u E CO (11) where 0 E (0,1). Then, p E C' (N) and (inf ess p)ry > II Dut I Loo, a = II Du I L- if u E Cl,* (Sl) where a = f3 if v = v < IIDulIL- and a is arbitrary in (0,3) if v = IIDuIILOO. In the case of the boundary conditions (6.42)-(6.43), or (6.42)-(6.44) with N2µ or (6.42)-(6.45) with > -N, the results contained in Theorem 6.16 still hold.

We conclude this section by giving a few examples of situations where we know that stationary solutions satisfy (6.166), i.e. are bounded from below. The first type of situation concerns steady flows that are close to constant flows (p constant, u - 0). Many results of that sort for various kinds of boundary value problems can be obtained using different methods and techniques. In particular, in some cases, one may use directly the implicit functions theorem and obtain directly "smooth solutions". We just sketch here one possible approach that we illustrate in the case of a bounded (open, smooth) domain Sl C RN with N = 2,3 with the boundary conditions (6.42)-(6.43), or (6.42)-(6.44), or (6.42)-(6.45)as in section 6.3-and we conclude the same assumptions as in Remark 6.15,

section 6.7. In addition, we assume that y > 1 if N = 2 and -y > 3 if N = 3. Then, we claim that if M is fixed in (0, oo) and if f, g are small in Lm (for instance) then the solution (p, u) with fn p dx = M built in section 6.7 which belongs to LOO (n) X W l,q (SZ) (for all 1 < q < oo) satisfies (6.166). Indeed, as shown in section 6.7, if 11f I I Loo , II9I I L- < 1, then we have for any q E (1, oo) IIPIIL00 + IIuIIw1,9 < C

where C > 0 depends only on M and q. In addition, by the energy identity, IIuIIHI goes to 0 as 11f 11L- + II9IIL- goes to 0, and thus IIuIIwI.9 goes to 0 as 11f 11L- + II9I I L- goes to 0 for any 2 < q < oo. From the momentum equation, one easily deduces that V pry converges to 0 in W-1,1 and that p, pry converge respectively to mew f2 and (meas cz )ry in L'' for all 1 < r < oo. In addition, by the arguments shown in section 6.3, we see div u - µ+E [pry - (meas )ry] converges to 0 in W i,' (SZ) as 11f II L- + 1191I L- goes to 0, for all 1 < r < oo. In particular, div u - $ [pry - (mess 57 )ry] converges to 0 uniformly on U. At this stage, it is easy to check that p is bounded from below on SZ as soon as f and g are small enough in Lc*. Indeed, we simply write U. V(log p) +

a P. =

µ+t;

a

M

p+l;

meas(Sl)

ry

+ cp > d > 0 ,

for some d E R,

if f and g are small enough in L°°. Then, the same type of comparison argument as the one shown in step 1 of the proof of Theorem 6.16 implies the following inequality d

P

(o+)a

1/ry

Regularity of solutions

157

and our claim is shown. Of course, this is slightly formal since we do not know a priori that log p is bounded but the lower bound we obtained can easily be combined with the existence proof. The second case we wish to mention concerns the situation of large mass, i.e. when fo p dx = M and M is large (all other data being fixed). In order to give one meaningful example of the approach, we keep the same boundary conditions (with the same assumptions) as in the example considered above and we assume now that N = 2 or 3 and that y is large enough (-y > 3 213 if N = 2 for instance). In order to keep ideas clear, we only present the argument when N = 2. Then, the strategy of the proof is to show that 11divu- E+ (pT - n p7) (IL°° = 0(M^9 as M goes to +oo. Then, we argue as above and write

e + divu -

u V(log P) + a Pry = a µ+C µ+l;

(X+

a

>

J

+l; a

>

M"Y

- p+ meas(SZ)'r

a p+1;

(Y_f PP7 a

divu - p+l;

p7

p7

+ o(M").

We then deduce a lower bound on p exactly as in the case considered above. Next, we follow the a priori bounds shown in section 6.7 and check easily that if -y > 3 and M > 1 then we have IIUIIHI(n) + 111011V-y(n) < CM

where C denotes various positive constants independent of M > 1. Then, we turn to the regularity proofs made in section 6.3, keeping track in all the bounds of the dependence upon M, and we obtain successively the following bounds for all M > 1

divu -

a

p,y

< CM1 LP

and then IIuIIwI,P < CM'

,

IIPIILP1' < CM

for all p E (1, 2y) where C depends upon a lower bound on 2y - p. Next, if r > 2 (close to 2), we deduce

II(Pu V)UIILr < CM2 IIDuIIL. < where

1 and 2 +

1

p

0

=s

CM2+e+(1-e)ry

Stationary problems

158

-

f-0 pry] in W 1,'' and thus in LOO by [p1' Since we can estimate div u +C (pu 0)u in L'', we only need to have the following inequality in order to conclude

2+0+(1-0)'y <'y,

i.e.

2 < 0(-y-1).

On the other hand, it is enough that this inequality holds when p = 2-y, r = 2 in which case we find by a straightforward computation that 2 < 0(y - 1) if and only if 'y >

13

3 %

6.10 Related problems This section is devoted to two topics related to the problems considered in the preceding sections. We consider first the case of general pressure laws, i.e. systems of equations where the pressure term apt' is replaced by a general function p on [0, oo) that we always assume (in order to simplify the presentation) to be normalized by p(O) = 0 and to be continuously differentiable on [0, oo) with p' locally Holder continuous on [0, oo)-even though this condition can be somewhat relaxed in particular at 0. The crucial requirement we impose upon p is the following

p is non-decreasing on (0, oo) . (6.183) We have already seen in section 5.5 (chapter 5) that this condition seems necessary for our approach to compactness issues for Cauchy problems, and this is also the case for stationary problems. Let us also mention that such a condition is meaningful from a physical viewpoint since it is a consequence of the principles of thermodynamics (see chapter 1 of volume 1). Let us finally observe that when the viscosity coefficients vanish (P = C = 0), i.e. when we consider the inviscid case (compressible Euler equations instead of compressible Navier-Stokes equations), the condition (6.183) is precisely the one that ensures the hyperbolicity of the system. We thus consider the following problem: i) time-discretized problems

div (pu) + cep = h, p > 0 , div (pu®u) + apu - POu - V div u + Vp(p) = Pf +g

(6.184)

and ii) stationary problems

div (pi) = 0 div

u

p>0 Du

-

7V

div u + Vp(p)

(6.185)

where h > 0, f and g are given. On each of those problems, we mention below various existence results without stating any theorem (or proposition), and we simply explain the main modifications and adaptations to be made in the proofs introduced in the previous sections. Let us begin with a general observation which yields the fundamental energy

identities that are basic to obtain some a priori estimates. We have (at least formally)

Related problems

159

with q'(t)

u Vp(p) = div[uq(p)] - div(pu) q(P) P

In particular, if fo

- qt t) = p'(t)

on

(0,00). (6.186)

ds < oo, then we may-and we will-choose q(P) = p

p

P'(s)

ds.

(6.187)

Jo

In general, we may choose q(p) = p fl ''s' ds for all p E (0, oo) and then q is no longer non-negative on (0, oo) while it clearly is in the preceding case ((6.187)). Obviously, if p(p) = ap"r and a > 0, y > 1 then (6.187) yields q(p) = p", while if -y = 1 (the isothermal case) we use the second rule and find p(p) = p log p.

Finally, if p(p) = app' and a > 0, 0 < y < 1, then the second choice yields q(p) = a 11,Y (1 - p-0-10); of course, since we assume p to be Cl on [0, oo), we shall no longer consider this slightly more singular case. In view of the regularity we assumed on p, the condition fo p 1' ds < oo is simply equivalent to p'(0) = 0. In everything that follows, we assume that p'(0) = 0. This assumption plays in fact no role in most of the results discussed below, except when 11 = RN. In this case, if p'(0) > 0, then we may choose q (p) = p' (0) p log p + p fo s s p ° ds

and the lack of positivity of q or equivalently of p log p can be handled by

somewhat more technical arguments similar to those introduced in our study of the isothermal case (in two dimensions) in section 6.6. We next discuss the existence and regularity of solutions of the time-discretized problems (6.184). The results contained in section 6.2, namely Theorems 6.1-6.3, are still valid for the problem (6.184) provided we assume that h E L°° n L' (SZ) and that p satisfies for some y > 1

lira inf p'(t) t-++oo

t-l-'-1)

> 0.

(6.188)

In addition, whenever in Theorems 6.1-6.3, we obtain some Lq integrability upon p then we have in addition in this case that pi/1' E Lq. Let us mention that the assumption upon h made above is only a simplifying assumption and that h E L°° may be replaced by various (depending on which of the analogues of Theorems 6.1-6.3 we consider) integrability requirements that depend on p in a more technical way. The proof of this claim follows closely the proofs of Theorems 6.1-6.3 made in sections 6.2-6.5. We only explain two points concerning a priori estimates and one concerning the crucial compactness properties of (sequences of) solutions. About

a priori bounds, one obtains obviously as usual the global conservation of mass which yields an L1 bound: fo p dx = a fn h dx; and, one obtains bounds on Du in L,2 on p in L", on q(p) in L1 and on (p+h)Iu12 in L' by the following (formal)

Stationary problems

160

energy identity easily derived from (6.184) upon multiplying the momentum equation by u and using (6.186)

f

DuI2

(P+h)l u12 + µI n 2

+ e(div

u)2

+ aq(P) dxx

(6.189) P

J12

The second modification concerns the bootstrap argument used to obtain "regularity" (that is the improved integrability) of p and Du. Using Lemma 6.3 (section 6.7) with cp = , together with the method introduced in section 6.3 to obtain similar estimates without assuming p bounded, it is then easy to adapt the arguments introduced in section 6.3. The final adaptation needed in the proof that we wish to mention concerns the compactness results obtained in section 6.4 and more precisely Theorem 6.4. Of course, the setting remains the same replacing (pn)7 by p(pn), assuming in addition that p(pn) is bounded in Lgl7. Then, we claim that Theorem 6.4 holds if a > 0 with essentially the same proof. The only modification concerns (6.67) which is now replaced by 9

a(pe)s"e+div {u)h1'0} > h+ 16 b {p(P)Pe - p(P) Pe}

(6.190)

Next, we use Lemma 5.3 (section 5.5, chapter 5) and we deduce

a(p8)119 + div {u()"°} > h.

(6.191)

Therefore, we deduce with the notation of the proof of Theorem 6.4

div {ur} + ar > 0 ,

where

r = p - (pe)", > 0

and we conclude easily that r = 0 a.e. and thus pn converges strongly to p. Let us emphasize the fact that this proof relies upon the strict positivity of a and thus we shall have to come back to this compactness issue when studying stationary problems (6.185).

We now turn to stationary problems (6.185) and we immediately mention that the analogues of Theorems 6.7-6.10 are still valid assuming (6.188) and that p is strictly increasing on [0, oo) (exactly as above, p E Lq is now replaced p E Lq and p(p) E Lq/7). The only new point in the proofs is the compactness analysis which corresponds to Theorem 6.4 in the case when a = 0. In this case, we wish to deduce, following the proof of Theorem 6.4, the strong convergence of pn to p from the following identity {P(P)PO - p(P) Pe

}

=0

a.e. on

{x,> 0}

.

Let us observe that (pn)e converges to 0 in L1({pe = 0}) and thus the preceding equality is clearly equivalent to

Related problems

p(P)PB

- p(p) pe = 0

161

a.e.

Let us then recall that Lemma 5.3 (section 5.5, chapter 5) implies that this quantity is non-negative as soon as p is non-increasing: the argument below will in fact yield a "different" proof of this fact. We claim now that, when p is strictly increasing, it vanishes if and only if pn converges strongly to p. Indeed, denoting we have (Pn)e = pi, pe = Pl, pl(t) = p(P)PB

- p(P)PB

= pi(Pi)P1 - p1(Pl)Pl

= [pl(P1) -pl(Pi)][pi - P11

or in other words this quantity is nothing but the weak limit of IIn = (pi (pi) i) which is clearly non-negative if pl is non decreasing and strictly P1(Pl)) (P1 positive for pi # pi if pi is strictly increasing. Therefore, the above expression vanishes if and only if IIn converges strongly to 0 (in L1) and our claim follows

-

easily.

Remark 6.39 The strict monotonicity of p is crucial in order to assess that bounded sequences of solutions of the problem (6.185) are compact (in some LP space). Indeed, if p is constant on an interval [a, b] (C [0, oo)) with a < b, then

we may choose f - g = u - 0 and p to be any (smooth or not) function taking only values in [a, b]

!

We now turn to stationary problems (6.185) set in unbounded domains and more precisely to the settings studied in section 6.8. Under the condition (6.188) and assuming of course that p is strictly increasing, then the results obtained in section 6.8 can be readily adapted to (6.185). The main modifications concern the integrability properties of p and p(p). Indeed, if we take, as an example, the case L2N/(N-2)(RN), of Theorem 6.11, then one obtains similar a priori bounds: u E Du E L2 (RN), p E L°° + Lq (RN) and p(p) E L°° + Lq/-' (RN) where q = if N = 3, q = N 2 (-y -1) if N > 4. If N = 3, then one shows also that p E L°O (R3 ) and, following the proof of Theorem 6.11 in section 6.8, we obtain a bound on p(p) - p(p°°) in (L3/2'°° + L3 (R3)) n LOO (R3) = L3(R3) n L°° (R3) . Then, if p'(pOD) > 0, we deduce a bound on p - p°O in L3(R3) n L°°(R3). If p' (p°°) = 0, the situation is slightly different. For instance, when we assume that p satisfies in a neighborhood of p°O : Ip(t) - p(p°°) I > Sit - p°O l"` for some m > 1, 6 > 0, then we deduce a bound on p - p°O in L3m(R3) n L°O Similarly, when N > 4, we obtain a bound on p(p) - p(p°O) in LN/(N-1),°° + L2 (RN) if ry > N -1 and in Lg17 + L2 (RN) if ^y < N -1. Therefore, if p' (p°O) > 0 then p - p°° E L2 n Lq(RN) while if p satisfies the above condition near p°O then P - P°° E (L' + L2m(RN)) n (Lq + Loo(RN)) Let us conclude this brief study of problems with general pressure laws by recalling that the regularity of bounded solutions can be studied exactly as we did in section 6.9 in the pure power case. The necessary modifications of the arguments introduced in section 6.9 are explained in Remark 6.36 (section 6.9). Let us observe that the analysis requires p' to be bounded from below on compact sets (R3).

Stationary problems

162

of (0, oo), a condition that can be viewed as one form of the strict monotonicity required for compactness results of solutions of stationary problems.

The second topic we briefly address in this section concerns the case of "Stokes" equations which correspond to neglecting the term p(u.V)u (or div (pu(& u)). In order to restrict the length of this presentation, we only consider the case

of stationary problems in a bounded open smooth domain S2 of RN (N > 2) namely

div(pu) = 0,

pf +g

p> 0,

with the normalization (for instance) fn p dx = M E [0, oo). It is easy to check that the arguments introduced in the preceding section yield the existence of a solution p E L2"Y(12), u E H' (S2) as soon as y > N+2 with either Dirichlet bound-

ary conditions (respectively one of the three variants (6.42)-(6.43), or (6.42)(6.44), or (6.42)-(6.45)). In addition, any such solution satisfies p E Ll (St), u E WWo` (12), (µ + ) div u - ap'r E Wl (S2), and curl u E WI q(S2) for all 1 < q < oo (respectively p E L°° (S2), u E W l,q (S2), (µ+e) div u - ap'' E W.(12), and curlu E Wl,q(S2) for all 1 < q < oo).

6.11 General compressible models We only wish to discuss in this section stationary problems for the full compressible Navier-Stokes equations (with a temperature equation, or a possibly nonconstant entropy) with Dirichlet-type boundary conditions in a bounded smooth open domain fl C RN (N >_ 2). We shall not address here time-discretized problems, which in fact may be written in various forms depending upon which unknowns are being used, like density, velocity and temperature or entropy or total energy, and of course everything we do below can be adapted to appropriate periodic cases as we did in section 6.7 (Theorem 6.8). We thus consider the following system of equations div (pu) = 0

in S2,

p> 0 in 2

,

(6.192)

div(pueu)-µ0u-eVdivu+V(pT) = pf+g in St , u-n = 0 on 812, (6.193) div (puT) + (-y-1)(divu)pT - div (kVT) = 6[2pldl2 + (e-p)(divu)2] in n (6.194) The vector fields f, g are given say in L°°(S2)N, -y is a constant in [1, oo), µ and

are positive constants and µ > 0, k is a non-negative constant or possibly a non-negative function of T and finally b will be either (-y-1) or 0. We denote, as usual, d = (Du + Dut), the deformation tensor, and of course T stands 2 for the temperature which will always be a non-negative function. Notice that this notation allows for the non-physical constant y = 1 and that 8 = (^y -1), -y > 1 correspond to the correct equations from a physical viewpoint while S = 0 corresponds to the classical (at least for non-hypersonic gases) assumption which consists in neglecting the heating due to viscous friction. Of course, the quantity

163

General compressible models

2pIdI2 + (l

- p) (div u)2 should always be non-negative and this is equivalent

to requiring that p and 2p + N(C - p) > 0. We have already assumed that p > 0 and we assume-in order to simplify the presentation and avoid further technicalities-that we have 2p + N(e- p) > 0, i.e. C > N2 p. We now have to present the boundary conditions we add to the system (6.192)-(6.195) and more precisely to the equations (6.193)-(6.194) for the velocity field and the temperature. For the velocity field u, we either impose Dirichlet

boundary conditions (u = 0 on aft) or one of the three variants used several times before, namely (6.43) or (6.44) or (6.45). Boundary conditions for T are slightly more delicate and we only mention one meaningful possibility here in order to illustrate our methods. We impose the following boundary condition

8 + A(T -To) = 0

on

8St

(6.195)

where A is a non-negative constant and To is a given non-negative function on

all say in L' (all) bounded away from 0. Let us recall that we denote by n (= n(x)) the unit outward normal to all at x, and of course, if k = 0, then we do not impose any boundary condition on T. Before we start discussing some results about the above system of equations (and boundary conditions), we wish to recall from chapter 1 (volume 1) some classical identities involving either the total energy or the entropy. In fact, at least formally-or in other words if p, u, T are smooth and T > 0-the equations we shall obtain either for the total energy or for the entropy are equivalent to (6.194), provided of course that p and u solve (6.192) and (6.193). First of all, multiplying (6.193) by u and writing [-p0u-e0 div u = -2p div ddiv u] we obtain div

pu

+

'Y

1T

- 2p div (du) - (C -p) V (u div u)

ILH-1-

-div(kVT)+ 1= pu f +

(6.196)

8

7-1

in

fl .

Next, we deduce from (6.194) and from (6.192)

log T+(-y-1)(divu)p - div

k TT =

kI T2I2 +

and thus we find

div(pus)-div

TT = kI TT (1c)

I2

+T [2pIdI2+(t;-p)(divu)2]

where we denote the entropy by s = log T - (-r -1) log p = log (per} .

(6.197)

164

Stationary problems

We are going to investigate several cases in this section. We begin with the case when b = 0 , k = 0. In this case (6.197) reduces to

div (pus) = 0 . Then, in view of (6.192), we may simply take s - so E R (the entropy is constant!). Then, we find, in view of the definition of s: T = apps-1 where a = es0, and we are back to the situation studied in the previous sections, namely pT is replaced by apl. Next, we consider the case when k = 0 , b = (7-1) > 0. We then claim that in general there are no solutions; indeed, integrating (6.197) over n, we deduce easily that u - 0. Therefore, the whole system (6.192)-(6.194) reduces to V(pT) = p f + g, and we reach a contradiction as soon as g is not a gradient vector-field (curl g # 0) and f - 0. Even if g = 0, there are no solutions or at least no non-trivial solutions (p # 0); indeed, if, for example, f is given by (0, x1), 0 E SZ and N = 2, then any solution would satisfy

0 = curl (pf) = -a(pxi) i.e. p =

for some a E V. Then, p E L' implies that a

a(x2)

a.e.

0 and thus p - 0. Another

case we wish to consider is the case when b = -f - 1. First of all, if we assume in addition that k = 0, then (6.193) reduces to

div (puT) = 0, and we may choose T to be a positive constant, i.e. T > 0 and we are led to the isothermal case, i.e. the case -y = 1 with the notation of the preceding sections. This problem was studied in section 6.7 if N = 2 or 3 where we obtained, with the boundary conditions (6.43) or (6.44) or (6.45), the existence of a continuum of solutions (p, u) with fn pP dx = M E [0, oo), p large enough (and p E L°°, U E W 1,4 (f) for all 1 < q < oo). In particular, see for more detail Theorem 6.10 in section 6.7, we obtain for some interval [0, Mo) and Mo E (0, oo], the existence of solutions (p, u) with the above regularity and fn p dx = M. Next, still in the case when -y = 1, we assume that k > 0 and is a constant even though we might consider as well quite general positive functions of T. First of all, if a = 0 (respectively if To is a constant), then we may choose T to be any positive constant (respectively T = To > 0) and exactly as before we are back to the isothermal case. Next, if A > 0 and To is not necessarily a constant, then T solves the following elliptic equation

inn , a +A(T-To)=0 on all,

(6.198)

and we deduce from the maximum principle that we have 0 < inff ess To < T < sup ess To an

a.e. in Sl

(6.199)

and multiplying (6.198) we easily deduce that T is bounded in H1(f) (independently of (p, u) satisfying of course (6.192) and (6.193)). Then, arguing as in

General compressible models

165

section 6.7 (Theorem 6.10), it is not difficult to show the existence of a continuum of solution (p, u, T) in LQ(S1) X W1,q(11) X W2,q(11) (for any q E [2, oo)) such curl u E W 1,q (!a) and that: p E L°°(S2), Du E BMO, divu - µ1 pT E W

T E W2,q(cl) for all q E (1, oo), T satisfies (6.199) and for each M E [0, oo) there exists a solution (p, u, T) such that fn pP dx = M where p is chosen large enough (see section 6.7 for more details). In particular, we deduce the existence of some Mo E (0, +oo] such that, for all M E [0, Mo), there exists a solution (p, u, T) with the above regularity satisfying fn p dx = M. We need to make precise that, as in Theorem 6.10, we assume that N = 2 or N = 3 and that we use one of the three boundary conditions (6.43)-(6.45) which allow us to obtain the "regularity" of

div u - ,+z pT and curl u "up to the boundary". The two last situations we study below where 'y > 1, k > 0 and 6 = -y - 1 or 5 = 0 are more delicate and also more meaningful from a physical viewpoint. For reasons similar to those recalled above, throughout the rest of this section we shall make all the same assumptions as the ones we just recalled above and more specifically we assume that N = 2 or N = 3 and we impose the boundary condition (6.44). And exactly as above or as in section 6.7 (Theorem 6.10), we shall study (6.192)-(6.194) with the boundary condition (6.195) and the following normalization:

in

pP dx = M

(6.200)

where M is arbitrary in (0, oo) and p will be chosen large enough later. Finally, we assume that A > 0 since otherwise there does not exist in general a solution. Our main result is the following.

Theorem 6.18 If N = 3 and 6 = ry - 1, we assume in addition that k depends upon T and satisfies (1+T)O1 > k(T) > v(1+T)°C

for all T > 0 with a > 1 , for some v E (0,1) .

(6.201)

Then, for any p large enough, there exists a continuum C of solutions (p, u, T) of (6.191)-(6.194) in LP(Q) x W1"p(Sl) x W2,P(S2) such that: p E L°°(f ), u E W 1,q (f1) for all 1 < q < oo, Du E BMO,

div u - pT and curl u E W 1,q (11)

for all 1 < q < oo, T E W2() for all 1 < q < oo and inf-ff T > 0. Furthermore, for any M E [0, oo), there exists a solution (p, u, T) E C such that fn pp dx = M. In addition, there exists Mo E (0, +oo] such that there exists for all M E [0, Mo) a solution (p, u, T) E C such that fn p dx = M.

Remark 6.40 In the above result, we have taken, when N = 2 or when N = 3 and 5 = 0, k to be a positive constant. In fact, the proof below yields the same result if k depends upon T and (6.201) holds (for example) for some a > 0. We do not know if this assumption suffices when N = 3 and 6 = 'Y -1.

Stationary problems

166

Remark 6.41 As in all the results of this type obtained in the previous sections, we do not know whether the critical constant (mass) Mo is finite or infinite. This is of course a fundamental open question.

Remark 6.42 We have already shown in section 6.9, Example 6.4, that in gen-

eral solutions (p, u, T) with the regularity stated in the above result are no longer regular if there is a vacuum (p ¢ C, u ¢ Cl). On the other hand, if there is no vacuum, i.e. p is bounded away from 0 (assumption (6.166)), then the analogue of Theorem 6.16 holds with almost the same proof; we skip the straightforward modifications and adaptations. The exponent 0 is now given by

0 = +£ (inf ess p) (mina T). In addition, T E C,b +2,« if a # 0, T E Cb+" for IL all ,8 E (0,1) if a = 0.

Remark 6.43 If we replace (6.44) by the boundary condition (6.43) (respec-

tively (6.45)), then we have to replace the right-hand side of (6.194) by 6[p(curl u)2 + ( +µ)(div u)2] (respectively S[j I DuI2 + (div u)2]). With these modifications, Theorem 6.18 still holds in these cases. In addition, if we impose (6.45) and if we keep (6.194) then Theorem 6.18 is still valid provided we assume

that A -kin > 0 on 811 where x is the curvature tensor. 0

Sketch of proof of Theorem 6.18. Since most of the proof is similar to the proof of Theorem 6.10, we only explain how to obtain a priori bounds. In addition, the case when 6 = -y - 1 being more delicate than the case 6 = 0, we detail the a priori estimates in the first case and then explain the modifications required to treat the latter case. Furthermore, we begin with the two-dimensional case and then analyse the three-dimensional one.

Step 1: A priori bounds when S = -y - 1, N = 2. We first integrate over 11 the energy identity (6.196) and the entropy identity (6.197). Using the boundary condition (6.196) and the normalization of p, we then deduce JTdS < C (1 + IItIILP')

n

I TT12 2

fn

+

IDT12

fn

,

dx < C,

(6.202)

(6.203)

where C denotes various positive constants independent of p, u, T. We are going to deduce from these two inequalities bounds on T in La (f') for all 1 < q < oo. Indeed, we first observe that (6.203) implies that V log(1+T) is bounded in L2(1l) and thus, by an easy functional analysis observation, log(1+ T) - fan log(1+T)dS is bounded in HI (Q). By classical Sobolev embeddings in two dimensions, we deduce for any q fixed in [1, oo)

C>

jexp q(1+T)

- fan log 1+T dS dx = I(1+T)q -C°

where Co = fan log(1+T)dS. Then, we remark that we have in view of (6.202)

General compressible models

0< Co = fan log(1+T)dS < log

f

167

n(1+T)dS < log{C(1 + IIUIILA')}

Therefore, we deduce IITIIL4 < C (1 +

(6.204)

IIuIILp')

In particular, we may choose q = 1 and we deduce from (6.203) for all e E

(0,1),e>0 2

IDuldx< In

2E

Tdx J IDTI dx+2fn

2 E

+ CeIIDuIILI

since we may always assume without loss of generality that p > 2 (and thus p' < 2). We then deduce from this inequality a bound on u in Wl"l(ft) and from (6.204) a bound on T in Lq(fZ) for all q E [1, oo). Going back to (6.203), we deduce for all r E [1, 2) IDTI2 n

IDul''dx < C

+T

dx < C.

At this stage, we have shown that T and u are bounded in Lq (1) for all

1 < q < oo and that u is bounded in W'() for all 1 _< r < 2. We then use (6.193) and elliptic regularity to deduce that u is bounded in Wl,'' for all

1
We next choose p > 2. In particular, u is bounded in C(SZ). In addition, we use (6.194) and we may apply elliptic regularity and the strong maximum principle (or Harnack's inequality) to deduce that T is bounded in C(Q) and that we have

miin T > C

(6.205)

.

This lower bound allows us to copy the arguments introduced in section 6.3 and to prove using (6.192) and (6.193) that u is bounded in Wlq(fZ) for all 1 < q < oo, p is bounded in L°° (ft), Du is bounded in BMO, div u - + pT and curl u are bounded in W 1,q (f Z) for all 1 < q < oo. Finally, we deduce from (6.194) using elliptic regularity that T is bounded in W2,q(fl) for all 1 _< q < 00 (we write div (puT) = pu VT in view of (6-192)). 0

Step 2: A priori bounds when b = ry - 1, N = 3. We follow the preceding proof and obtain in place of (6.202)-(6.203) because of the growth condition (6.201)

fan T1+` dS < C (1 + IIUIILp') an

,

(6.206)

Stationary problems

168

2

fT2IDTI2dx+f

dx

C1+

f

n

TadS

(6.207)

.

Therefore, we deduce in view of (6.207) and (6.206) IITIIi3? <

CIITa/2IIH1

< C [IIDTa/2IIL2 + IITa/2IIL2(an)] 1/2


L TadS

Hence, if p > 3 (and p' < 2) and if a > 3

< r IDTI2

Jn

+Tdx < C 1+Ja

L) < (1 +

(1 +

n

TadS+IIuIIflDul

In particular, u is bounded in W1,1 and T is bounded in L3a. In fact, if a > u is also bounded in W1,' where 22r = 3a. We next wish to prove that u is bounded in H1 and pT is bounded in L2. Without loss of generality we may assume that a E (3, since the case when 3] of all, multiplying a > 3 is easier (as can be seen from the proof below). First (6.293) by u, we deduce easily that we have 31

IIuIIH1 < C(1 + IIPTIIL2) < (1 + IITIIL2p,(p-2))

,

(6.208)

.

(6.209)

and, (6.194) yields

II - div(kVT) + (pu . V)TIILI < C (1 +

IITIIL2pi(p-2))2

We next claim that we have IITIIL3c1+ag>,co < C

II

- div(kVT) + (pu V)TIILa + 1

.

(6.210)

We have made in chapter 3 (volume 1) several arguments of a similar type and

this is why we only sketch the proof of this claim. Multiplying (-div(kVT) + (pu V)T) by T A R for any R > 0, we find denoting M = II - div(kVT) + (Pu V)TIIL1

A R) (T -To) dS < C R M 1 kIVT121(T 1

in

V (T A R)1+a/2I2 < CRM + fan k(T A R) (T -To)1(T
General compressible models

169

< C(RM+1) < C(1+M)R. Therefore, we have for all R > 1 using the bound on T in L3a IIT A RIIi +3a < CII(T A R)l+a/2IIH1

< C (uv[2' A R)1+a/2JIIL2 + II(T A R)1+a/2IIL1) < C(1+M)112 R1/2 + R(1-5a/2)

< C(1+M)112 R1/2

and thus meas {R > T > R/21:5 C(1+M)3 R-3(1+a). This inequality also holds for R < 1 since T is defined on St which is bounded. Then, we deduce for all

A>0 00

meas {T > Al _

meas

{2(1)X > T > 2n A }

n=o 00

< C(1+M)3 A-3(1+a)

2-3(1+a)n

n=O

< C(1

+M)3-3(1+a)

.

This inequality implies our claim (6.210). We then combine (6.209) and (6.210) and we obtain IITIIL3(1+a)..

C (1 + I I T I I i(pi P-2) )

We may choose p > 4 and thus, if a E (3 , 31 , 3a < 2 < p-2 <- 4 < 3(1 + a) . Since T is bounded in Lea, the above inequality yields

IITIIL3p,(p-2) < C IITIIi3(1+a),, <_ C (1 + IIT'IILPI(P 2)) 20 = p-2 2p and thus l+a = 2 C1 - 3a(12 - 1)) p . Notice that if 20 20 p = oo and a = 13 then 1+a = 1 and that 1+a is decreasing with respect to a. 9 where 11=9 3a + 31+a)

Therefore, for any fixed a > 3, we may choose p large enough (p > 3«. 1) such that 1+a < 1. We thus deduce a bound on T in L3(1+a)'°° and thus in particular in L4, on pT in L2 and in fact in L2p/(p-2) and on u in H1. At this stage, it is now straightforward using elliptic regularity and the arguments introduced in section 6.3 to prove, as in step 1 above, all the bounds corresponding to the regularity stated in Theorem 6.18.

Step 3: A priori bounds when 6 = 0. We briefly mention how to obtain such bounds when N = 3, the two-dimensional case being similar and easier. First of

Stationary problems

170

all, we immediately deduce from the energy identity integrated over 11, using the boundary condition (6.195) and the normalization of p (fo pp dx = M) that u is

bounded in H' and thus in L6, and that T is bounded in L'(81Z). We are going to show that T is bounded in L3'°° (and DT is bounded in L3/2'O°) if p > 6. In order to do so, we argue by contradiction and we assume that we have a sequence of solutions Tn satisfying the boundary condition (6.195) and

T,,>0 in11,

in1Z,

(6.211)

where c,,, is bounded in L2p/(p+2) (like div (pu)) and bn is bounded in L6p/(p+6) and div bn = 0 in 11 (like pu)-in fact, we could replace 2p/(p+2) by any exponent

greater than 2 and s by any exponent greater than 3-and we assume that +p

IITn1IL2p/(p-2) goes to +oo as n goes to +oo while IITnhLLI(an) is bounded. We

then set Tn

In = IITnfiL2p/(p-2)

and observe that Tn solves the same equation as Tn namely (6.211). In addition,

cnTn is bounded in L'. We may then apply the same argument as in step 2 (taking a = 0) and we obtain for all R > 1

fn

I°TnI21(Tn
IITnARIIHI
IITnIIL3'ae <- C.

(6.212) (6.213)

In addition, (6.212) yields by an argument introduced in chapter 3 of part I (volume 1)

IIVTnIILj,. < C.

(6.214)

Extracting a subsequence if necessary, we may assume that Tn converges weakly in W 1'q to some T for all 1 < q < 2 where VT E L312'°°, T E L3'°° and thus Tn converges strongly in L' to T for all 1 -<,r < 3. In addition, we may pass to the limit in equation (6.211) and in the boundary condition (6.195) to deduce 0

T>0 in St

inn, (6.215)

,

an+AT=0

on 1911,

where we denote by b and c respectively the weak limits (extracting subsequences if necessary) of bn, cn respectively. Obviously, b E L6p/(p+6) and div b = 0, and cE L2p/(p+2). Next, we observe that TIac = 0 and that IITIIL2p/(p-2) = 1 since

P 2 < 3 if p > 6. In particular, T 0 and thus the strong maximum principle shows that mine T > 0 (T is in fact continuous on 1)). The contradiction shows a bound on T in L2p/(p-2) . Then, (div u) pT is bounded in L' and as above we

General compressible models

171

deduce a bound on T in L3'1 and a bound on VT in L3/2'°° thus proving our claim.

Once these bounds are shown, the rest of the argument is quite similar to what we did above and, in order to restrict the length of this chapter, we leave to the reader the straightforward conclusion of the argument.

7

EXISTENCE RESULTS FOR CAUCHY PROBLEMS 7.1 A priori bounds In this chapter, we are going to prove the existence of a solution (p, u) of (5.1)(5.2) (with p > 0) in three situations : i) the periodic case, ii) the case of the whole space and iii) the case of (homogeneous) Dirichlet boundary conditions. Other situations of interest will be studied later on in this chapter. The precise notion of solution we use is detailed in section 5.1 together with the notation and the conditions on the data (force f , initial conditions po, mo) that we shall use.

Let us also recall from section 5.1 the natural a priori estimates satisfied by solutions, bounds which follow from the energy identity (and possibly Sobolev embeddings). We always assume that po, mo satisfy

poEL1nL7(fl),

po>0a.e.in 11, po00

mo E L2-'/(-'+1)(S2) , mo = 0 a.e. on {po = 0} IMo12/po E L1(0) (defined to be 0 on {po = 0}).

(7.1)

We may now recall these "natural" a priori bounds that we write simply as "p, u(...) E X" where X is some functions space. What this really means is the fact that we have a bound in X that depends only on bounds on the data (po, mo, f) in the spaces they are assumed to belong to. First of all, in the case of Dirichlet boundary conditions, we assume that f satisfies

f E L1(0, T; L2-rl ('r-1) (c)) + L2/(1+a) (0, T; Lr(c))

(7.2)

where a = (2 - -y)+, r +-!(1- 2) + 2 + 1 g« = 1 and q= i2N2 if N > 3, q is arbitrary in [2, oo) if N = 2. Then, we have u E L2(O,T; Ho(c)

,

p E L°°(0,T; L'(&1))

,

PIu12 E L°°(0,T; L1(st))

.

(7.3)

Next, in the periodic case, we assume that f r= L1(0, T; L27/(,'-1) (SZ)). Then, (7.3) holds with Ho (SZ) replaced by H1 (0) provided ^y >- N+2 if N > 3.

A priori bounds

173

Remark 7.1 It is possible to combine (in a rather technical way) Remarks 5.1 and 5.4 and obtain a slightly more general condition on f than f E L1(0, T; L2-r/(7-1)(SZ)), which still ensures (7.3). But the condition we can obtain is very technical and less general than (7.2) and thus we do not know if (7.3) holds in the periodic case under condition (7.2) only. 0 We now consider the case when fl = RN and N > 3. Then, under assumption (7.2)-we can of course in this case replace L27/(7-1) by L27/(-r-1) + L°O-we have L2N/(N-2) (RN)), Du E L2 (RN x (0, T)) , u E L2 (0, T; (7.4) p1u12 E L°O(O,T; L1(RN)) , p E L°°(O,T; L1 n L-'(RN)) .

Finally, if SZ = RN and N = 2, we assume that f E L1(O,T; L2-r/(-f-1)(SZ)) + Li (0, T; L' (f?)). Then, we have u E L2 (0, T;L2 (BR)) for all R E (0, oo), Du E L2 (R2 x (0,T) ), 2)). pluI2 E L°°(O,T; L1(R2)) , p E L°°(O,T; L1 n 1

(7.5)

Throughout this chapter, unless explicitly mentioned, we always assume that the various conditions detailed above on the data (which lead to (7.3) or (7.4) or (7.5)) hold, and we shall not recall them. We may now turn to the crucial a priori estimates we want to derive in this section.

Theorem 7.1 Let (p, u) be a solution of (5.1)-(5.2). We assume that f E Li (0, T; L3/2) if N = 3 and -y > 6 and f E L1(0, T; L2) if N = 2 and -y > 2. We assume in addition that p E Lp(K x (0, T)) with j5 = max(p, 2) and p = y+ N 'y -1, K = SZ except in the case of Dirichlet boundary conditions where K is an arbitrary compact set of 11. Then, p is bounded in LP(K x (0, T)) in terms of bounds on the data only.

Remark 7.2 A somewhat technical extension of the proof allows us to prove the above result assuming only that p E L'10i We then have to show that p E LP (if p > 2) and prove the bound in LP by essentially the same proof as the one given below, but using first some truncations of the power functions introduced below in a way which is similar to what we did in the regularity arguments introduced in section 6.3 (chapter 6).

Remark 7.3 It is possible to provide some "explanation" for the exponent p occuring in the above result. Indeed, on one hand p E Lr(Lt) ("energy estimate") and on the other hand, if we simply expect p'y and pu ® u to have the same integrability, we deduce that pry-1 should belong to Lt (LN/(N-2)) since

u E L2 (Lx /(N-2)) by Sobolev embeddings (again the "energy estimate"). These

two bounds imply by interpolation that p E Li,t with p = 'y + N y - 1; notice by the way that if N = 2, we cannot quite say that u E Lt (Lz°) but nevertheless we can reach the exponent p. The heuristic argument leads to the following "conclusion" : the derivation of improved bounds on p requires an improvement

Existence results for Cauchy problems

174

of the "energy estimates", and this is obviously a fundamental question. Let us mention however that we cannot be too optimistic about bounds on p as can be seen from the following remark. Remark 7.4 A recent example by V.A. Weigant [550] indicates that, even with rather smooth data, we cannot expect bounds on p in L' for p large, at least if -y < 2 when N = 2. Notice that the exponent p in Theorem 7.1 is greater than

yif y> N,i.e.y>1if N=2. D Proof of Theorem 7.1. Of course, there is nothing to prove if y < 2 since p E L'(0, T; L1 n L'w). Therefore, in all that follows, we assume that y >

Step 1: Proof of Theorem 7.1 in the case when 11 = RN, N > 3 or in the periodic case. We present the proof in the case when St = RN, N > 3, and indicate briefly afterwards the modification to be incorporated in that proof

in order to treat the periodic case. We let 9 = p - -y and we recall from the proof of Theorem 5.1 (chapter 5, section 5.3, see in particular identity (5.53)) the following identity

ap7+e = at

[(pe)(-A)-1

div(pu)] +

+ (µ + t;)(divu) pe + (9-1)(divu) +pe[R=R,(putu3)

- u2RRR(pui)] -

div[u(pe)(-0)-1

div (pu)]

pe(-0)-1

(7.6)

div (pu)

pe(-0)-1

div (pf).

Then, we recall that plr and pIuI2 are bounded in L°"(O,T; L1) while Du is bounded in L2(RN x (0, T)) and u is bounded in L2(0,T; L2NI(N-2) (R N)). In particular, pu is bounded in L°O(0, T; L27/(7+1))nL2(0, T; L") withr1 = N-2 2N + 1 and thus, pe(-A) div(pu) E L°°(O,T; L1 n L3) where + 2ry1 77 _ If N + 1 27 < 1. Integrating (7.7) over RN x (0, T), we then deduce

-

-

1T1 PA d x

dt

C 1 + fdx fRN dx [Idiv u(1 + I (-O1 div (pu) I) + PBIRtiRj(Puiuj) - ujR=Rj(pu;)I +

peI(-A)-1 div (p.f)I,

(7.7)

We next observe that the three first terms in the integral of the right-hand side of (7.7) are bounded in L1 (RN x (0, T)), except when N = 3, y > 6 where the first term is bounded by CII PII ii

P

: indeed, Idiv uI I(-0)-1 div (pu)I pe is bounded

C L1. since 2+,-1.-N+e = 1. This argument is incorrect when r > N and this is possible only when N = 3 and y > 6 (recall that N > 3 in this step). In that case (of marginal interest) we argue as follows: we write III div ul

pe(-A)-1

div (Pu)IIL1,t

IIDuIIL=,, IIPIILP II pull LiP/(P-se)(L=P/(SP-BB))

A priori bounds

< C IIPIILz t g < CIIPIILzjt

175

IIPUIIL(1oy-6)/(y+3)(L3x (1oy-6)/(13-(+3))

2(2-y-3)/(57-3)

(7+3)/(5-y 3)

6))

IIPUIIL$°(L?.y/(7+1))

< CIIPIIL=,i IIPIIL /(L=)3) IIuIIL2(Le)(5ry-3) IIVPUIIL-(L=)/(5ry-3) C IIPIILz,,

+ p-29 = 1, 1 + e + 5p-66 1 = 1, 6p = 3 107-6 < 6p 5p-69 137+3 2 p 2p /(N-2)) 3. Similarly, p°R%R;(pu2u,) and peuiR4-RR(puj) are bounded in L' (Lx Lt°(Li) Lt°(Lile) C Lt'z since (again) N 2 + It = NN-2 + N2 = - 1.

since we have 2i +

- 3-

p

Finally, we consider p0(-0)-1 div (pf ). If N > 4 or if N = 3 and 2 < 7 < 6, pe(-A)-1 div (pf), in view of (7.2), is bounded in Lto(L7/e n Lz) +L?(-A)-1/2{Lz'La}] c L'(L2nL')+L2(Len

Lz)wherea1-72N -TNa X

c

'v

^1

+-

'

271 2=

1

77

=-L+1--L <1 and =N+1-27 (Lz'/(''-1))

+ . + a - N = N + 2 - 'r < 1. Notice that the proof for the Lt

part of f works if N = 3 in all cases. Next, if N = 3 and 2 < y < 2, we only need to consider the L2/(3-7) (Lx) part of f where a = 31 , and we find that p°(_A)-1 div (pf) is bounded in Lto(Lile n Lx) .

whereb-ry+ry+a

Lt/(3-7) ((_A)-1/2(Ly

'

Lx)) C

Lt/(3-7)(Ly

n Lx)

33+a-1<1.

Finally, if N = 3 and -y > 6, we assume that f E L1(0, T; L3/2 (RN)) and

thus pe(-0)-l div (pf) is bounded in L1(RN x (0, T)) since !+-1+23 - 3 =

0+1 +3=1.

In fact, there is a single term that we have not yet estimated, namely (div u) pe.

If 8 >_ 2 (i.e. y > 4 N), the bound is obvious. On the other hand, in the case when Q = RN, this rather simple term presents a technical difficulty when 8 < 2 since we do not know in that case if (div u) pe E L1(SZ x (0,T)). One way to solve this difficulty is to replace pe by pe 1(p> 1) . Then we obtain an estimate on p7+0 1(p>1) in L1(SZ x (0,T)) and we conclude easily since 0 < pl+e 1(p<1) <- p on SZ x (0, T).

-

In the periodic case, the proof is exactly the same except that ap7+e is replaced by a[p'y+e - pe(f-n pry dx)], and the additional term pe (fn p7 dx) is obviously bounded in L°° (0, T; L7/e). The bound follows. Let us recall that

in that case means the periodic solution with zero mean whenever cp is a given periodic function (or distribution) with zero mean; and, of course,

= (-A)-18zaj. O Step 2: Proof in the periodic case when N = 2, SZ = R2. In this case, most R;,R3

of the proof given above in step 1 carries over except for the slightly more delicate

terms A = (divu) pe(-0)-1 div(pu), B = pe[RRj(pujuj)

- uzR2R,(puj)] and

Existence results for Cauchy problems

176

C = pe(-A)-1 div (pf ), and we recall from the above proof (see 7.7) that we have

T

fdtfdx pp < C 1 +J

T

dt

Jn

dx {IAI + IBI + ICI}

(7.8)

.

We begin with the straightforward term C. If -y < 2, then (7.2) yields that f E Lt (L2y/(y-1)) n Lt/(3-y)(La) where a > yy l . Therefore, C is bounded in

(-A)-1/2{Lr(L-()

LeOO(Lx/e)

.

Lt/(3-y)(Lz)]}

(Lxy/(y-1)) +

[Li

C Lt (Lx) +L /(3-y) (Lx )

t

where 1=e+-1+

2=1-2y
2-f

-

since 8 = y - 1, a > -fry 1 and 'y < 2. Therefore, C is bounded in L' (fl x (0, T)). Next, if 7 > 2, we assume that f E L' (0, T; L2). Therefore, C is bounded in

Li°(L'y/e) (_A)-1/2 {L°°(Lz) Lt (Lz)} C L2,t 0+1

since

=1.

The two remaining terms, namely A and B, can be handled as follows. First

of all, pu = f f u can be estimated as follows IIPU

IIL,P(LyP/(P+1)) < C

IIPIIL

,,

Then, we have IIAIIL=,t < C IIPIIL=,t

since 1 +

e

+

p P1

-

2

IIPUIIL,P(L=P/(P+1))

= 2 + 20+1 = 1 and 2 + 2 + 2p = 1. Hence, (7.9) yields IIAIIL=

< C IIPIIL

(1/2)

(7.10)

Next, we use the fact that u is bounded in L2(O,T; H1) and thus in L2(O,T; BMO). Then, by the Coifman-Rochberg-Weiss commutator theorem [1111, we have for almost all t E [0, T) IIRiR;(Puiuj) - uiRiR;(PU;)IIL2P/(P+1) < CIIUIIBMO

IIPuLL2P/(P+1)

Therefore, we have I)RjR.i(Puiu.i) - ujRiRR(Puj)IIL2p/(P+1) t
IIPuIIL2P(L2P/(P+1))

Thus, we obtain IIBIILz,

since 2P +

2p

CIIPIIL=

IIPuIIL=P/(P+1)

= 1. Using once more (7.9), we deduce finally

.

A priori bounds

IIBIIL1

< C'IIPII

177

t112)

(7.11)

In view of the Lz,t bound on C and of (7.10)-(7.11), (7.8) yields IIPIILs t <-

C(1 + IIPIIL (12)1

and the Lz t bound on p is proven since 0 + z = -y - a < 2-y - 1 = p.

We finally observe that this proof is also valid in the case when SZ = R2 since

we also know in that case that u is bounded in L2(O,T; BMO) in view of the bound on Du in L2(R2 x *(0, T)). Indeed, we have for any cube Q in R2

u-f uldx < fQ u-f u12d2 < C J IDu12dx Q

Q

< CJ IDuI2dx , a.e. t E (0,T) where the constant C is obviously independent of Q by a trivial scaling argument. 11

Step 3: Proof in the case of Dirichlet boundary conditions. Exactly as in the proof of Theorem 5.1 (chapter 5, section 5.3 and 5.4) and for the same reason, the proof in the case of Dirichlet boundary conditions involves the localization of the preceding proofs. 'Alen N = 3, we have to localize the proof made in step 1 while, if N = 2, we make the same localization with the adaptations introduced

in step 2. In order to do so, we introduce for any fixed compact set K C St, a cut-off function cp E Co (S2) (for instance) such that cp = 1 on K and cp > 0 on SZ.

We may then follow the argument in section 5.4 and we obtain easily

t

(`p

div(pu)) + 8ij (cppuiuj) - (µ+)O(co div u) + aA(pcoy) = div(cpp f) + F

where F is given by F = (ei, cp) puiuj + 28icp8, (puiu3)

-

u)

.

+aOcppy-2(µ+C)VW V divu+2aVco Opy-pf VV.

(7.12)

The above equality then yields the following "local" version of (7.6), namely acppy+e =

[p°(-0)-1(p div(pu))]

at + div[upe (-O)-1(cp div(pu))J

-

(7) .13

+ pe{R=R3(wpuiu3) - u V(-0)-1(Wdiv(Pu))} + (0-1)(divu)pe (-0)-1(cpdiv(pu)). Then, if N > 3, the proof made in step 1 immediately yields an L1,t bound on copy+a and thus on fK pP d2 since all the terms, except the new one, namely

Existence results for Cauchy problems

178

pe(-0)-1F, are handled in exactly the same way. We thus only have to obtain an Lit bound on pe(-A)-1F. One then checks easily that, of all the terms

entering the definition (7.12) of F, only the term 2aVcp Op1' requires some analysis. Indeed, all the others are in fact "smoother" than the terms handled in step 1. If -y < N, we estimate pe (-A) (Vv - V pry) in Lr (La) using the bound on p in Lr(L t) where a is arbitrary satisfying 1

0

a

-y

+

N-1 N

=

2

1

N

y

+1-

1

1+

N

1

1

N

y

and in particular we may take a = 1. Notice that this argument also applies in the two-dimensional case. If -y > N, we first write p9(-A)-1(Ocp Op'') _ p°(-A)-1(div (p70cp) p1Ocp). Then, we estimate the two terms as follows

-

II(-0)-1div(p"V

with

)IIL:/,(L=) < CIIP1IV l"ILP1

div(p1' 4')IIL'/-1 (Lb) :5 CIIP''IAWHIL=/e

with

1 _-y a

p

1

-y

b

p

1

N' 2

N*

We then deduce from (7.13)

f

T cppP dx dt

C1+

(fTf

dt

(7.14)

providedy + 1a <- 1, k-f + b < 1. Obviously, we only have to check that We then write

e

7

1

0 1_21 -y

a

11 + (N+2),y-N N-y

+ .1 < 1.

'Y+ (N 'Y-1) N

N

and

0+1<1 a

'y

is equivalent to (N - 2)-y2 + 3Ny - N2 > 0 which is true if -y > N. We next claim that we can choose cp in such a way that IV IP/ry + < Ccp. Indeed, we replace W by V5t where m is a large enough integer ("` 1 < P) and i satisfies the same conditions as W. Then we have IocIP/'Y

IoVIP/ry + IzWIP/7 <

CVl(m-1) P/'Y

+

Cil(m-1)P/7 < CV)

(recall that 1V )12 < CV) since i/& is smooth and non-negative). We then deduce from (7.14) I VpP dx d< C 1 +

(fT cp pP dx dt

I and the desired bound is shown in all cases when N = 3. J

A priori bounds

179

Finally, we consider the case when N = 2. In that case, we need to handle the terms [pe{R%Rj(cppuiuj) - uV(-0)-'(cpdiv(pu))}] and [divupe(-0)-1

(cp div(pu))]. First of all, we write cp div(pu) = div(cppu)easily that we only need to take care of

and, one checks [pe{RjRj((ppuiuj)

- u2RiRj((ppuj)}] and of [diva pe(-0)-1 div(cppu)]. In order to do so, we are

going to multiply (7.14) by Wm for some m to be determined later on. This multiplication does not affect the bound on all the other terms in (7.13) except for T

f dtJin

mdiv[upe(-i)-1(cpdiv(pu))]dx pT

=J

dt

f (-u

Ocpm)pe(-0)-1(Wdiv(Pu))

dx

p70cp) if y > 2. The first of these two and the term cp"'Wo(-0)-1(div terms is easily bounded provided we choose m and cp in such a way that we have

Ipcp"`I < C.

(7.15)

We then argue as in step 2 and we find, provided (7.15) holds, the following estimate

1T1

+m pP dx

< C 1 + IIIIL/t IIJ IIL+ IIwmpe(-o)-1(div

(P''VV) - P"I(P)IIL=,t

.

Hence, if we choose m such that mp/B > 1 + m and 2p > 1 + m, then the preceding inequality yields T

f0

dt

J

dx cpl+m pP dx

< C 1 + II

mpe(-0)-1(div

(p7vcp)

II(-o)-1(div

C 1 + IIcomPolILs/o

C 1 + IIVmPOII L=/t IIP''IIL=t

< li

1 + IIcmPeIILP/e z,t

-

(p''V) -



IIV(1+m)1PPIILP

z,t

if we choose (again) m such that mp/B > 1 + M. In conclusion, the proof of Theorem 7.1 is complete provided we choose m and cp such that (7.15) holds and 1 + m < 2p, mp/6 > 1 + m. In particular, we may take m = 1, then (7.15) automatically holds and we conclude since p > 1

andp=2-y-1 > 28=2(ry-1). 0

Existence results for Cauchy problems

180

Remark 7.5 The proof above (step 3) is entirely local and only requires (p, u) to be a solution satisfying the following local bounds: plul2 and p1 are bounded in L°O(O,T; L C(1)), and u is bounded in L2(0,T; Hoc(S2)). Then, we obtain a bound on pin L7 (O, T; LP (1)) . 10C Remark 7.6 In the case of Dirichlet boundary conditions and when N > 3 and 2 < 'y < N, the proof made in step 3 shows that we may in fact choose cp to be strictly positive on SZ and equal to dist (x, 01) in a neighbourhood of act. Thus, we obtain in this case the following estimate T

dt

J

f

n

dx pp dist (x, & l) dx <_ C.

(7.16)

Existence results In this section, we state our main existence results and we make several 7.2

remarks about them including various outstanding open questions. Let us recall from the preceding section that we always assume (7.1), that f E L'(0, T; L27/(-y-1)(cZ)) except in the case of Dirichlet boundary conditions where we assume (7.2), that y > i2N if N > 3 in the periodic case and that the precise notion of solution we use is detailed in section 5.1. Roughly speaking, by solution we mean (p, u) satisfying (5.1)-(5.2) in the sense of distributions, such that p > 0, p E LOO(O,T; L1 n L1) n C([0, T]; L'') for all 1 < r < -y, plu12 E L°O(0, T; L1),

u E L2 (0, T; H1) (or Ho in the case of Dirichlet boundary conditions) except when ct = RN in which case Du E L2(RN x (0,T)), u E L2(0,T; HOC(RN)) and L2N/(N-2) (RN)) u E L2(0, T; if N > 3, pu E C([O, T]; L27/(7+1) - w). Finally, as in Theorem 7.1, we assume that f E L1(0, T; L3/2) and p E C([0, T]; L3/2) if N = 3 and -y > 6, and f E L1(0, T; L2) if N = 2 and y > 2. We may now state our existence result (valid as usual in the three situations of Dirichlet boundary conditions, of periodic boundary conditions or of "zero conditions at infinity" in the whole space). Theorem 7.2 Under the above conditions and if we assume that y > 2 if N = 2, -y > s if N = 3 and y > 2 if N > 4, there exists a solution (p, u) of (5.1)-(5.2) satisfying the initial conditions (5.6) and such that p E LP (Q x (0, T)), except in the case of Dirichlet boundary conditions where p E LP (K x (0, T)) for any compact set K C SZ, where p = -y + N -y - 1. In addition, (p, u) satisfies the following energy inequality for almost all t E [0, TI: fpu2

+


no 2

ds fn dx plDu12 + (div u)2 f a 1dx + f0t ds fi2dx pu f 0

pry dx + Po

z

+ "Y

po

(7.17)

.

Remark 7.7 Many questions are still open. In fact, the above result can be seen as the analogue of J. Leray's celebrated work ([325-327]) on incompressible Navier-Stokes equations for compressible (isentropic) Navier-Stokes equations in

Existence results

181

three dimensions. Indeed, we obtain the global existence of a weak solution (in L2 (0, T; H')) satisfying an energy inequality, and exactly as in the incompressible case (see chapter 3; volume 1 for instance), the uniqueness of solutions, the regularity of solutions and the fact that this energy inequality (7.17) might (or should) be an equality are outstanding open questions.

Remark 7.8 The restrictions on ry may look a bit unnatural-and they certainly are from a physical viewpoint. Our condition should really be understood as requesting that i) -y > 2 and ii) p E L; The second condition is needed in order to apply the compactness results shown in chapter 5 (it was needed there for somewhat technical reasons and it is plausible that it might be relaxed). If C.

N>4and y> 2,then pEL,tsince pEL°°(0,T;Lw).But, ifN=2 or N = 3, the L2 bound is obtained using Theorem 7.1 as soon as p = 'y + N ry - 1 >_ 2 i.e. -y > 2 if N = 2, ry > s if N = 3. Therefore, improving the requirements on p in the compactness results of chapter 5 or improving the bounds on p in Theorem 7.1 would lead to improved conditions on -y in two or three dimensions.

However, the first restriction on 7, namely -y > 2, is absolutely essential to our analysis. Let us give one example of the manner this condition is used in our proofs (see also chapter 5 for similar arguments) p E LO°(0,T; L") and u E L2 (0, T; L2N/(N-2)) (Sobolev embeddings) imply that pu ® u E L' (0, T; LP)

for some p > 1 if and only if 7y > E.. Improving this restriction is a fundamental

open question: for instance, can one prove an analogue of Theorem 7.2 when -y > 1 and N = 2 or 3?

Remark 7.9

7 if N = 3, y > 6 if N = 4, the proof of

Theorem 7.2 yields the following (version of a) local energy inequality in the sense of distributions say in the case = 0 (in order to simplify notation) 8 (.1plU12 (PIuI2+

at

a

py

+ div u p

ry-1

- µA

2

2 -v

ary 2

-1 pu.f ^f

p'

(7.18)

where v is roughly speaking-we do not wish to detail here the precise meaning of what follows-a bounded non-negative measure in t with values in LN/(N-2) (or in LP for any p > oo if N = 2) supported on the set l p = 01-the vacuum. Let us mention by the way that we do not know if p remains strictly positive when po is strictly positive. In particular, we can check that v = 0 if the set {p = 0} has zero measure, and, exactly as in Remark 7.7, the equality in (7.18)-which is to be expected at least formally-is another interesting open problem.

Remark 7.10 Theorem 7.2 is the only known global existence result for general initial conditions. Many references are given in the bibliography which prove the existence of smooth (or relatively smooth) solutions-with uniqueness results in some cases-in special regimes. For instance, locally in time smooth and

182

Existence results for Cauchy problems

unique solutions do exist-see also the following remark for further details on that aspect-and global in time solutions close to an equilibrium (i.e. p close to a strictly positive constant and u close to 0) exist. For the latter type of results, probably the most general result is the recent work by D. Hoff [2511 which shows that if po is sufficiently close (in the L°° norm), then a global solution exists with

(p, u) staying close in L°° x Li to (p, 0). 0

Remark 7.11 We wish to mention here the recent examples by V.A. Weigant [550 presenting the formation of singularities in finite time of "smooth" solutions

(p, u) of (5.1)-(5.2). More specifically, if -y < 1 + N (and thus if 1 < y < 2

when N = 2), one can find f E LQ(l x (0, T)) for some q > N such that the maximal, local in time, unique solution (p, u) of (5.1)-(5.2) with Dirichlet boundary conditions (f is the unit ball) such that u E W 2',t ,4, p E LO ° (Wz ,4 ),

inf p > 0 for all t in the existence interval, 7 E Lr(Li), blows up in finite time. In other words, the maximal existence interval [0, To) is finite (T0 < oo) and (sup p)(t) goes to +oo as t goes to To. These examples show various facts of considerable interest: i) first of all, the special regimes briefly described in Remark 7.10 above which allow the global existence of smooth solutions do not capture some really non-linear and delicate phenomena for this system of nonlinear partial differential equations, and ii) we have to be careful with the type of regularity we may (or can) expect for p and u. 0 7.3 Existence proofs via regularization Both this section and the next section are devoted to the proofs of Theorem 7.2. In fact, we present two different proofs, one in each section. Of course, both involve approximating and "regularizing" the system of equations (5.1)-(5.2) but the type of approximation we propose in each section differs substantially. In section 7.4, we make a natural time discretization of (5.1)-(5.2), which allows us to use the results of chapter 6 and we then check that we can pass to the limit and recover a solution of our original system. In this section, we build solutions with the properties listed in Theorem 6.2 by a convenient approximation (and regularization or simplification) of (5.1)-(5.2). In fact, the approach here is much more in the spirit of what we did on time discretized problems in chapter 6. However, unlike what we did in chapter 6, we present various regularization procedures which keep some of the basic properties of (6.1)-(6.2) like non-linear transport for the density p, compactness analysis as in chapter 5, bounds on p as in section 7.1 above. These procedures can be seen as a collection of "tricks" rather different from those introduced in chapter 6 for stationary problems. Of course, not only do we not claim that this approximation, namely the one developed below, is the only one (or even the simpler one!) that yields Theorem 7.2, but we want to make clear that several other approaches are possible, some of which use some ingredients of the proofs below. We decided to present a particular one even though it is a "multi-layered approximation" for several reasons: first of all, as stated above, it has no similarity with the

Existence proofs via regularization

183

particular approximation strategy introduced in chapter 6; next, we believe that some of the tricks mentioned below might be useful in other contexts; and finally, it also makes contact with some problems discussed in chapter 9. As indicated above, our approximation involves several layers and we thus split our presentation into several steps (one per layer at least). This is why most of the proof of Theorem 7.2 presented in this section will be made "backwards": if equation(s) (B) approximates equation(s) (A), we first explain how solutions of (A) can be obtained from solutions of (B) before explaining how to obtain the latter-possibly via solutions of (C). Finally, we should warn the reader that some of the details (in the various steps) are quite tedious, so we have tried to present the main ideas. Anyway, there is always the possibility of having a glance at the approximation tricks introduced in this section and immediately jumping to the next section! Finally, we begin our first proof of Theorem 7.2 by considering the periodic case and then we treat the other cases, namely the case of Dirichlet boundary conditions and the whole space case.

Step 1: Periodic case; preliminary reductions. We first claim that it is enough to prove the existence of solutions satisfying (7.17) in the case when f is smooth in (x, t) and periodic, po is smooth, periodic and positive on SZ = 11N 1 [0, Tti] (T1,... , TN are the periods), and mo is smooth and periodic on st. Then, for general data (f, po, mo) (as in Theorem 7.2), we build a sequence of data (f", po, mo) such that f" converges to f in L1(0, T; L27/(7-1)( ))N, po converges to po in L" (St) and mo = po v" where v" is smooth and periodic, in L2(f)N. v" converges to VJW Then, if we have shown the existence in the case of smooth data, we obtain a solution (p", u") of (5.1)-(5.2) satisfying (7.17) with (f, po, mo) replaced by (f ", p0 n, M0 n) Next, we deduce easily from (7.17) (and from (5.1)) as in section 5.1 of chapter 5 that p" is bounded in L°° (O, T; L'Y(f) ), p" lu" I2 is bounded in L°O(O,T;L1(11)) and u" is bounded in L2(O,T;H'(11)). Furthermore, p" E LP (11 x (0, T)) with p = y + N -y - 1 > 2 by assumption (see Remark 7.8

above) and we may apply Theorem 7.1 to obtain a bound (uniform in n > 1) on p" in LP(1 x (0, T)). We may then apply Theorem 5.1 (section 5.2 of chapter 5) to deduce, extracting a weakly convergent subsequence of (p", u") in L'' (11 x (0, T)) x L2 (0, T; Hi (St)) for instance, if necessary, that p" converges strongly to p in C([0,T]; L''(Q)) for all 1 < r < y while p" u" converges strongly

to pu in L-(0, T; L"(11)) for all 1 < s < oo, 1 < r < 2, and (p, u) is a solution of (5.1)-(5.2) satisfying the properties listed in Theorem 7.2 including (7.17) thus proving our claim. We also wish to observe that, when N = 2 and -y > 2 , it is enough to prove the existence of a solution (p, u) of (5.1)-(5.2) such that p E LQ (S2 x (0,T)) for all 1 < q < p = 2-y -1. Indeed, we then claim that, in fact, p e LP(S) x (0, T)). In order to prove this claim, we inspect carefully the proof of Theorem 7.1 (step 2) replacing p by q arbitrary in ('y, p) and 9 = -y -1 by 9 = q -p. We may then copy

Existence results for Cauchy problems

184

the proof made in step 2 (Theorem 7.1) and find a bound on p in L9 (SZ x (0, T) ) independent of q E (-y, p). This obviously implies that p E L"(fZ x (0, T)) and we conclude. In addition, we only have to consider the case when -y > 2 in two dimensions. Indeed, once we have obtained a solution (p, u) for -y > 2 with p E L21-1(SZ x

(0, T)) satisfying (7.17), we may deduce a solution (p, u) for ry = z with p E We only have to use L2(SZ x (0,T)) satisfying (7.17) upon letting ry go to Theorem 7.1 (and its proof) which yields a uniform bound on p in L2' I ' (IZ x 2 (0, T)). Then, observing that 2 , we may apply Theorem 5.1 as we did above and recover a solution in the case when -y = Let us finally explain one aspect of the proofs that we skipped until now, namely the possibility of passing to the limit in (7.17). This is indeed straightforward since (7.17) is equivalent to 2+.

2.

fo dt ffi dx WW

2PIu!2

+,y a 1P7+ f

ds[µIDu(x,s)12+t;(divu)2(x,s)-pu f]

(f(t)dt) f

f22

Jtno po

+ rya 1 PO d2

for all cp E Co (0, T), cp > 0. We may then pass to the limit in this inequality as soon as Du converges weakly in L',t, p converges in Lz,t and Ct(Lz) for some

p > 2 (in particular this is the case if it does so for all p < 'y and y > 2 !), po converges in L2, po converges in L7 and pu is bounded in the dual spaces of those to which f belongs, which is always the case in view of the assumptions made upon f and the bounds proven on (p, u)!

Step 2: Periodic case; smoothing pat + pu V. As seen in the preceding step, we may assume from now on that all the data are smooth (and that when N = 2 and when -y > 2 , we only have to build a solution (p, u) with p E L4 (SZ x (0, T) )

for all 1 _< q < 2-y - 1). In order to build a solution (p, u) of (5.1)-(5.2) as in Theorem 7.2 (except when N = 2 as just recalled) including the inequality (7.17), we are going to approximate the system of equations (5.1)-(5.2) by (recall that all functions are periodic)

+ div (pu) = 0

,

p > 0 in RN x (0, T)

div ((pu)E ® u) - AAu - V div u + aV p" at (p£u) +

(7.19)

=pEf inRNx(0,T),

where e E (0,1) and cp, denotes, for an arbitrary periodic function cp, cpE = W * nE

where rc£ _ - ,c(), K E S(RN), ic > 0 and in RN and fRN rk dx = 1-for

Existence proofs via regularization

(27r)-N/2 a-1x 2/2

instance !c = have obviously

and is =

(29re2)-N/2

185

exp -(IxI2/262). Since we

+div(pu)E = 0, we may write equivalently (at least formally) the second equation in (7.19) as 8u

pf inRN x (0,T).

peat +(pu)£

Therefore, the approximation we use consists in smoothing in the momentum equation the "total derivative" (P At + Pu V). Let us assume temporarily that we have already built a solution (p, u) of (7.19) satisfying plt=o = po, ult=o = mo/(po)e and the same integrability requirements as in Theorem 7.2-in fact, we shall build in step 3 below a solution (p, u) with p E LP((SZ x (0, T)) and p = (1 + N)-y if N > 3, p E [1, 2-y) if N = 2-and the following analogue of (7.17) for the system (7.19):

f

n

2 pelul2 +

7

<

f

a

J

In,

ImoI2 2

P0

rt

P' dx +Jof dsdx[µjDuj2+1;(divu)2] oJn

(7.20)

+ alPodx+ rt -y-

Vt E [0,T].

0

We next claim that we can let a go to 0+ and recover a solution of the original system. First of all, we need to prove some a priori bounds. Of course, in view of (7.19), w e obviously h a v e a bound on pin C([0, T]; L1(11)) (f n p(t) dx = fn po dx

for all t E [0,T]). Then, (7.20) immediately yields a bound (as in section 5.1, chapter 5) on p in L°°(0,T; L-' (St)), on peI uI2 in L°°(0,T; L1(St)) and on u in L2(0,T; H1). Since pE is also bounded in LO°(0,T; L'r), we deduce easily that L21'/(-t+l)) + if N > 3, pcu is bounded in LOO (0, T; n L2 (0, T; Lr) with 1r = N-2 2N

1
We next claim that p is bounded in LP(St x (0, T)) where p = -y + N 'y - 1 if N > 3, 1 < p < 2ry - 1 if N = 2. The proof of this claim follows the proof of Theorem 7.1. In particular, we have with the notation of the proof of Theorem 7.1 (and 0 < ry - 1 if N = 2) instead of (7.6) ap7+e =

49

[(pe)(0)-ldiv(pcu)]+div[u(pe)(o)-ldiv(peu)] + (µ+l;)(divu)(pe) + (0-1)(divu) p"(-0)-1 div(p£u)

+PB[ R; ((Pu:) u ) -- uu Rj R, (p6 u -pe(-0)-ldiv(pef)+a E

CfnPrydx

T

]

PB

(

7.21

)

J

In view of the bounds shown above, we only have to repeat the argument made

in step 1 (N > 3) of the proof of Theorem 7.1 to conclude that the bound we claimed above holds uniformly in e E (0, 1]. When N = 2 (see step 2 of the proof of Theorem 7.1), we also conclude easily since we have chosen 0 <

Existence results for Cauchy problems

186

ry - 1. Indeed, choosing p = -y + 0, we obtain in exactly the same way that [(divu) pe(-A)-1 div(pEu)] is bounded in L1,t in terms of IIPIILe/2-in fact, for this particular bound, we could still take 0 = y-1. All the other terms are treated as in step 2 of the proof of Theorem 7.1 except for the term { pe [RjRj ((pui)Eu j) uj RjRj (pEui)] } for which we can no longer use the commutator argument. We then bound separately each term

-

II P°RiRj((PUi)EUj)II L'

<- II PeIILr(L=ie) IIPIIL-(L=) II UII L2(L=)

IIP°UjR%Rj(PEUi)II L=,< < II PeII LL°(L=ie) IIPIIL-(Ls) IIUIIL2(L=)

where s (< oo) is determined by

+ ti +

1, i.e. s = try/((ry - 1) - 0). Our

claim then follows.

Finally, we then wish to pass to the limit as e goes to 0. First of all, we may find a sequence of solutions (pk, uk) of (7.19) corresponding to a sequence of e--still denoted by e in order to avoid tedious notation-such that uk converges weakly to some u in L2 (0, T ; H') and pk converges weakly to some p > 0 in LP ( 1 1 x (0, T)) and in L°° (0, T; Llf) (- weak *) where p = -y + nr 'y -1 if N > 3, 1 < p < 2-y - 1 if N = 2. Obviously, pk also converges-for the same topologies to p-since a goes to 0. Since, at = div [(pkuk)£], we may still use Lemma 5.1 (section 5.2 of chapter 5) in order to deduce as in the proof of part (1) of Theorem 5.1 (section 5.2 also) that pkuk and pkuk both converge weakly in L2(0, T; L'') to pu where r = 7 + 2N if N < 3, 1 < r < -y if N = 2. In addition, since pkuk is bounded in L°°(0,T;L2"r/(7+1)), we also deduce that pkuk converges weakly to pu in LO°(0,T; L27/(7+1))-weak*. In order to conclude, we have to check that (pkuk)£ ® uk converges (say in D') to pu O u and that part (2) of Theorem 5.1 also remains true. And in fact we only have to check that assertion (5.19) in part (2) of Theorem 5.1 still holds and even more precisely that we can adapt the proof of part (2) of Theorem 5.1 made in sections 5.3-5.4 of chapter 5. Both facts involve the same difficulty: replace (pkuk)£OUC by pkuk®uk or [RiRj((pkuj)Eui)-uiR2Rj(pkuj)] (for some small enough 0 > 0) by (pk)e[R4 Rj(p9 uj ui) uiRR,(pk uj)]. If these substitutions are possible, then we conclude easily adapting the proof of Theorem 5.1. It turns out that these substitutions are straightforward once we observe that we have (dropping the index k in order to simplify notation) for all 0 < 0 < 1

-

(PU) E (x, t) - (PE U) (x,

t)1 = JRN P(y, t) [u(y, t) - u(x, t)] r.E (x -y) dy 1-e

< [(P119)E]0 [JRN

I U(y, t) -U(x,

t)1i/(1-e)

r e (x-y)dy

Therefore, if we take p E [2, N N2) (p < oo if N = 2) and define r by T = we obtain for almost all t E (0, T) choosing 0 = 3 7

-}- P1

187

Existence proofs via regularization

II ((Pu)E - (Pcu))(t)IILr

r/P r [IRN

fn

Iu(y, t) -u(x, t)I P/roc (x-y)dy

dx Iu(y,t)-u(xt)IP/rKE(x-y)dyjr)

1/P

f dxl f N

< II(P1/r)EIIL <-

1/r

n

1/p

IIPIIL4 Jndx fRN dy Iu(y,t)-u(x,t)Ipce(x-y)

In addition, we observe that we have for almost all t E (0, T) with p = 2 + (1cY) 2N (P = a if N = 2) 1/p

fn dx fRN dyl u(y, t) -u(x, t) I P KE(x-y) 1/p

(,fRN dz x, (z) fn dx Iu(x-z) - u(x)Ip «p/2

<

fRN dzice(z) fn dx Iu(x-z) - u(x)12 (1-«)P/2 1/P

fn dx IDu(x-z) - Du(x) 12 + Iu(x-z) - u(x)I2 «/2

<

fRN dz rc,(z) fn dx I u(x-z) -u(x)l2

ffN dz n, (z) fn dx I Du(x-z)-Du(x)I2 + Iu(x-z) - u(x)11 <

(fjt,dzr,(z)fjdxju(x-z)

- u(x)I2

(4IIuIIii)

We thus obtain II (Pu)£

- (PEI) II L2(O,T;Lr)

(f

rT

_< C

0

(7.22)

r r dx l u(x-z) dz icE(z) dt Jn JRN

- u(x)I2 1)

«/2 ,

where C denotes various positive constants independent of C. We then claim that we have the following inequality in H1

f dzic,(z) f dx lu(x+z) - u(x)I2 n RN

1/2

< CeIIuIIHI

(7.23)

for some C independent of u in H1. This inequality is standard if rc E Co and it can be easily seen that it also holds if rc E S (or more generally if I zl2ic E V) since we have JRN

dz rc(z)

Jn

dx l u(x+z) - u(x) I2

Existence results for Cauchy problems

188

rdz

r rce(z) JRN

dx J Du(x+sz) - z ds

Jn

f ds f dx I Du(x+sz)12 1

dz rce(z)Iz12 JRN

o

IIDu1I2

2(n) E2 J

(N rc(z)I Z12 dz

.

Combining (7.22) and (7.23), we finally deduce that we have II (Pt)e

- PeUIIL2(0,T;Lr) < CEa

and thus (pkuk) e - pE uk converges to 0 in L2 (0, T ; L'') as E goes to 0+ for all the exponents r defined above. This suffices to adapt the convergence analysis and we conclude. E

Step 3: Periodic case; non-linear damping of p. We now wish to build up solutions of (7.19) as limits of solutions of the following system

+div(pu)+6p"=0, p>0, 49

(Peu) + div((Pu)e (9 u) + b(p')eu - pLu -6Vdivu+aVp"" = pff in RN x (0,T) )

(7.24)

where p > 1 + -y will be determined later (close enough to 1 + 'y), b E (0, 1). Of course, p and u are periodic in x and satisfy the following initial conditions: PI t=o = Po UIt=o = mo/(Po)e

We are going to build in step 4 a solution of (7.24) (at least for 6 and e E Lq (0, T; W -l (1)) and small) (p, u) such that: u E Lq (0, T; W 1 ,q (SZ) ), p E C([0,T];Lq(SZ)) for all 1 < q < oo, pe E C1([0,T];Ck(SZ)) and (pu)e E C([0, T]; Ck(SZ)) for all k > 0. In fact, the system (7.24) admits a (unique) smooth solution (p, u E COO (Si x [0, T])); see for more details Remark 7.12 below.

We wish to construct solutions of (7.19) letting 6 go to 0+. In particular, we first need to check that, as b goes to 0+, p is bounded in L°O(0,T; L1') fl LP (Q x (0, T)) where p = -y + 72-1 -y if N > 3, p E [1, 27) if N = 2, and u is bounded in L2 (0, T; Hl) fl LOO (0, T ; L2 ). In order to do so, we first write PC

au + (pu)e - Vu - p0u - t;0 div u + at

aVp-r

= Pef

from which we deduce easily the following identity

a at

u1 2

pE

I

2

+

ap -y-1

+b(pp)EIuI2 + ab

+ div (Pu)e Pp+7-1

+ aIDuI2 + t; (div u)2 = Peu -

f

u1 2 I

- pA

2

12

+ 2

ay

up7

-y-1 - ediv (udivu)

(7.25)

.

The justification of these computations presents no difficulty in view of the regularity of (p, u) stated above. Integrating over the domain of periodicity SZ the

Existence proofs via regularization

189

identity (7.25), we deduce (as usual) that p is bounded in L°O(O,T; L7), u is bounded in L2 (0, T; H 1) and p£ I u 12 is bounded in L°O (0, T; L') . Furthermore, we have the following bounds T bodtn dx (p9)eIUI2 < C and

(7.26)

j

dx pp+I-i < C. (7.27) n Here and below, we denote by C various positive constants which are independent b

dt

J

of 5-they may depend upon e even though the two preceding bounds do not. We next claim that we have for b small enough

pf > 1/C

on St x [0, T] .

(7.28)

Indeed, we simply write for all t E [0, T]

f

n

p(t) dx =

f

ds

(Po)E dx - b T

inn

f

o

in

fo

Po dx - b

f Po dx - 6 J ds T

t

ds o

dxp" dx >

Jft dx

pP+--'

n

1(p>,) - b

T

J

dxpn in

dxp 1(p<,)

ds fro,

Po dx -XT - bT J Po dx n

in view of (7.27) and since fo p(t) dx < fn po dx for all t E [0, T]. Therefore, choosing b small enough (0 < b < min (4' fn po dx, L)) , we obtain for all t E [0,T]

Jn

p(t) dx > 1

fo

po dx.

Then, (7.28) follows since p,(x, t) = f p(y, t) i£(x

- y)dy > (infzE2-ff r,(z))

(fn P(y, t)dy). Of course, the inequality (7.28) and the bound on p£IUI2 yields a bound on u in L°° (0, T; L2) . This bound then implies that p, is bounded in LOO (f2 x (0, T) ) while (pu)E is bounded in L°° (SZ x (0, T)) if ry > 2, T; Loo (1)) if L41'/3(2-ry) (0,

7 < 2 and N = 3, L'(O,T; L°°(SZ)) for 1 <s <

if y < 2 and N=2. It only

remains to obtain a bound on p. We thus follow once more the proof of Theorem 7.1 and we have now instead of (7.6) or (7.21) apti+0

= at [(PO(-0)-ldiv(peu)]+div[u(pe)( 0)-idiv(peu)] + (p+t;)(divu)(pe) + (O-1)(divu) p°(-0)-' div(peu) + P°RiR? ((Pui)E uj)

-

+

- peu,RjR; (PEuj)

p°(-A)-1 div(Pef) bp0(-0)-1

+a -f p dx

div((p')£u) +

pe

9bpp+e-1(-A)-1 div (PEu)

(7.29)

Existence results for Cauchy problems

190

We then wish to choose 0 = N y if N > 3, 0 < 0 < -y if N = 2. In order to

simplify notation, we skip the case N = 2 where L2N/(N-2) has to be replaced by L" where r is arbitrary large-this is why we do not seem to be able to choose 9 = 'y when N = 2. We next check easily that pa(-0)-1div(pu) is bounded in Idivul pe(1

u(P°)(-0)-1div(peu),

LO°(0,T; L1),

PBRjRj((pui)euj), PeujRjRj(Peuj),

+ I(-0)-1div(pu)I),

pe(_A)-1 div (Pef), (fn pry dx) pe are bounded

in L1 (Q x (0, T)) using the bounds on u in L2 (0, T; L2NI (N-2) ), p in L°° (0, T; L-f ),

pe and (pu)e in L°°(1 x (0, T)). The only "tight" bound is in fact the one on PBusR,Rj(Peui) IIP0 uiRjRj(Peui)IIL=,t < 11P°II L- (LNI2)

IIUIILe(L2N/(N-2)) IIPellLr IIUII

LL(L2N/(N-2))

< CIIPeIIL-(L"/2) = C11PIILt°(Ly)

< C.

It only remains to estimate the last two terms of the right-hand side of (7.29), and we write, using (7.26) and (7.27) pe(-A)-1

bll

div((P")cu)IIL=,t

< C6 I PII Lr(LZ) II V

<

Cbb-p/2(p+1'-1)

IIL2(P+,-1)/P(L-) II

b-1/2

(Pp)e ullLs,t


-

since on one hand 1 + 2 p+1 < 1 and on the other hand 2 N + 0 = 1. pp+e-1(-A)-1 div (pcu): first we observe that peu is Finally, we estimate 6 61lpp+e-1IlL(P+ry_1)/(P+e_1) is bounded and bounded in L'(L2) n Lt (L2N/(N-2))

x,t

1

Next, we remark that if p = 1 +'y, p = ry+2ry = N+2 > 1 1. Therefore, if p > 1 +'y is close to 1 +'y, we deduce not while 2N +pB p+-1 = bpp+e-1(-0) -1 div (pu) but we also show that the L1 only a bound in Ll t on norm of this expression goes to 0 as 6 goes to 0+. Let us observe by the way that the same is true for the expression bpe(-A)-1 div ((pp)eu) by the estimate

by Co b-

shown above.

In order to complete the proof of the passage to the limit as 6 goes to 0+, we only have to check that we can copy the proof of Theorem 5.1, using the fact shown above on the additional terms 5pe(-t)-1 div ((pp)eu) and b pP+e-1(-o) -1 div (peu). In fact, the analysis is much simpler once we remark that (p6)E converges respectively to pe in L''(St x (0, T)) (strongly) while u6 converges to u in L'' (0, T ; L2 (Q)) for all 1 < r < oo as 6 goes to 0+. Of course, we

denote by p6 and u6 the solutions corresponding to 6 and by (p, u) their weak limits (or weak limits of appropriate subsequences). Indeed, (p6) a is obviously bounded in L°O (0, T; Ck (St)) for all k > 0 while a

& ` = -div (p6 u6 )E - b [(p1)p] e

Existence proofs via regularization

191

is bounded in L(7' -1)/"(0, T; Ck (1)) for all k > 0. Therefore, (pa)E converges to p in C([0, T]; C" (?i)) for all k > 0. Next, u6 is bounded in L2 (0, T; H') while we have (skipping S) au

l;

1

div((pu), ®u) + µ 1 Au + PC

-div

it div PE

+ ((PU)eV!) PE

+aV

P

a

a W, + f -

1

div u

PC

PE

- Vu + l V

u-µ O (!).v) u PE

CV

1

S

6 (PI )Eu

- aV p

+f

PE

divu

P£

(p")eu,

p Pe

V div u -

PE

and thus at is bounded in L2(0,T; H-1) + LOO(O,T; W-1,1). This implies that ub converges in L1(1 x (0, T)) to u and thus in L'' (0, T; L2 (11)) (Y r < oo) or in L2(0,T; LP (11)) (d z

< 2 2)

Step 4: Periodic case; truncation of the pressure. In this step, we propose our final approximation (or final layer of approximation). We are going to construct solutions of (7.24) (with the regularity mentioned in step 3) by taking limits of solutions of the following system of equations

ap+div(pu)+Sp"=O, p>0 U1;

PC

au

at

l

-µ0u-eVdivu+aV(pAR)1 (

(-7 30 )

= PEf in RN x (0, T) ;

all functions are periodic in x; (p, u) satisfy the following initial conditions: PI t=o = Po, ult=o = mo/(Po)e; and R > 1 is a truncation parameter. Let us recall that a A b denotes min(a, b). We are going to explain in the next step why there exists a smooth solution (p, u) of (7.30): p E C([0,T]; W1,q) and u E W2,1,q = {cp E Lq (O, T; W2,q) with

E Lq (St x (0, T)) } , for all 1 < q < oo. In fact, if we replace p A R by a smooth truncation [i.e. (R(P) = R((p/R) where C(0) = 0, S' E Co (R), ('(t) > 0, ('(t) = 1 if It) < 1, and C'(t) = 0 if Itl > 2], we can even check there exists a smooth solution (p, u) E C°° (S2 x [0, T]) .

Next, we explain how one can recover of solution of (7.24) satisfying the

properties listed in step 3 by passing to the limit as R goes to +oo in the system (7.30). In order to do so, we have to explain (as in the previous steps) how to pass to the limit and that we have the following a priori estimates: p is bounded in LO° (0, T; Lq), u is bounded in Lq (0, T; W 1'q) and is bounded in Lq(0,T; W- 1,q) for all 1 < q < oo. We begin as usual with the bounds (that may of course depend upon S but are independent of R). First of all, the energy identity in the case of (7.30) takes the following form as is easily checked by elementary computations:

Existence results for Cauchy problems

192

a

p+ div IuI

+S(pp)EIuI2-µs

= pfu f

[(PU)e_+aU7R(P)]

+a5o' (P) Pp

2

(div u)2

(7.31)

in R x (0, T)

dt and ryR (P) = /3R (P)p, hence /3R (P) _ X 11 (p A R)'1-1P + R11-1(P - R)+, QR(p) = 711 (P A R)7-1 and 7R(P) = fll (p A R)1-1p. where 13R (P)

= P fop

As usual we deduce from the equation satisfied by p and (7.31) that u is

bounded in L2(0, T; H1), p and OR (p) are bounded in L°° (0, T; L1) and thus pE is bounded in L°° (0, T; Ck (S2)) for all k >_ 0, pp is bounded in L1(S2 x (0,T)), p' Iu I2 is bounded in LO° (O, T; L1), (pp) E I uI2 and f3(p)p" are bounded in L1(f Z x (0,T)).

We next claim that (7.28) holds uniformly in R > 1 and in S as long as S is chosen as in the preceding step. Indeed, we only need to observe that the argument made in step 3 still applies replacing pP+'r -11(p> 1) by pp (p A R)'' (p> 1) (= y-f 1 pP 1O(p) Therefore, u is bounded in LOO (0, T; L2) and, in particular, (pu)e is bounded in LP (0, T; Ck) (for all k > 0) in view of the bound on pP mentioned above-recall that p > 1 + oy > 2. We are now going to prove that p is bounded in LOO (0, T; P), u is bounded in

L'(0, T; W l,) and au is bounded in L' (O, T; W-14) for all s E (1, oo). Indeed, we have for any q E (1, oo)

a at P4 + div (up') +

g5pp+q-1

= (q-1) div u pq .

Therefore, we obtain, denoting s = (p + q -1)/(p - 1), IIP(P-1)

II PII

<- C IldivuLIL=,, < CIIDuIIL=,,

IIL= t

(7.32)

-

Next, writing ((pu)E V)u = div((pu)E (gu) [div(pu)E]u and applying the results of Appendix F, we deduce from the equation satisfied by u that IIUIIL.(O,T;W1.s)

< c 1 + II(p A

II(PU)E

II[diV(Pu)E]uIIL-(nx(O,T))

This inequality obviously yields the following one valid for any r E Is, 00] II utIIL-(O,T;W-1.a) + II

PUII

Lr,/(r--)(O,T;L1)

II LII Lr(O,T;L8)I

Since p > 1 + -y, we deduce from (7.32) and (7.33) the bounds we claimed above

provided we can bound the last term of the right-hand side of (7.33) for all s E (1, oo). We then check this by a bootstrap argument: in view of the bounds

Existence proofs via regularization

193

shown above, we can take s = 2, r = oo and the above argument yields a bound on pin L°° (0, T; LP- 1) and thus in L°O (0, T; L'Y). In particular, if 'y > 2, then pu is bounded in L°° (0, T; Li ). We may then choose r = s and deduce from (7.33) IIUIIL.(O,T;wi')


[1

(7.34)

+

Our claim then follows easily in that case. Next, if ry < 2 (that is a < "y < 2

andN=2or s <-y<2andN=3),wemaychoose s=p+43,r= 3 -

so

that rs/(r - s) = p if N = 3 (s < -k- if N = 2) and we obtain a bound on p in L°°(0,T; Lq) where q = (s -1)(p - 1). Since q > 2, we have thus obtained a bound on pu in L°° (0, T; L') and we may conclude as in the preceding case. The passage to the limit as R goes to +oo is now quite easy in view of all the bounds proven above. In fact, the argument sketched in the previous step based upon the proof of Theorem 5.1 also applies here (and is in fact simpler). The only observation that we need to mention is the following: for any 9 > 0, the weak limit of pe (p A R)'Y coincides with the weak limit of pO+'Y (as R goes to

+oo). Indeed, we just note that we have P°(P A R)'

- pe+7IIL1(nx(o,T)) < <

[dtf 1 R

dx

P0+71(p>R)

T

dt f dx o

pO+'r+1

n

In this way, we recover a solution (p, u) of (7.24) satisfying: u E Lq(0, T; W 1,q), au E Lq(0,T; W-1,q) and p E C([0, T]; Lq) for all 1 < q < oo ; pE E W1"1 (0,T; C' (-Q)) and (pu)e E LOO (0, T; Ck(Si)) and thus pf E C1([0, T]; Ck(Q)) (for all

k > 0). Indeed, pu E C([0,T];Lq) (for all 1 < q < oo) and thus (pu) e E C([0, T]; C' (11)) (for all k > 0). 0

Step 5: Periodic case; solution of (7.30). We only explain how to obtain a priori estimates on a solution (p, u) of (7.30). We have already obtained in the preceding section estimates on p in C([0,T]; Lq(SZ)), u in Lq(0,T;Wi,q(SZ)) and on i in Lq (0, T; W -1 ,q (S2)) for all 1 < q < oo. We show here that it is possible to obtain bounds on p in C([0,T];W1,q(Sl)), at in C([0,T];Lq(SZ)), U in Lq (0, T; W 2,q (SZ)) and on au in Lq (SZ x (0, T)) for all 1 < q < 00. With such bounds, it is relatively easy to build a (unique) solution and we leave it to the reader. Let us also recall (from step 4 above) that we may even obtain C°° solutions if we replace p A R by a smooth truncation. In order to prove the bounds we just stated, we adapt and extend a method of proof due to V.A. Weigant and A.V. Kazhikhov [552]; see also A.V. Kazhikhov [294]. We first prove that p is bounded in L°°()). Indeed, we write

Existence results for Cauchy problems

194

at

(log p) + u

/(log p) + div u + bpp-1 = 0

(7.35)

and

div u =

µ

-

+ (P A R)ry

-

+

(-0)-1 div (b(pP)Eu)

Jt

(-A)-i

(p A R)ry +

div (P£f )

- RiRj ((Pui)Euj) - a

(-L)-1 div

(PEu)

All the terms, but the last one, of the right-hand side of the preceding inequality are clearly bounded since pE f , (p")Eu are bounded in LOO (SZ x (0, T)) and (pui)E is bounded in L°° (0, T; Ck (Ii)) for all k > 0 while u is bounded in L°° (0, T; Ca (SZ))

for all a E (0,1) by Sobolev embeddings. Similarly, -D = bounded in L' (n x (0, T)). Therefore, we deduce from (7.35)

a

-5j (log

p+P)

+bpp-

1

div (peu) is

= XF

where W is bounded in L°° (SZ x (0, T)) since u V4D = uiRiRj is bounded in L°O(11 x (0, T)). Since 5P-4 > 0, this equation implies, using the maximum principle, a bound from above on log p + This yields the desired bound on p in L' (f2 x (0, T)). Let us observe, in passing, that the above equation also yields a bound from below on p. Next, we differentiate the equation satisfied by p and we obtain easily for all q c- (1, oo) using the bound on p in L°O(SZ x (0, T))

at I VPI9 + div (uI VPI4) < C [I DuI I VPIq + ID2uI IOPIQ-1]

.

This inequality implies that we have on (0, T) dt IIVPIILq(n) <_ C [II DUII L- IIVPIILq + IIDZUIILq}

(7.36)

We then use the equation satisfied by u and deduce first from standard parabolic estimates for all t E [0, T] II D2uIIL4(nx(o,t)) < C[1 + IIV(PA R)ryIILq(nx(O,t))]

hence IIDZUIILq(nx(o,t)) < C11 + IIVPIILq(nx(o,t))]

.

(7.37)

Next, we use an estimate from [5521 on the heat equation extended in Appendix F and find for all t E [0, TJ choosing q > N IIDuIILOO(nx(O,t)) < C 10g(1 + oma t IIV(P A R)''IIL9(n))

therefore

Existence proofs via regularization

195

IIDuIIL-(nX(o,t)) < C 1og(1 + o
.

(7.38)

Combining (7.36), (7.37) and (7.38), we deduce, denoting X (t) = maxo<s
X(t) < C 1 +

r

t

X(s){1+log(1+X(s))}ds

Jo

As is well known, this differential (or integral) inequality implies a bound on X on [0, T] hence a bound on p in LO( 0, T; W 1,q (Q)) and thus, in view of (7.38), a bound on Du in L°O(SZ x (0, t)) and, in view of (7.37), a bound on u in Lq(0,T; W2,4(fZ)). Finally, the bound on at in Lq(SZ x (0,t)) is then easily deduced from the equation satisfied by u. 0 The regularity analysis performed in step 5 above can also be applied directly

to the system (7.24) and used to obtain the existence of a (unique) smooth solution (p, u) of (7.24) ((p, u) E Coo (SZ x [0, T]) ). Indeed, the only modification to be made concerns the derivation of the L°° bound on p. We write in this case div u = + p7 - Tot, (f) + W where W is bounded in Lc* (SZ x (0, T)). Then, (7.35) yields as in step 5 above

p7 = W

5i (log

where' is bounded in L°°(1 x (0, T)). This equation implies, using the maximum principle, bounds from above and then from below on log p+l and thus on log p. This shows a L°° bound (and a bound from below) on p. 0

Step 6: Dirichlet boundary conditions. The proof made in steps 1-5 cannot be (or we believe it cannot be) immediately adapted to the case of Dirichlet boundary conditions, and we have to make a somewhat different argument. One possibility consists in deducing the existence of solutions in the case of Dirichlet

boundary conditions from the existence result already proven in the case of periodic boundary conditions by using the so-called domain penalization method.

More precisely, we choose a "periodic box" 1 = fN 1(ai, bi) (ai < bi) such that SZ C fl-without loss of generality, making a translation of the origin, we may take ai = 0 and set Ti = bi - ai for 1 < i < N. Then, we introduce p E COO (RN ), periodic in each xi of period Ti for each i E {1, ... , N}, such that p = 0 in SZ, P > 0 in f2 - l We then consider the following problem in RN x (0, T) where the solution (pe, uE) is required to be periodic in xi of period T2 (V i) and e E (0, 1] i9pe

+div(puE)=0, pE'>0

a (p u ee) + div (pEE u ® uE) 1

+ aV (p')7 + pue = P.f ,

µ0u'_ V div uE

(7.39)

Existence results for Cauchy problems

196

where f is an extension of f to fl x (0, T) by 0 and is then extended periodically to RN x (0, T). The proof made above (steps 1-5) immediately adapts to the system

(7.39) and thus yields a solution (ps,uC) of (7.39) satisfying all the properties listed in Theorem 7.2 in the periodic case replacing obviously (7.17) by

f

1(Pe)7

1 PE Iuf I2 +

a

dx +

'Y

< fc] 2

Imo Po

I2

a

+

Jo

po dx +

'Y

ds rt

J_

tt

dx

ds

Jo

n

[IDuf2 + l;(div u,)2 + dx pu"

EPIteI2

a.e. t E (0, T)

(7.40) Of course, p£ I t=o = Po, peui I t=o = mo and (p"o, rno) are periodic extensions of

(po, mo) by 0. Exactly as in step 1 above, it is enough to consider the case when f is bounded on RN x (0, T). This inequality (7.40) yields, as usual (by now), bounds on (p`)l and p1Iu2I2 in L°°(0,T; L' (fl)), on uE in L2(0,T; H1(SZ)) and on EpjU I2 in Ll(St x (0,T)) uniformly in e E (0,1). In addition, the proof of Theorem 7.1 (see in particular Remark 7.5) in section 7.1 yields a bound on pC in LP (K x (0, T)) for any compact set K C SZ where p = 'y + N'y - 1. Extracting subsequences if necessary, we may assume that p£ converges weakly in L''(0, T; L'r (fl)) for all r E (1, oo) as a goes to 0+ to some p (> 0) E LO°(0,T; D (fl)) n LP(K x (0, T)) for any compact set K C S1, and we may assume that u` converges weakly in L2 (0, T; H1(!)) as E goes to 0 for some u. In addition, since T J ue is bounded in L2(11 x (0, T)), VfP- u£ converges to 0 in L2 () x (0, T)) and thus lp- u =- 0 in fl x (0, T). Therefore, u = 0 a.e. in (St - St) x (0, T). Since U E L2 (0, T; H1 (fl)) and n is smooth, this implies that u E L2 (0, T; H01).

Finally, in order to conclude, we simply observe that Theorem 5.1 (in chapter 5, section 5.2) can be applied in our context inside the domain fl x (0, T). Indeed,

the proof of part (1) of this result is still valid here while the proof of part (2) also applies as one can easily check observing that we only need the limiting velocity namely u to belong to L2(0,T; Ho (SZ)).

Step 7: St = RN. Let us explain briefly one possible way to deduce the existence

of solutions in the whole space case from the results shown in the preceding steps. First of all, as explained in step 1 above, we only need to consider the case when f E L' fl L°° (RN x (0, T)). We then consider the solution (pR, uR) of (5.1)-(5.2) set in the ball BR(X (0, T)) with Dirichlet boundary conditions on BBR. We choose R > Ro so that fBR Po dx > 0. The existence of such a solution (satisfying the properties listed in Theorem 7.2) is ensured by step 6 above. In particular, we have for almost all t E (0, T) JBR

PRIURI7 + a 1(PR)dx +

ds J J ft1 < R

BR

dx pI DuRI2 + Z; (div uR)2

Existence proofs via time discretization

197

The conservation of mass (equation (5.1)) and this inequality yield, as usual, bounds uniform in R on pR in L°O(0,T; L' n L1' (BR)), on ARI URI2 in LOO (0, T; Li (BR)) and on DuR in L2 (BR x (0, T)). If N = 2, we deduce, as in Remark 5.1, a bound on uR in L2(O,T;Lq(BM)) for all 1 < q < oo, M E (0,00) and in L2(0, T; BMO)-considering UR as a function on R2 by extending it to R2 by 0. Next, a careful but straightforward examination of the proof of Theorem 7.1

shows that pR is bounded in LP(BM x (0, T)) for any M E (0, oo), with p = -y + N 7 - 1, uniformly in R > 1 + M. It only remains to apply the compactness analysis of chapter 5 (Theorem 5.1 in particular) in order to recover a solution of (5.1)-(5.2) in the whole space. 0 7.4

Existence proofs via time discretization

This section is devoted to the presentation of another proof of Theorem 7.2. This proof is based upon a simple time discretization of equations (5.1)-(5.2). Let M > 1; we denote At = T-r and define by induction for k > 1 (pk, Uk) to be the solution of Ot pk + div pkuk

(pkuk) = Qt pk-1

+ div(pkuk ® Uk) - µiuk - Vdiv uk + a0(pk),'

(7.41)

(7.42)

1

= pk fk + 1 Mk-1 and mk = pkuk, po = po, m° _= m0, f k is a convenient discretization of f to be detailed below. More precisely, since the uniqueness of solutions of such problems is not known, we shall in fact simply take one solution whose existence follows from the results of chapter 6, and Theorem 7.2 will follow upon letting At go to

0 (or M go to +oo).

Step 1: Preliminaries. Exactly as in the proof of Theorem 7.2 given in the previous section, we shall only consider the periodic case since the other case can be "deduced" from it, even though the proof presented below can be adapted to

the other cases directly. Similarly, we may assume that f is smooth. Then, we solve (7.42)-(7.43) requesting all unknown functions and all data to be periodic in each x2 of period Ti > 0 (V 1 < i < N). And we may simply take f k to be f (x, kOt); if we were not assuming f to be smooth, we would have to take f k to be (of 4k-1)ot f (x, s) ds for instance. In addition, we may assume that po is smooth and positive on RN and that uo is smooth (or simply L°°). Finally, when N = 2, we have explained in the preceding section why it is enough to study the case when -y > 2 and to prove the existence of a solution (p, u) satisfying p E L2 ,t for all q E [1, 2-y -1). We may now use the results of chapter 6, namely Theorems 6.1-6.3 (section

6.2), which ensure the existence of (pk, uk) (for 1 < k < M) periodic, solving

Existence results for Cauchy problems

198

(7.41)-(7.42) and satisfying, for 1 _< k _< M, uk E H1, Pk E Lq with q =

1 fN>4orN=3and<3 ,q=+ooifN=2orN=3andry>3,

uk E W l,a for q E [1, oo) if N = 2 or N = 3 and y > 3, and the following energy inequality ffZ 21

(pk-lIuk12+pkIukI2) +

Lt('y-1) ary Otmk-1.uk +

< j Pkuk.fk +

(Pk)ry + uI Duk12 + e(divuk)2 dx 1

At

pk-1(pk)7-1 dx. (7.43)

Of course, integrating over the period the equation (7.41), we obtain for 1 < k <

M pkdx =

Jpk_ldx

[3

.

(7.44)

in

Step 2: A priori bounds. Obviously, (7.44) shows that pk is bounded in L1 uniformly for 0 < k < M. Next, we write (7.43) as follows

f2(pklukl2-pk-llUk-l12)dx+ a .Y

+

2 pk-11uk-uk-ll2

dx +

(pk)y - (pk-1)7dx l

1J

, -a 1

+ (pk-1)ry _ ,y(pk)ry-l pk-1 dx + At < At fn pkuk . fk

( Y -1)( P k)ry

}

(7.45)

tilDukI2 + Z;(div uk)2 dx it

Then, we observe that ('y-1)al + if' - ^yalf -1b >_ 0 for all a, b > 0 since (t H V ) is convex on [0, oo), and summing (7.45) from k = 1 to k = Q, we deduce easily the following bounds sup IIP`IILv + IIPeIutI2IIL1 < C

O
k=1 and

f

[k1 Iuk_uk-112]+[('Y-1)(pk),+(pk-1)ry_

(Pk)-"-1pk-l]dx

(7.46)

< C (7.47)

1

M At IIDukIIL2 <- C, k=1

where, here and below, C denotes various positive constants independent of k, At and thus M. We then deduce easily the following bound M

of Iluk IIHI <- C. k=1

(7.48)

Existence proofs via time discretization

199

Indeed, since -y > 1 if N = 2 and ry > NN if N > 3, we have, using the above bounds, M

M

OtIIukIIL2 <

ukl

IIDukIIL2 + k=1

k=1

M

2

rr


ukl dx

k=1


M


f

pkl uk I

ukl2

k=1 M


Sup IIP`IIL-r 0

At IIPkIILQ

(7.49)


k=1

> N when N > 3 and -y > 2 when N = 2. In order to prove (7.50), we follow the proof of Theorem 7.1 and we write for k > 1

where q = -y + 1 if

a(pk)7 = a (pk), + (p+t;) div uk +Rj (Pkukuj)

(7.50)

+ At [(K(pkuk) - K(pk-luk-1)1 - K(Pk.fk) denoting by K the operator (-A)-l div. We then multiply (7.51) by pk and remark that we have

1 [K(pkuk) At

-

K(pk-luk-l)]pk

= 1 [pk K(pkuk) Yt

-1

- pk-1K(pk-luk-1)1

J

l (Pk

- pk-1) K(pk-luk-1)

,

and

Y

(pk

Pk-1) K(pk-luk-1) = -dlv(ukpk

Therefore, we deduce from (7.51)

K(Pk-Iuk-1))+pk uk

Rj(Pk-1 ug-1)

.

Existence results for Cauchy problems

200

a(pk)ry+i 1

= apk (fn(Pk)ry) + )Pk -ukR7(Pk-lug-i)} - K(pkfk)pk +Pk{RiRj(PkUkuj) + [pkK(pkuk) _Pk-1K(Pk-luk-1)+div(ukpkK(Pk->uk-1))] Ot

(7.51)

Multiplying (7.51) by At, integrating it on fl and summing this inequality from k = 1 to k = M, we deduce easily our claim (7.49) exactly as in the proof of Theorem 7.1. In fact, introducing for any function (cp°, cp1, . , cpM) the piecewise constant (respectively piecewise affine) function cp (respectively cp) defined by

cp(x, t) = cpk(x)

if

kit < t < (k+1)ot

(respectively cP(,t) = c0k() + (t-kOt)(Wk and (7.51) as

na,0k)()), we may rewrite (7.50)

a(p)ry = a f (p)"1 + (µ+e) div u n

+R;.R;(P'ujfij) +

(7.52)

a K(Vu) - K(pf)

nd a(p)^'+1

= ap

(],3u1)

+ (µ+e)(div u)p

+ '9{R,R,(puj-uj) - utiR;RR((PUl)( - At))} - K(Pf)P

+

(7.53)

(pK(pu)) +div[upK((pu)( - At))]

are bounded in LOO (0, T; L7) : gfi12, pJu12 are bounded in L°° (0, T; L 1) and thus pu, pu are bounded in L°O (0, T; L2ry/ (-y+ 1)) ;

In addition, (7.46) imply that

while u and u are bounded in L2 (0, T; H1). These observations allow us to copy the proof of Theorem 7.1 in the case when 0 = 1 and ry > N if N > 3, -y > 2 if

N = 2.

Step 3: Passage to the limit when -y > N. We assume in this part of the proof that -y > N. We first observe that (7.41) and (7.42) may be written as

at + div (pfi) = 0

at

(Vu)

+ div (pfi (9 fi) - µ0u - V div u + aV (p)ry = p f .

(7.54) (7.55)

Next, we remark that (7.47) yields in particular the following information

p - p( - At) , p - p converge to 0 as At goes to 0+ , in Lq (0, T; Lry) for all 1 < q < oo.

(7,56)

Indeed, we first observe that the following elementary inequality holds (since -y > 2) for all a, b > 0

201

Existence proofs via time discretization

(-y-1)a' + P - rya'1-1b > via-bI"

for some v > 0.

,

Therefore, (7.47) yields M

1113-13(' - Ot)II L7(Ot,T;L1) =

E At lip, - P,-IIIiy < COt k=1

and (7.56) follows easily recalling that 13 and 13 are bounded in LOO (0, T; Lt).

We also claim that (7.47) yields

pu - (3i) ( - At) , 1311-1311 converge to 0 as At goes to 0+ , 2^y in Lq(O,T; L'') for all 1 < q < oo, 1
and forall1
1

=

1

(7.57)

2

+

2N

This convergence is obvious since 1311 is bounded in L°O (O, T; L27/(7+1)) n pkuk-pk-luk-1 = (pk-pk-1)uk+pk-1(ukL2(0,T; Lr°) and since we may write uk-1), and (7.57) follows using (7.47) and (7.56). In addition, we claim that 1311®u - Fu ®11 converges to 0 as At goes to 0 in L1(SZ x (0, T)) (for instance). Indeed, 13(11®11) is bounded in L1(0, T; L''1) where

= 7 + N 2 if N > 3 (1 < ri < 'y if N = 2) and in L°° (0, T; Li). Furthermore, we have I11IIpu-(au)('-ot)

113(11®11)-1311®111 <

and our claim follows easily from (7.57). In fact, this argument shows that 13(11® u) - Vu ®11 converges to 0 as At goes to 0 in L1(0, T; L') n Lq (0, T; L1) for all

1 3, 1
p > 0 and 11 converges weakly in L2 (0, T; Hl) to some u. Then, (7.56) implies that p` also converges weakly in LOO (0, T; L7) (weak-*) and in L7+1(SZ x (0, T) )

to the same p. In view of the bounds obtained above and (7.54), we deduce as in the proof of part (1) of Theorem 5.1 (section 5.2, chapter 5) that 1311 and thus 311, (1311) ( - At) and 1311, using (7.57), converge weakly to pu in LOO (0, T; L21'/(7+1) )

(weak-*) and in L2 (0, T; Lo) (where ro < y if N = 2, o = y + zN if N > 3). Then, using once more the bounds shown above and the proof of part (1) of Theorem 5.1, we deduce from (7.55) that Fu ®11, and thus 1311®11, converges weakly in L'(0, T; L'') n Lq (0, T; L1) (1 < r < r1 i 1 < q < oo) to pu ®u.

In order to pass to the limit in (7.54)-(7.55), we need to obtain the strong convergence of 13 in L7 (S2 x (0, T)) (say) and we shall do so adapting the proof (of part (2)) of Theorem 5.1. More precisely, we adapt the proof made in section 5.3 (chapter 5) in the case when we have bounds in Ly 1 n L°° (0, T; L3) for some

t

s > N, which is precisely the case here. We use the notation of this proof and denote by 7 the weak limit of 0 for any W.

Existence results for Cauchy problems

202

First of all, as shown above, we have on RN x (0, T) (with periodicity)

+div(pu) = 0, a(Pu)

1

+div(pu(9 u)-j u-l;Vdivu+aV pT = pf.

Next, we observe that we have for 1 < k < M (with an easy justification) k-1

k

(log pk + 1)

P

+ div (ukpk log pk) + (div uk)pk = 0 I

Lit

and since (1 + log pk) (Pk o f -1)

At

pk log pk - pk-1 log pk -1, we deduce

(pk log Pk _ pk-1 log pk-1) + div (ukpk log pk) + (div uk)Pk < 0

on RN x (0, T) . (7.59)

This information combined with the convergences established above and (7.50)-(7.53) allow us to copy the proof of Theorem 5.1 and yield the desired strong convergence of p, provided we show that we can pass to the limit in p[R%Rj(,Mifij) - 4iR{Rj((p4Lj)( - At))]. This is the case since we just have to remark that JP[uiR%Rj(Pfij) - i2 R%Rj((Puj)( - At))] I <_ pIiI IRjRj(,3u

- (pu)( - ot))I

which converges to 0, as At goes to 0, in L1(11 x (0, T)) (and in fact in L1(0, T; LQ )

where 1 > . > 7 + NZ, recall that -y > N) in view of (7.57). In conclusion, we have shown the existence of a solution (p, u) of (5.1)-(5.2) satisfying the properties listed in Theorem 7.2 except that we have only shown that p E L-Y+ 1(cl x (0, T)) instead of L4(1 x (0, T)) with q = -y + 1 -y - 1 (and q < 2-y -1 suffices for N = 2).

Step 4: p E L4(f x (0, T)). We explain in this step how it is possible to recover the integrability of p a posteriori without assuming it as we did in Theorem 7.1. The possibility of doing so was mentioned in Remark 7.2, where we discuss an even more general possibility than what is needed here. We thus follow the proof of Theorem 7.1 (steps 1-2 in particular) but instead of writing (7.7), we replace pe by an appropriate test function /3(p) to be determined below satisfying as least /3 E C1 ([O, oo)), 8 > 0 on [0, oo), 0 and /3' are bounded on [0, oo). We obtain the following identity

ap'P(P) = a fn Pry

i3(P)

+'

[3(p)(-0)-1

div (pu)]

+ div [u/3(p)(-0)-1 div(pu)] + (p+1;)(divu),0(p) div (pu) + (0'(P)P-l3(P))(div +Q(P)[IZ Rj(Puiuj) uiRjRj(Puj)] 0(p)(-0) u)(-0)-1

-

-

div (P.f) (7.60)

Existence proofs via time discretization

203

Next, if N > 4 or if N = 3 and 3 < ry < 6 (recall that we assume that -y > N), we may check easily that p)` ,Q (p) is bounded in L'(1 x (0, T)) provided fl satisfies Cte-1 fort > 0. 0 <,8(t) < Cte , I f'(t)I < (7.61) In addition, the L1,t bound on p((3(p) is uniform in 0 provided (7.61) holds for

some fixed positive constants C. We then conclude choosing (3(t) = (t A R)e for

R > 1-we neglect the lack of smoothness of /3 at t = R which can be easily circumvented; indeed, we deduce that 3(p) (p A R)e is bounded in Li (C x (0, T)) and we let R go to +00. In the case when N = 3 and y > 6, or N = 2 and ry > 2, the above argument yields the following inequality for all R > 1 II p''(P A R)ellL'Z,t

<-

C(l + II(p A

R)B(divu)(-0)-1 div(Pu)IILzjt)

where C denotes various positive constants independent of R. We then follow the corresponding proof in step 1 of the proof of Theorem 7.1 in the case when N > 3 and we deduce IIP"(P A R)°IIL=,< <- C 1 + lip A RIILx

and we conclude easily in this case. In the case when N = 2, we adapt that proof and obtain also II(P A R)e(divu)(-o)-1 div (pu)IILi,t < I I Dull L=,, lip A RII Ly

<- Clip A R110

to

IIPuIIL

IIPuII2 L

+e)/

since

'Y

+0 <2

(L(-'+0)1-1)

s) IIPuIIL


< ry where B

O

6 + 7+20e

-Y+1

_ e, i.e. 6 = 'y('y - 6) ('y2 - e(ry + 1))-1 < -y since +

ry

At this stage, we have proved Theorem 7.2 in the case when ry > N and we conclude this alternative proof by considering the remaining case when y < N.

Step 5: y < N. The idea of the argument is to approximate (5.1)-(5.2) by the same system of equations where we simply replace ap"( by app + cpN+l (e E (0,1]). Then, we observe that the proof made in the preceding steps immediately apply and yields a solution (pE, uE) of this system of equations, satisfying all the properties listed in Theorem 7.2 except that pE E L' 1 x (0, T)) with q = N + N (N + 1) if N > 3, 1 < q < 5 if N = 2, and (7.17) is replaced by e ry jPIUcI2 + y - 1 (P) + (p) N1 dx +

Jo

ds

n

E2 dx p I Du I +e (div ue)2

Existence results for Cauchy problems

204


Imo l2

Jn 2

Po

t

a

+ 7

1

Po +

N

Po +1

+f

a.e. t E (0, T).

ds Jst dx Peue . f

o

We immediately deduce from this inequality uniform bounds in e on pE in L°O(O,T; L'r), uE in L2(0,T; Hl), pEIuEI2 in L°°(0,T; L1) and E(pE)N+1 in L°°(0,T; L1). Furthermore, following the proof of Theorem 7.1 with 9o = N 7 -1, we deduce easily bounds on pE in Lq (St x (0, T)) where q = -y + Oo = y + ('y - 1) and on E(pe)N+l+eo in L1(SZ x (0,T)). N We finally conclude passing to the limit as e goes to 0+, following and adapting the proof of Theorem V.1. The only new term involves E(pE)N+1+e where

9 > 0 is small enough. In particular, we may choose 0 in (0, Oo) and we thus deduce from the preceding bound on e(pc)N+1+9o the fact that E(pe)N+1+e converges to 0 in Lx,t (say) as e goes to 0+. 13

Remark 7.12 In this remark, we present an observation on the regularity of the solution (which follows from the existence proof given in the preceding section). We assume that u E L2,'/(-'-1) (0, T; W 1,P) and pu E L°° (0, T; L") for some p > N and we take f = 0 (or smooth) in order to avoid irrelevant technical details. Then,

we claim that p E Lx t and inf ess p > 0 assuming of course 'that po E L°° and inf ess po > 0. Indeed, let us recall that we have at(log P-p)

p-cp) +

µ+a pry

(7.62)

= RiR? (Puiuj) - uiRi=R7 (Puj )

where cp =

µ+C(-0)

div(pu) E L°O(11 x (0, T)).

Then, we observe that, if N < q < p, for almost all t E (0, T) II RjR.7(Puiuj) - u=RiR7(Puj)II LZO < CII [RiRj,ui](Pua)jjw=,9

< C IIuIIw=,p IIPuIILo < C IIuII ,,i,p IIPIIL< C IIuIIW=., exp(2IIWIILoo) [IIPeILO eXP(-IIcPIILo

a

<_

<2

+

Il e-"A II L4 exP(-YIk2IILo) + CII

-1) exp

try (1IIcoIILoo)

exP(-7IIcuILo)

Next, we apply the "maximum principle" on equation (7.62) we and find for all t E (0, T) [sup ess (log p- cp)] (t) < sup ess (log po -cpo) + C

/'t

J0

ds IIuII W; p 11 <_ C

hence, p E LOO (SZ x (0, T)). We may then apply again the "maximum principle"

and deduce for all t E (0, T) [inf ess (log p-cp)](t) > inf ess (log po-coo)

205

General pressure laws

-C

f ds (IIPIILIIuuV.P + IIPII7c) > t

-C,

0

therefore, p is bounded from below and we conclude. 0 Remark 7.13 The above proofs also show that we may add in Theorem 7.2 the following energy inequality

JpIuI2+

1

dt n 2

prydx + f PjDu12 + (divu)2dx < fn Pu f dx inD'(0, T). n

7.5 General pressure laws In this section, we discuss the analogues of the results shown in the preceding sections in the case of general pressure laws, i.e. when we replace (5.1)-(5.2) by

j +div(pu) =0, p>O 8

)

1

+ div (pu ®u) - pAu - l; V div u + Vp(P) = Pf

and p is a non-decreasing continuous function on [0, oo). As usual, we consider either the periodic case, or the case of (homogeneous) Dirichlet boundary conditions or the whole space case and we assume that p > 0, P + l; > 0. In order to simplify the presentation and avoid unnecessary technicalities we assume in addition that f E L°° (0, T; L' n L°°), leaving to the reader the straightforward extensions and adaptations to more general force terms (combining arguments introduced above in chapter 5 and in the preceding sections with those developed below).

Throughout this section we shall make the following assumption

lim inf p(t)t-^' > 0

(7.64)

for some -y > 1.

Then, replacing Theorem 5.1 by Theorem 5.2 (section 5.5, chapter 5) in the proofs made above of Theorem 7.2, one can check easily that Theorem 7.2 (and Theorem 7.1) holds in the case of Dirichlet boundary conditions or in the periodic case, with the same proofs under the same conditions on y and on the initial conditions (and on f) and that the solution (p, u) satisfies in addition p(P)Pe E L'(11 x (0, T)) with 0 = y + N 'y - 1. The case when fl = RN requires a more careful analysis. We begin with a priori bounds, recalling and extending some observations made in section 5.5 (chapter 5). First, we recall the local energy identity cat

P (2 2

+ q(P)

+ div u p 12 2

+ (p+q)

u2

-POI 2 - div(u div u) + pjDuI2 + l;(div u)2 = pu f where q is defined (up to a constant) by

4 (q(t))/t) = p(t)/t2 for t > 0.

(7.65)

Existence results for Cauchy problems

206

As we shall see, the a priori bounds depend upon the behaviour of p near 0. First of all, substracting a constant from p if necessary, we may assume without loss of generality that p(O) = 0 (and thus in particular p > 0 on [0, oo)). Then, we set q,(t) = t fi p(s)s-2 ds fort _> 0 (q,(0) = 0 since p(0) = 0) and we observe that q1 = f 1l p(s)s-2 ds + p t is non-decreasing since we have (when p is smooth, the general case follows by density) qi = P tt >_ 0. Thus, qi is continuous and convex on [0, oo). In particular, qi(0+) = limtjo+ qi(t) exists in [-oo, +oo). ds < oo. Indeed, if this We then claim that qi (0+) > -oo if and only if fo ds. Conversely, we set a(t) = f tl p(s)s-2 ds integral is finite then qi (t) _> - fo for t E (0,1] and we remark that, if ql(0+) > -oo, then q'1 is bounded on (0,1) and thus ta'-a = t2 (i )' is bounded by a positive constant C on (0,1). Therefore, we have on (0,1]

0 < a(t) < a(1) + C t (t

-

-

1

thus a is bounded on [0, 11, and our claim is proven. We shall then consider separately the two stituations where fo p(s)s-2 ds <

oo in which case we choose q(t) = t fo p(s)s-2 ds for t > 0, and where f f p(s)s-2 ds = +oo in which case we choose q = q1. The reason why we need to make a distinction between the two cases is the fact that p vanishes at infinity and thus, in the second case, we cannot choose q in such a way that q(p) > 0 on RN.

We begin with the first case namely when ff p(s)s-2 ds < oo and q(t) _ t fo p(s)s-2 ds > 0 for t > 0. Then, integrating over RN the first equation in (7.63) and (7.65), we deduce easily that p is bounded in L°° (0, T; L' (RN)), pJuI2 is bounded in L°°(0,T; LV(RN)), Du is bounded in L2(RN x (0,T)) and u L2N/(N-2)(RN)) is bounded in L2(0,T; if N > 3 (respectively u is bounded

in L2(0,T; H1(BR)) for all R E (0, oo) if N = 2), and q(p) is bounded in L°°(0,T;L1(RN)). From this last bound and the conditions (7.64), we deduce easily that p is bounded in LO° (0, T; L1 (R')) since we have for some v E (0,1), for some R E (1, oo) and for almost all t E (0, T) JRN

p" (t) dx < !RN

R-'-1p(t)1(p(t)
< R-Y-1 f p(t) dx + v N

J

N

q(p(t))1(p(t)?R) dx

q(p(t)) dx .

Next, we follow the proof of Theorem 7.1 and deduce a bound on p(p) (1 + pe) in L' (St x (0, T)) where 0 = N -y -1, and thus in particular p is bounded in L4 (St x

(0, T)) with q = -y + 0. Adapting the proof of Theorem 7.1 is straightforward once one notes that p(t)/t is bounded on (0,1). Indeed, we have on (0, 1) 1

j

1

p(s)s-2 ds >

itJ

1

p(s)s-2 ds >_ p(t)

Jt

s-2 ds

= p(t)

t -1

.

207

General pressure laws

Finally, we may follow and adapt the proofs of Theorem 7.2. We obtain in this way the existence of a solution (p, u) of (7.63) satisfying all the properties listed in Theorem 7.2, and in addition p(p) (1 + pe) E L1 (RN x (0, T)), with (7.17) replaced by

f

N

pu2 + q(p)dx(t) + 1lmol2

fo

ds f dx{ pIDuI2 + (div u)2 }

rt < LN2PO + q(po) dx +J ds / dx pu f, a.e. t E (0, T). o

(7.66)

ff

Of course, we need to assume that the initial conditions (mo, po) satisfy in addition to (7.1): q(po) E Ll (RN). We now turn to the other case, i.e. when fo p(s)s-2 ds = +oo. Let us recall that we choose in this case q(t) = t fl p(s)s-2 ds. We then assume that, at least formally, we have for any function p >- 0 in L' fl L°O(RN) : 2

at

PI 2 + [q(P) -- q(p) - q'(p)(P-p)]

- p0 122 +div u [pL2-+(p+q-q1(7i)p)] 2 + pIDuI2 + l;(div u)2

-

div (u div u)

(7.67)

= pu (f - 0(q'(p)))

which yields d

d

f +

22

N

I

pI

+ [q(P) - q(p) - q'(p)(P-p)] dx iIDuI2+t;

N

(7.68)

(div u)2dx = JNpu (f -V (q'(p))) dx. IPo s

E L1(RN) and {q(po) Therefore, if we assume that po E Ll n L1(RN), q (-p) - q' (p) (po - p) } E Li (RN), then we deduce a priori bounds on p, pI uI2 and {q(p) - q(P) -q'(p)(po - p)} in L°O(0,T; L1(RN)), on Du in L2(RN x (0,T)) and L2N/(N-2) (RN)) if N >_ 3, provided we assume that p satisfies on u E L2(0,T; (for example)

V(q'(p)) E LOO(RN) Let us give a few typical examples of q and p.

(7.69)

Example 7.1 p(p) = p + app', a > 0, 7 > 1. Then, we have: q(p) = p log p + 7 -r (p'r - p). We may for instance choose p = exp (- < x >a) where 0 < a < 1, <x>= (1+IxI2)112, and (7.69) clearly holds since we have IVq'(p)I =

p I

+a7#7-2VpI < CID log pI < C.

;

Next, we compute q(p)--q(p)-q'(P)(p-P) = [p log(P/p)+7i-'p]+a[PI+(7-1)pry-

Ip -'p] and we deduce easily that this quantity belongs to L', assuming that

Existence results for Cauchy problems

208

p E L1, if and only if p E Li' and p log (p/ p) + p - p or p log p + p < x >01E L'.

We then claim that this latter requirement is also equivalent to p log p and p < x >a E L'. Indeed, we have

f Ip log p+<x>a Idx > I

p(log p+<x>*)dx

J (P?pl/2)

RN

>;

p<x>adx

J

and

p <x>a dx < LNh12 <x>a dx < oo . O jp<1/2) Example 7.2 p(p) = p6 + ap", a > 0, 0 < b < 1 < 'y. Then, we have: q(p) = 116(p-p6)+ ryl(p1'-p) and thus Vq'(P) = b7i(6-2)op+a,yp'-20p. We may then choose p =< x >-a with a = 116, and we check easily that (7.69) holds with such a choice. Of course, p E L' if and only a > N, i.e. 6 > NN 1 In addition, we have: q(p) q(p) q'(P) (P-P) = [P6 - 1-6 p6 + 1-4'p] + a[pi' + ('y-1) p -71p-1p]. If b < N 1 (and b > N-2 for instance), one can check that q(p) - q(p) - q'(p)(p-p) E L' contradicts the fact that p E V. This is why we assume from now on that 6 > N i First of all, if b > N+1, then P6 E L'. We then claim that, in this case and if p E L', q(p) - q(p) - q'(p)(p-p) E L1 is equivalent to p5, ap'y and <x> p E V. Indeed, if q(p) q(p) q'(P)(P-p) E L1 then (p - 116p6 + 1b6p -1p) and a(p1' + ('y -1) p ' - ryp'-1 p) E L'. Since p6 E L', py E L' and r-1 E L°°,

-

-

-

-

we deduce that ap E L1 and p6 - 6 < x > p E V. Finally, we have 6 <x> p - p6 > p6 on {x / 6 <x>

p1-6 > 2},

oo, and our claim is proven.

f(P.

)

P6 dx < fRN C6p6 <

Finally, if N1 < b < N+1 and p E L1, then q(p) - q(p) - q'(p)(p- p) E L1 is obviously equivalent, in view of the argument above, to ap E L1 and (p6-116p6+1a6bp6-1p) E V. And, (p6-116p6+166p6-1p) E L' is easily shown + to be equivalent to the rather delicate constraint < x >0 I p P6 1(P?3 /2) + p61(PS7/2) E L', where 0 = (2 - 6)/(1 - 6). We now continue the derivation of a priori bounds and we assume from now on that (7.64) holds. We first show that the bounds obtained above yield a bound

on p in L0(0,T; Li'). Indeed, introducing q, (t) = q(t) for t > 1, = q'(1)(t - 1) for 0 < t < 1, we see that qi and q - q1 are convex on (0, oo) since q'1 (t) _ max(q'(t), q'(t)) and (q - q1)'(t) = (q'(t) - q'(1))1(o
- q(P) - q'(p)(P-p) ? q1(P)

- qi(P) - gi(P)(P-P) > 0

and thus qi(P) -q1(P) -q (P)(p-P) = q1(P) -q'(1)(P-1) = (q(P) -q'(1)(P-1))+ is bounded in L°°(0, T; L1). Hence, (q(p) + q'(1))+ _ (q(p) + p(1))+ is bounded in

.

Other boundary-value problems

209

LO°(0,T; L1). Next, we deduce from (7.64) the existence of Ro E (1, oo), v E (0,1)

such that

q(p) + p(1) > vp7

if p > Ro.

Therefore, p" 1(p>&) is bounded in L°° (0, T; L') and we conclude since we also have

JNP
R.o-1

fN

pdx

.

At this stage, it is now straightforward to adapt the proofs of Theorem 7.1

and 7.2 and these results hold under the same conditions on y, and on the conditions stated above on p and on the initial conditions (mo, po). In particular, we obtain in this way the existence of a solution (p, u) of (7.63) satisfying all the properties listed in Theorem 7.2, with (7.17) replaced by (7.66), and in addition

p(p)(1 + pe) E L1(RN x (0,T)) (9 = N'Y - 1), and q(p) - q(P) - q'(P)(P-P) E L°°(0, T; L1(RN) )

7.6 Other boundary-value problems In this section, we wish to mention the adaptations and extensions of the results shown in sections 7.1-4 in the case of various boundary-value problems. We shall begin with the case of non-homogeneous Dirichlet boundary conditions in, say,

a bounded smooth domain fl. Next, we shall consider other problems set in unbounded domains like exterior domains, or tube-like domains. We thus begin with non-homogeneous Dirichlet boundary conditions. We want to solve (5.1)-(5.2) in a bounded smooth open domain in RN (N > 2) and we prescribe (p, u) on aSZ as follows

u=uonacZ x (0, T), p = p on {(x,t) E all x (0,T) /u(x,t) n(x) < 0}

(7.70)

where, as usual, n(x) denotes the unit outward normal to aSZ at x. The boundary data (u, p) are assumed to satisfy: (7.2) holds with f replaced by au-

at

and

(St) u E L2(0, T; H' (n)) n L°°(0, T; , diva E L'(0,T; LO°(Sl)) , Du E L1(O,T; L°°(S2))

p E Ly (ail x (0,T);

dt)

,

(7.71) (7.72)

(or in other words, f f dt fan dS p ' (u n) - < oo). At this stage, we have to clarify a rather technical point namely the meaning of (7.71): we shall consider (weak) solutions of (5.1)-(5.2) satisfying the same properties as those detailed in section 5.1 (chapter 5) or in section 7.1 above. In particular, u E L2 (0, T; H1(il)) and thus u = u on all x (0, T) simply means

that the traces of u and u on aSi x (0, T) (that belong to L2(0,T; H1/2(8St))) coincide or equivalently that u - u E L2(0, T; Ho (S2)). The boundary condition on p is more delicate and we do not wish to detail all the tedious aspects of this

Existence results for Cauchy problems

210

question: a simple way to understand how one can indeed prescribe p in such a manner consists in recalling that if cp is a smooth function on 1 x (0, T) and if (5.1) holds, then we have obviously T

t f n dS

p(x,T)cp(x,T) - p(x, 0)cp(x, 0) dx + fo d

jTj

Jo

= 0.

This elementary integration by parts combined with the fact that p E LOO (0, T; L-1 (1)), / u E LOO (0, T; L' (11)) and u E L2(0,T; H1 (f2)) allows us to check that p(u u) makes sense on ast x (0, T) as an element of the dual space of the space consisting of traces on aft x (0, T) of the functional space E C([O,T];L-YI(-r-1)(&)) / at E L' (0, T; L-fl ('Y- 1) cP

E L' (0, T; W1,2-t/(-1-1)(n)) + L2(0, T; W1,r(Q))

2+N--1 ifN>3,0< r <1-,1-y ifN= 2; r iswell defined where 1 = provided 'y > 2N if N _> 3, otherwise we drop the L2(0, T; W l,r (Sl)) part in the above definition. Then, the boundary condition on p really means that p(i.n)- on an x (0, T). In addition, we shall impose a solution (p, u) to satisfy p E L' (OSt x (0, T) ; lu n1dS (& dt).

Since our proofs in chapter 5 or in section 7.1 are essentially local, the only new argument needed concerns the energy bounds, i.e. the bounds on p in LOO (0, T; L'f (1l)), on fpu in L°°(0,T; L2 (Q)) and on u in L2(0, T; H, (11)). In

order to obtain these bounds, we multiply (5.2) by (u n) and obtain, at least formally, with some straightforward computations

a

p

+

2

- µ0 )u 2ul

a p7 ry-l

+ µI

+ div u p D(u-u)I2

2

+a

ry

ry-

l p7

+ a(div u)py (7.73)

-l; div[(u-u) div(u-u)] + l;[div(u-u)}2 5t

=pf-

) (u-u)+((p .V)E.(u-u))+µdiv((u-i)vu)

+l; div((u-u) div u) -µ D(u-u) Du - div(u-u) div u Therefore, integrating by parts over SZ, we deduce

.

Other boundary-value problems d

f f f

pIu-uI2 +

dt n

+ =

n

2

pI

a ry

an

dSa

211

ry- 1 (a.u)+ P 'Y

D(u-u)I2 + l;(div(u-ii))2 dx

r rc)+ f P f - aU--) (u-2c) - 1(u

(7.74)

n dS a ry -1

(u-U) - a(div-u)p1' - pD(u - u) - Du - e div(u-u) div u dx. +

The first term of the right-hand side is bounded because of (7.72), the last two terms are easily estimated by the Cauchy-Schwarz inequality, and we argue for the second term as in section 5.1 (chapter 5) or as in section 7.1 since we made

the same assumption on y as on f. Finally, we estimate the two remaining terms easily using the bound on Du in L1(O,T; LOO (11)) and writing

fo

in

(div u) p1' dx < C II DiuI L- (n) fn p'r dx ,

(pu.V)u(u-u) dx < IIDuIIL-(n) [fn p iu-ul2dx + IIPIILI IIuIIL21/(-y-1)

.

We then deduce easily from Gronwall's lemma the desired bounds. Notice that we also obtain a bound on p in LY(8it x (0, T) ; dt). Having thus derived the "energy bounds" it is now easy to adapt the proofs of Theorem 7.2, and we obtain the existence of a solution (p, u) satisfying the same properties as in the case of homogeneous Dirichlet conditions and, in addition, p E L'(8ft x (0,T) ; ®dt). We now turn to problems set in unbounded domains. Let us immediately mention that we shall set homogeneous Dirichlet boundary conditions, even though it is rather straightforward to combine the arguments developed below with those introduced above and thus to treat as well non-homogeneous Dirichlet conditions on the boundary of the unbounded domains we are going to consider. We shall consider two types of situations: i) the exterior case when it = Oc, O is a bounded, smooth, open domain in RN-we agree that we can allow 0 to be empty in which case fZ = RN and no conditions are imposed on 8O!, and ii) the tube-like case when fl = R x w and w is a bounded, smooth, open domain in ISBN-1. We have already mentioned and studied these situations in chapter 5, section 5.6. In both cases, we wish to solve (5.1)-(5.2) and look for a solution (p, u) of (5.1)-(5.2) in fl x (0, T) satisfying the initial conditions (5.6), p > 0 and the following boundary conditions. In the exterior case, we impose

u=0

on 80 x (0, T)

(p,u)(x,t) --+ (p"0, u') as IxI -- +oo, for all t E (0,T) where p°° > 0, u°O E RN

(7.75) (7.76)

Existence results for Cauchy problems

212

In the tube case, we request that (p, u) satisfies

u=0

on (R x ow) x (0, T)

(P, U) (Xi, x', t) - (P+, a+) as xi - +oo , (P, u) (xi, x', t) -- (P , u-) as xi -- +oo, f o r all x' E w, t E (0, T) ,

(7.77)

(7.78)

where p+, p- > 0, u+ = (ui , 0), u- = (ui , 0), ui , ui E R. Both (7.75) and (7.77) are to be understood in the sense of traces of functions which are in H1 (essentially, see the definition of the precise space below), while (7.76) and (7.78) will be understood in a weak "integral" sense.

More precisely, we look for a solution (p, u) of the above boundary value problems, satisfying with the notation of section 5.6 (chapter 5): i) p - p E - p =- #1 in the exterior case, and p E Q[0, T]; LP (11 n BR)) n L°°(0,T; L2-1 C([O,T];L7(QnBR)-w) for all R > 1, 1 < p < -y, ii) u-i E L2(0,T; H1(IlnBR)) for all R > 1, u- E L2 (0, T; H1(1)) in the tube case or in the exterior case when if N > 3 in the exterior pO° > 0 and N > 3, and u-u E L2(0,T;L2N/(N-2)(S2))

case, iii) plu-112 E L°° (0, T; L1(St)) and iv) p(u-u) E C([0, TJ ; L2 + L' (11) - w)

where r=3 if-y>2,r=-

if'y<2.

We have already explained in section 5.6 (chapter 5) how one can adapt the usual energy bounds and thus derive all the a priori bounds corresponding to the above list of requirements on a solution (p, u). Let us only mention that

we deduce from the bounds on p - pO°, and plu - ul2 and on u - u a bound on u - u in L2 (S2 x (0, T)) in the exterior case if N > 3 and p°O > 0. Indeed,

1(p)p-/2)(U-U) is obviously bounded in L2 (S2 x (0,T)) while u - u is bounded in L2(0,T; (fl)) and sup esstE(o,T) meas{x / p < pO°/2} < 00. Let us also mention that these bounds were proven in section 5.6 in the case L2N/(N-2)

when f - 0 only. It is not difficult to check that the proofs can be adapted provided we assume that f E L1(O,T; L1 n (it is possible in fact to extend slightly this condition in a manner somewhat similar to condition (7.2)). We conclude by observing that Theorem 7.1 can then be easily adapted and yields a bound on p in LP(K x (0, T)) for any compact set K C 1 where p = 'y+ N'y-1. Similarly, following the methods of proofs of Theorem 7.2, we obtain the existence of a solution (p, u) of the above problems such that p E LP(K x (0, T)) for any compact set K C 11, under the same restrictions upon ^f as in Theorem 7.2. Of course, the crucial compactness result namely Theorem 5.1 used repeatedly in the proofs of Theorem 7.2 is to be replaced by Theorem 5.3 (section 5.6, chapter 5). L27/('f_1))

RELATED PROBLEMS 8.1 Pure transport of entropy In this chapter, we shall consider various related problems, many of which (but

not all of them) can be studied with the methods introduced in the preceding chapters. We begin in this section with the study of models where entropy is purely transported (along particle paths). From a physical viewpoint, see for instance chapter 1, this amounts to assuming that the thermal conduction coefficient can be taken to be 0 and that one can neglect the heating due to viscous dissipation-an approximation which is often made except for hypersonic gases. Then, the entropy s solves the following equation P

as

or, if (p, u) solve (5.1), equivalently (at least formally)

a

(ps) + div (pu s) = 0 .

In the ideal gas case, the pressure is then given by p = RpT = Rprle'lc' where R > 0, C > 0, y > 1 are given. Of course, replacing s by s/C,,, we may assume without loss of generality that C = 1 and take p = Rp''e' (and R = -y - 1). Therefore, we look for a solution (p, u, s) of (8.2),

at + div(pu) = 0, p > 0 8pu

4.div(pu®u)- Au-POdivu+77,n

=f

and

p = Rp7 e' .

(8.4)

As usual, we assume that p > 0, p + > 0, and we consider as in the preceding chapters the case of Dirichlet boundary conditions (u = 0 on 811 x (0, T)) where the equations hold in 1 x (0, T) and 11 is a bounded smooth open domain in RN

Related problems

214

(N > 2), the case of the whole space where the equations hold in RN x (0, T) and (p, u) vanishes at infinity and the periodic case where the equations hold on RN x (0, T) and all unknowns (and data) are assumed to be periodic in each xi (for 1 < i < N) of period T= > 0. Let us immediately warn the reader that, unless explicitly mentioned, we shall always consider these three cases for all the models and equations studied in this chapter. We also prescribe initial conditions Plt=o = Po

,

Putt=0 = mo

,

in 1

Pslt=o = So

(8.5)

where po, m0 and So satisfy po

- OEL'nL'f(SZ), 1PO 4

(8.6)

E L1(SZ) , mo E L27/(ti+1) (SZ) and So E L°O (St) and I So 15 Cl Po a.e. in SZ 2

for some C1 > 0, and mo = 0 a.e. on {po = 0}, as is defined to be 0 on {po=0} and p000. I

Obviously, we may expect from (8.2) a LOO bound on s and more precisely (8.7)

IISIIL-(cx(0,T)) < C1.

It may be worth remarking (once and for all) that s is not really well defined on the set where p vanishes since the equation (8.1) degenerates completely on this set. On the other hand, this does not affect the pressure p given by (8.4) and we may ignore completely this difficulty, agreeing for instance that s - 0 on {(x, t) / p(x, t) = 0}. In fact, if s" E L' (P x (0, T)) solves (8.2), then replacing the values of s by 0 on {p = 0} and denoting by s the resulting function, one sees that s is still a solution of (8.2). Next, we observe that we have at least formally

8

pJU12

at

2

+ -y-1 R P

+div u

2

p_u + 2

Rry

y-1

p-f e'

(8.8)

+µ0l22 +µfDuI2-Cdiv(udivu)+t;(divu)2 = pu f. Indeed, we have clearly

at (p'ie') + div (uppe') = (-y-1)(div u)p7e'

.

Then, because of (8.7), we deduce exactly as in section 5.1 (chapter 5) the same a priori bounds on (p, u) under the same conditions on f . Similarly, one can

check that the proof of Theorem 7.1 (chapter 7, section 7.1) can be adapted mutatis mutandis and thus the same result holds. All these observations lead to the following existence result.

Pure transport of entropy

215

Theorem 8.1 Under the same conditions on f and -y as in Theorem 7.1, and if (po, mo, So) satisfies (8.6), there exists a solution (p, u, s) of (8.2)-(8.3)-(8.4) with s E L°O(fZ x (0, T)) , ps E C([0, T]; LP (11)) for all 1 < p < oo, satisfying the 1, initial conditions (8.5) and such that p E LP(1 x (0, T)) with p = y + except in the case of Dirichlet boundary conditions where p E LP(K x (0, T)) for any compact set K C ft. In addition, (p, u, s) satisfies the following energy inequality for almost all t E (0, T) y -N

J

+Rpresdx+ <

ImoI2 n 2

+ R

fJ

dx, c

dx +

ds Jo

po

r dx pu f

(8.9)

fo

Remark 8.1 Let us also point out that it is possible to adapt in a similar way the results mentioned in section 8.6 concerning other boundary value problems.

0

The proof(s) of this existence result is essentially the same as the ones of Theorem 7.2 given in chapter 7 but for the systematic use of Theorem 5.1. We thus have to discuss the analogue of Theorem 5.1 (section 5.2, chapter 5) in this new setting. First of all, we consider, exactly as we did, a sequence (p", u'y, s') of solutions of (8.2)-(8.3) with f replaced by fl and make the same assumptions on (pn , un , f') as in section 5.2. Furthermore, we assume that s" converges weakly in LOO (Q x (0, T)) - * to some s. Then, part (1) of Theorem 5.1 holds with the same proof, and we claim that part (2) also holds assuming that po converges to po in L1 (ft) and that So converges to So in L1(0) (L' (ft n BR) for all R E (0, oo)

if ft = RN). More precisely, we claim that (5.20)-(5.22) then hold and that in addition p" /3(s") converges to p,3(s) in C([0, T]; LP(f )) n L'(Ki x (0,T)) for all 1 < q < r, 1 < s < q and for any continuous function 8 on R. In particular, s" converges to s in LP (((fZ n BR) x (0, T)) n {p > 0}) for all 1 < p < oo, R E (0, oo) .

In fact, exactly as in the proof of Theorem 5.2, we only have to show this last claim together with (5.20): the other statements then follow. Next, several proofs of these claims are possible and we shall present here the simplest one even though we shall need a much more general argument below. A simple argument consists in defining p3 = pes/7: then, p = Rpl' and p solves (5.1): we are thus back to the situation studied in the preceding chapters. In particular, we deduce from

Theorem 5.1 that pn converges to some p in C([0,T]; LP (fl)) n L3(K1 x (0,T))

for all 1 < p < r, 1 < s < q. Furthermore, we observe that we have for any continuous function Q on R

a

(pna(s' )) + div

0.

(8.10)

This identity is obvious formally, at least if p, s,,6 are smooth. The fact that it holds for any smooth 6 follows easily from the regularization argument used

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216

several times in the previous chapters (see in particular the fundamental regularization Lemma 2.3, section 2.3, chapter 2, volume 1): indeed, we have, dropping the index n to simplify notation and denoting cpE = cp * ,c£ where x, _ £

. 79K(E), r. > 0, Ic E CO (RN), fRN rcdz = 1

5i(pe) + div(up£) = r,

5j(ps)£ + div(u(ps)E) = rE

where re, r"E converge to 0, as c goes to 0+, in Lm(0,T; L m(1)) with ,L = 2 + . . Therefore, denoting S. = (ps)£/p£ if p£ > 0, sE = 0 if pE = 0 (sE E L°D(f x (0, T)) and is bounded in LOO (fl x (0, T)) uniformly in e E (0,1) ), we obtain in the sense of distributions as£ PE

at

+ PEu .

We + rESE = T£

and thus (by one more regularization justification that we skip) p£

3(S£) + P£uV 3(`SE) + r,P'(` e).£ = #'(`SE)r£

or

5i (PEQ(sE)) + div(pEuf(SE)) + rE f Q'\S£)SE

-i

(SE) = Q/(SE)r£

We then recover (8.10) upon letting e go to 0+: indeed, sE converges a.e. on J p > 0} to s and sE is bounded in L°° (Sl x (0, T)). Therefore, f3(9E) converges in L9 (((l n BR) x (0, T)) n J p > 0}) to 3(s) for all 1 <_ q < oo, R E (0, oo), and, we

deduce easily that p,/3(§,) converges to p,3(s) in L1 , (say) as c goes to 0+. At this stage, we have checked (8.10) when /3 is smooth and the general case follows by density. Then, we deduce from (8.10) and the proof of Theorem 5.1 (or directly from Appendix B) that we have, for any non-negative continuous function ,(3 and for some 9 > 0 (small enough), (p

i3(Sn))B

div un - µR

- (P3(S))B (divu

- µR

(P)

weakly, denoting by 7 the weak limit of co (recall that we are using the same notation as in chapter 5). In view of the strong convergence of p' shown above, we deduce that we have

(p/3(s))° div u" -n(p/3(s))e div u weakly (in 12' say). We may then adapt the proof of part (2) of Theorem 5.1 to deduce the strong convergence of pn/3(sn) to p/3(s) (in C([0, T]; LP (Q)) n L'(K1 x (0, T))

for all 1 < p < r, 1 < s < q) for any non-negative continuous function /3 and

Pure transport of entropy

217

thus for any continuous function /3. Let us briefly sketch the argument: denoting f n = p",8(sn), we have :

5,(fn)e+div(un(fn)e) _ (1-0)(divun)(fn)0; thus we deduce letting n go to +oo

of +div(uf) = 0,

+div(ufe) = (1-0)divu(fe)

and we obtain

a(7-(fe)1/e)+div(u(f--(f9)"°)) = 0. Hence, fo f - (fe)'1e dx(t) = 0 for all t >_ 0 and our claim is shown. Taking /3 = 0, we obtain the strong convergence of pn, and the proof of the compactness claims is complete. 0 We conclude this section with the case of a general state equation for p (and s). For the same reasons as those mentioned at the beginning of this section, we solve (8.2) and (8.3) but replace (8.4) by (8.11)

P = AP, s)

where p is a given continuous function of p E [0, oo) and of s E R. As is natural from a physical point of view-it is indeed a consequence of the second law of thermodynamics, see section 1.1, chapter 1, part I, volume 1-we assume that

p is non-decreasing with respect to p for each s fixed and, in order to avoid ambiguous definitions on the vacuum, we assume that p(0, s) = 0 for all s E R. Finally, we assume that p satisfies for some y > 1 lim inf ,sl
s) t-7 > 0

for all R E (0, 00),

(8.12)

and in order to avoid the technical difficulties associated with the behaviour of p near p = 0, we assume in the case when 11 = RN that p satisfies 1

fo

p(t, s)t-2 dt < +oo ,

for all s E R.

(8.13)

Let us also mention in passing that it is also possible to analyse more general situations than (8.13) using the ideas and methods developed in section 7.5 (chapter 7), and we denote q(p, s) = p fo p(t, s)t-2 dt. Of course, in (8.6), we replace po E L7 by q(po, so) E Ll.

Theorem 8.2 Under the above conditions and the same assumptions as in Theorem 8.1, there exists a solution (p, u, s) of (8.2)-(8.3)-(8.11) satisfying the initial conditions (8.5) such that s E LOO (SZ x (0, T)), ps E C([0, T]; LP(f )) for all

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1
Nry-1

except in the case of Dirichlet boundary conditions where p E L'r+e (K x (0, T) ) and p(1 + p°) E L' (K x (0, T)) for any compact set K C Q. In addition, (p, u, s) satisfies the following energy inequality for almost all t E (0, T)

pII2 + q(p, s)dx + I/'t dr (dxpIDu2 + (div u)2

f

ImoI2 1 2

Po

n

+ q(Po, so)dx +

(8.14) It

dr

in

dxpu f

0

Exactly as in the proof of Theorem 8.1 which we sketched above, there is a crucial point that we need to explain. All the rest of the proof follows what we did above and in chapters 5-7. This crucial point concerns compactness properties

of p and s and we detail it in the periodic case (or in the case of Dirichlet boundary conditions); the case iZ = RN is then easily adapted. We thus consider a sequence of solutions (pn, un, s') of (8.2)-(8.3)-(8.13) such that s" is bounded in LO° (fl x (0, T)) and weakly converges in L°° (SZ x (0, T)) (weak-*) to some s, un is bounded in L2 (0, T; HI (11)) and weakly converges in L2 (0, T; H' (1)) to some u, pn is bounded in L°° (0, T; L t (1)) n L1'+e (SZ x (0, T)) and weakly converges in D+0 (11 x (0, T)) (for instance) to some p > 0, and pn(unI2, p(pn, sn)(1+(pn)0) are bounded in L°O (0, T; Li (0)). Furthermore, we assume that po = pn It=o converges [(Pnn)It-o] (p0n)-1 in L' (and thus in LP for 1 < p < 7) to some po and that so = (defined to be 0 for instance on {po = 0}) converges in L' (and thus in LP for 1 < p < oo) to some so. Finally, without loss of generality, we may assume that

p'nsn converges weakly to some function that we denote by ps in L'Y+e(11 x (0, T))

(for instance). Obviously, 1731 < Cp a.e. for some C > 0 and we set s = p-1 ps on {p > 0}, = 0 on {p = 0}. As shown below (Theorem 7.3), p(pn, sn) - p(pn, s) converges to 0 in L' (f x (0, T)) and for any 6 > 0 (small enough) p p(pn, sn) - pnp(pn, s) converges to 0 in L 1(Q x (0,T)). This allows us to adapt the proof of Theorem 5.1 (section 5.2, chapter 5) and we conclude, for example, that pn converges to p in C([0, T]; LP (Q)) for all 1 < p < y and in LQ (SZ x (0, T)) for all 1 < p < 'y + 0. We then easily deduce that pni3(sn) converges to po(s) in the same topologies for any continuous function /3. Indeed, pn/3(sn) satisfies (8.10) and we deduce from the above bounds and convergences that pj(s) converges to p/3(s) in the same topologies (as those in which Pn converges to p). Finally, the convergence to 0 of pn(/3(sn) -0(9)) follows from Theorem 7.3 below. These compactness properties allow us to follow the proofs made in the preceding chapters, and we conclude our discussion of the proof of Theorem 7.2. 0 We still have to show some (new) properties of solutions of linear transport equations that were used in a fundamental way in the above argument. One property, which could be called conditional compactness, seems to be an interesting phenomenon in itself and this is why we slightly extend the framework in which we wish to state it. More precisely, we consider pn > 0, un, sn satisfying

Pure transport of entropy

219

W + div(pnun) = 0 , apnsn

+ div(p"unsn) = 0 in fI x (0, T) . 8t We only consider here the periodic case in which case we assume that un is bounded and converges weakly in LP(0, T; W l"P(St)) to some u where 1 < p < 00, or the case of Dirichlet boundary conditions in which case we replace Wisp by W0

11P.

We skip the case of the whole space (11 = RN) because of the many

possible assumptions we can make ("at infinity") on (p, u, s) even though all of them are straightforward adaptations of the arguments below. In addition, we assume that pn (respectively sn) is bounded and converges weakly in LQ (11 x (0, T)) (respectively LOO (11 x (0, T))) to some p > 0 (respectively

s) where 1 < q < oo. Obviously, whenever one of the exponents p, q is infinite, what we mean by weak convergence is really weak-* convergence. We are interested in compactness properties of sn and we call them conditional compactness since, in fact, S' = pnsn and pn are both solutions of the same transport equation governed by un and we are interested in the compactness of We shall assume that po = pn l t=o and So = Sn l t=o satisfy: po converges weakly in L1(Q) to some po > 0, and So /po converges in L'(11) (strongly) to some so. Finally, we assume that we have .1 + q < 1. Let us observe that the proof made above to derive (8.10) immediately adapts to our more general setting and (8.10) holds for any continuous function Q. Let us also mention that the assumptions on pn, un, sn and p, q allow us to check easily that pn (respectively, pnsn or png(sn) for any continuous function (3) is bounded in L°(0,T; L'(11)) and, in fact, belong to C([0, T]; L'(S1)) where 1 + -L < 1 if Sn/pn.

q < oo, m < +oo if q = +oo. We shall not need this fact below except that it clarifies all the issues related to initial conditions. Before stating our main result, we shall need one more notation. Without loss of generality, we may assume that pnsn converges weakly in LQ(S1 x (0, T)) to a function denoted by ps and we set p = 1 Ts if p > 0, = 0 if p = 0. We may now state our "conditional compactness" result.

Theorem 8.3 With the above notation and conditions, we have T

f dx]

dt pn1/3(sn)-Q(s)I -' 0

J dx J

dt I

f2

f2

(8.16)

n

0

p(pn, sn)

0

_p(pn, s)I -' n 0

(8.17)

whenever Q E C(R), p E C(R2; R) satisfies sup Ip(t, s)I Isl
(tl_Q

-+ 0 as I tI -* oo,

if q < oo, and p(0, s) = 0 for all s E R.

for all R E (0, oo)

,

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220

Remark 8.2 Let us briefly mention some extensions of the preceding result. First of all, we may allow right-hand sides in the equations contained in (8.15). Also, it is possible to replace the "isotropic" LP(1 x (0, T)) spaces in the above conditions by Lp' (0, T; Lp2 (S2)) spaces. Finally, we may replace the L°° bounds on sn by Lt bounds. In each case, conditions have to be imposed that we do not wish to detail here.

Remark 8.3 Let us emphasize the fact that in general s and s do not coincide. Remark 8.4 In the above sketch of the proof of Theorem 7.2, we oversimplified the use of Theorem 8.2. As it stands, our argument is correct provided we assume

that p(t, s)It1-(l'+e) - 0 as Itl - oo uniformly in s bounded. In general, the argument needs a slight technical modification. It is in fact sufficient to observe that we have :

f

T

Sup

f dX P(Pn, n)-P(Pn, Sn)1(p^
1(1+(pn)l) -- 0 as R -- +oo

where 0 < 6 < N ry - 1. We may then apply Theorem 8.3 and we obtain for all

0>0,

IP(pn,

sn)-P(Pn, s)I1(P^
Combining this information, we deduce p(pn, Sn) (p' )e - (pn )e p(p', sn) > 0 which is precisely the information needed for the compactness analysis.

Remark 8.5 Exactly as in the proof of Theorem 5.1 (chapter 5), it is possible to deduce from (7.16) the fact that pn (sn - s) converges to 0 in Li (12) (or even in Lm(SZ), m being defined in the observations preceding Theorem 7.3) uniformly in t E [0, T1.

Remark 8.6 Let us mention, without stating the precise conditions we need, another extension of the previous result. We may replace (8.15) by I

at

+ div(vn) = 0 ,

8

atn + div(vnsn) = 0.

(8.18)

Proof of Theorem 8.3. We first prove that p pn/32(sn) = (pn(3(sn))2 for any bounded continuous function 3 on R. Here and below, we denote, as in the proof of Theorem 5.1, by 7 the weak limit of Wn. Let us mention in passing that, in view of the bounds on pn and sn, it is a simple exercise in functional analysis to deduce the existence of a subsequence n' for which pe'7(sn) converges weakly say in L' (S2 x (0, T)) for any continuous function -y. We simply ignore this detail and write Wn for cpn'. In order to prove the above claim, we deduce from (8.10) letting n go to +oo a (p/3 (s)) + div(u p,8(s)) = 0 (8.19)

at (p,31 (s)) + div(u TO' (s)) = 0.

Pure transport of entropy

221

This passage to the limit presents no difficulty in view of the assumptions made

upon u", p" and s" and of Lemma 5.1 (section 5.2, chapter 5). Next we set

Q= Ppf(s)ifp>0,=0ifp=0,'82= Pp/32(s)ifp>0,=0ifp=0. Therefore, we may rewrite (8.19) as

a (p,3) + div(pu,3) = 0 , while we have of course (take Q

e

(pat) +

div(pu(132))

=0

(8.20)

1)

+div (pu) = 0.

(8.21)

Then, by the same argument as the one used to derive (8.10), we deduce from (8.20) and (8.21) the following equation [p('82

at

- (Q)2)] + div [au(#2 - (Q)2)] = 0

(8.22)

.

In addition, we have in view of the assumptions made upon the initial conditions

pO(s)It=o = po/(so), p/32(s)It=o = pof2(so) . Therefore, we have p(/32 (4)2)It_0 = p/32 (p8)21t=o = 0 in Sl. Since, we

-

-

obviously have 02 > (4)2 a.e., we deduce from vthis initial condition and (8.22) (which we integrate over Sl) the fact that p(j32 (4)2) = 0 a.e. and our claim is

-

shown.

In particular, we have T

dxp"IQ(sn)41 < C

dt

fT dt

o

=C

dt o

Jn

r dxpfI$(Sn)_al2 n

dx p"($(s"))2 - 2pn/3(sn)Q + pn(Q)2

n 0.

In particular, we may choose p(t) - t and (8.16) is shown. Next, in order to prove (8.17), we observe that we have for all S E (0,1) and for all R E (1, oo)

f 0

T

/'

dt J dx Ip(p', Sn) _p(p", s)I [1(p^<5) + f2

< C sup{p(t,s) /0 < t < S, Isl < Ro} +C sup P(t I

It > R, Isl < Ro Iq

where Ro = supn> 1 I I s" 11 L:°t . In view of the assumptions made upon p, we deduce that the above integral converges to 0 uniformly in n as S goes to 0+ and R goes to +oo. Next, we conclude, remarking that we have T

J0

dt

f

t2

dx

p(pn,s"')_p(pn,sn)11(b
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222

<

1T

dt / dx i(p.>b) sup 0

O
f2

while

We conclude this section with a long remark about a related but different model.

Remark 8.7 We briefly present here what we know and what we do not know about the model where we still take the thermal conduction coefficient to be 0 but we incorporate the heating due to viscous dissipation. Then, see chapter 1 for more details, (8.1)-(8.2) are replaced by p

as at +

2T

IDu+DutI2 + eTP(divu)2

(8.23)

or in conserved form

a (ps) + div(pus) = 2L IDu+Dut I2 + at

7,

(div u)2

(8.24)

and we keep of course the system (8.3) with, for instance, the ideal gas relationships: y = 1 + (R, C > 0) and

T = Pry 1e'/C

p=RpT.

(8.25)

In particular, T solves the following equation p

aT + pu-VT +( y-1)( diva)P T= µ at 2Cv

t12 Du+Du + e-µ( div a )2. C, v

( 8.26)

We shall assume p > 0 and N + (2 - N)µ > 0. In particular, we then have 2 IDu+Dut12 + (l;-p)(divu)2 > vIDu12 for some v > 0 independent of u. We first list the a priori bounds we know on this system of equations. Assuming that pI t=o = Po, pul t=o = mo and pT I t=o = eo satisfy Po, eo >- 0 ,

po E L1 n L"(SZ)

eo = mo = 0 on {po = 0} ,

,

eo E L1(SZ) ,

fmoI2

E L1(SZ)

(8.27)

PO

where we agree that

IPo 2

= 0 on {po = 0}, we obtain easily the following

bounds: p, pIu12, ps and pT are bounded in LOO (0, T; L' (Q)) and DT 2 is bounded in L1(SZ x (0, T)).

Pure transport of entropy

223

Next, we remark that we have

pa(Tl/(7-1)) + pu.V(T1/(7-1)) +

(divu)pT'/(7-1)

= 2C T(2-7)/(7-1)IDu+Dutl2 + C

µT(2-7)/(7-1)(divu)2

v

and thus, at least formally,

(T r) + div (uT r)

p

Hence, if we assume that we have, denoting by To =

a = inf ess

(Tth/Po)

> 0.

v

>0

(8.28)

(in other words, we assume that po < Ceo for some C > 0), then we deduce from the maximum principle

T74 > ap

a.e.,

(8.29)

an inequality from which we deduce a bound on p in L°° (0, T; L7). Next, we observe that we have

8

E (y-1)

(8.30)

[IDu+DutI2 + ( -µ)(divu)2J

hence

vIDuI2-y(divu)p > -Cp2

5 and 49

(inf essp) + C(inf essp)2 > 0.

Therefore, if we assume that we have

inf esspo = R inf ess (poTo) = f3 > 0

(8.31)

we deduce a lower bound on p and thus on T using (8.29) inf essp >_

inf essT >[(1+i3ctr'] R

(7-1)17

(8.32)

We may also deduce from (7.30) the following equation

a (pl/7) + div u 1/7

2 - 1 [LIDu+Dutl212

div u

2

1/7

1

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224

> ry - lvp (1--1r)IDuI2. y Since p is bounded in L°° (0, T; L1), we deduce, integrating this inequality in x and t, a bound on IDuI2 p-(1-(1/7)) in L'( SZ x (0, T)). Furthermore, we may write IDuI2''/(27-1)

=

(()) IDuI2

p

from which we deduce a bound on Du in L2(O,T; L27/(27-1)(S2)). Let us mention that this is the "best" bound we can obtain on Du and in particular we are unable to obtain a bound on Du in L2 ((fl x (0, T)). This bound does not seem to be "strong" enough to implement the strategy of proof of Theorem 7.1 (section 7.1, chapter 7) in order to prove some Li,t bound on p for some q > 1 (or some Li,t bound on p for some q > y). Indeed, the LOO(Lt)

bound on p and the above bound on Du does not yield, even when N = 2, a 1 better bound on pu ®u than Lz,t; indeed, we have, if N = 2, 1-r + 2 (2-y2-1 - 2 = 1 ! The second obstruction we encounter in proving an existence theorem for the problem we are studying here is the compactness analysis of solutions: indeed, even if we postulate Ll bounds on (p, p) (or simply Li,t x Lz,t bounds for some q > -y, r > 1), the proof of Theorem 5.1 (chapter 5) can be adapted to yield the following information

div u - p] Q(p, p) =

div u - p Q(p, p)

but we are unable to conclude from this information any compactness of p or p. 11

8.2 A semi-stationary model We consider in this section the following system of equations i9p

+div(pu)= O , p> 0

(8.33) 0

with the same boundary conditions as before (see section 8.1), where a > 0, µ > 0, µ + t; > 0, ry > 0. Let us observe that we have not included force terms (right-hand sides for the second equation) in order to avoid unnecessary (and straightforward) technicalities. There are various motivations for the study of the model (8.33). First of all, we have shown in chapter 5 (section 5.2, Remark 5.8) how solutions of this system of equations allow us to build solutions of our initial system (namely (5.1)-(5.2)) which exhibit persistent oscillations. The second motivation is the model derived in W.E. [165],[166] for the dynamics of vortices in Ginzburg-Landau theories in superconductivity, which is precisely of the above form. As it stands, the above model is slightly ambiguous in two cases: i) when

St = R2 and ii) in the periodic case. In the periodic case, u is defined by the

A semi-stationary model

225

second equation up to a constant and we thus need to add one more constraint like for example

In

dx u(x, t) = 0,

for all t > 0.

(8.34)

In the case when fZ = R2, requiring that u vanishes at infinity needs some explanation (while it is an obvious requirement if 11 = R" and N > 3). The simplest way to argue is to write the explicit integral relationship between u and p7 (assumed below to be in L' (R2) for all t > 0) namely

u

I

X

for all t > 0.

in R2 ,

* P7

(8.35)

Of course, we need to complement the above system of equations with an initial condition on p namely (8.36)

P!t=o = Po ? 0

where po E L' (0), po E L7 (f) if y > 1 and po I log po I E L1(11) if y = 1. Before stating our main results, we wish to make a few observations. First of

all, if 0 = RN (N > 2), we see that we have a div u =

µ

+ p7

,

curl u = 0 in RN

,

for all t > 0

(8.37)

C

while we have in the periodic case

div u =

µ+ p7 _3:p1 dx

curl u = 0 in RN

,

(8.38)

for all t > 0 .

We may now state our main results.

Theorem 8.4 (The periodic case). Let y > 0. Then, there exists a solution (p, u) of the above problem satisfying: p E C([0, T]; L1), p E C([0, T]; L7) n L2'(1 x

(0,T)) if y > 1, p E L1+7,°°(f x (0,T)) if y < 1, pl log pl E L'(0, T; L1) if y = 1, u E L2(0,T; H1), and u E LOO (0,T; W1+1/7) if y < 1, for all T E (0, oo). Furthermore, we have for any ,Q E Co ([0, oo)) and for all T E (0, oo)

8) +

di v(u,Q (p)) + ( div u)[Q' (p)P-P(p)] = 0 ,

II P(t) II too t-11''

is bounded on (0,T)

(8.39) (8.40)

If po E LP (Q) for some p E (1, oo) (resp. p = oo)

then p E Q0, T]; LP) n LP+7(1 x (0,T))

(8.41)

(resp. E L°° (f x (0, T )))y

If in less po > 0, then inf oes s) p > 0 for all T E (0, cc), fl x

(8.42)

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226

If y > 1 or if y < 1 and inf essn po > 0, then p E C([O,T];W1"p) (resp. L°O(O,T;W1"°°)) whenever po E W 1,P for some p > 2 (resp. p = oo).

(8.43)

Finally if y > 1 (resp. if -y < 1 and inf ess po > 0) solutions p in L°O (0, T; L°° n W 1,P) (resp. which are in addition bounded from below) are unique if p = N when N > 3, p > 2 when N = 2.

Theorem 8.5 (SZ = RN). Let -y > 1. Then, the preceding result holds except for (8.42) and provided we replace u E L2 (0, T; H1) by u E L°O(0, T; Lwt-4-1'°°),

Du E L2 (RN x (0, T)) and Du E L' (0, T; L1/7)if y < 1.

Theorem 8.6 (Dirichlet boundary conditions). Let y > 0. Then, there exists a solution (p, u) of the above problem satisfying: p E C([0, T]; L1), p E C([O, T]; L'1) nL21(SZ x (0, T)) if -y > 1, p7 E C([O, T]; L1) nL2(SZ x (0, T)) and p E L1+1r (0, T; Li + ' (11)) if -y > 1, pI log PI E LOO (0, T; L') ify = 1, u E L2 (0, T; Ho ), u E L°° (0, T; W 1,1/7) if -y < 1, curl u and div u - 14+t p- E LOO (0, T; LOO ), for all

T E (0, oo); and (8.39) holds. Furthermore, we have for any compact set K C SZ and for all T E (0, oo): IIP(t)IIL-(x)

is bounded on (6,T), for allb > 0,

(8.44)

If po E L oC(SZ) for some p E (1, oo) (resp. p = oo)

then p E C([O,T];LP(K)) and p E LP+-'(K x (0,T))

(8.45)

(resp. LOO (K x (0,T))).

Remark 8.8 Let us make a few remarks on the above statements: i) First of all, in the case when SZ = RN, we have restricted ourselves to the case when y > 1 in order to avoid the technicalities associated with y < 1 although this case can be treated by a convenient adaptation of the considerations introduced in section 7.5 (chapter 7). ii) In the case when SZ = RN, it is possible to treat other situations with different "conditions at infinity" adapting the arguments of section 7.6 (chapter 7). Then, whenever the condition (inf essRN po > 0) makes sense, the property (8.42) also holds.

iii) The fact that p E Ly can actually be deduced from the bound on ti/'tIIPIIL and on II PII L-(L=) Indeed, we have, for all R > 0, denoting Co = sup[o,T](t1/^'IIPIILo),

meas {(x, t) / p(x, t) > R} <

f

(Co/R)-'

dt meas {x / p(t) > R}

0

(Co/R)'

1

0

< Co

SUP (IP(t)IILz R-(1+y) t E [0,T]

.

227

A semi-stationary model

iv) Next, we claim that p E V+7(11 x (0,T)) as soon as po (log po)+ E L1, the proof being analogous to the proof below. v) The equality (8.39) really means that p is a renormalized solution of

a + div(pu) =

+ u - V p + (div u) p = 0.

vi) In addition, in view of (8.40) (or (8.44)), it is clear that (8.39) holds for any ,3 E C' ([0, oo); R).

vii) Let us also observe that it implies the following energy identity a1

-5t

p'1

+ div

a'Y1

up

- p0I 2

I2

(8.46)

-l;div(udivu) +µIDuI2 +l;(divu)2 = 0 and a

t

f p"Y dx(t) + ds fn pIDuI2 + (div u)2 J a = 1 I P-O , for all t > 0 .

(8.47)

y-

Indeed, we have easily since u E L2(0,T; Hl) and pr' E L2(S1 x (0, T)) u

2 Du -}-AA I22 - div u( div u) + pl I2 l; (div u) +a div u

a div u )pry = 0

and we conclude using (8.39) letting,8 converge to p7. When y > 1, we just have to replace by p log p. viii) Next, we observe that all bounds on p are easily translated into bounds on u using elliptic regularity. ix) Finally, we have to clarify the meaning of pu in (8.33) at least when y < 1. Then, we observe that

p E Lr(L')nL'(L2) where 1