On the Existence of Globally Defined Weak Solutions to the Navier Stokes Equations

Transcription

1 J. math. fluid mech. 3 ( /1/ $ / c 21 Birkhäuser Verlag, Basel Journal of Mathematical Fluid Mechanics On the Existence of Globally Defined Weak Solutions to the Navier Stokes Equations Eduard Feireisl, Antonín Novotný and Hana Petzeltová Communicated by I. Straškraba Abstract. We prove the existence of globally defined weak solutions to the Navier Stokes equations of compressible isentropic flows in three space dimensions on condition that the adiabatic constant satisfies γ>3/2. Mathematics Subject Classification (2. 35Q3, 35A5. Keywords. Compressible Navier Stokes equations, global existence, critical adiabatic exponent. 1. Introduction We prove the global existence of weak solutions of the Navier Stokes equations of an isentropic compressible fluid: ϱ t + div(ϱ u =, (1.1 (ϱu i t + div(ϱu i u+a(ϱ γ xi =µ u i +(λ+µ(div u xi,i=1,2,3. (1.2 Here the density ϱ = ϱ(t, x and the velocity u =[u 1 (t, x,u 2 (t, x,u 3 (t, x] are functions of the time t (,T and the spatial coordinate x where R 3 is a bounded regular domain. The viscosity coefficients µ, λ satisfy µ>,λ+ 2 3 µ, a>is a positive constant, and the adiabatic constant γ is subjected to the constraint γ> 3 2. (1.3 We prescribe the initial conditions for the density and the momenta: ϱ( = ϱ, (ϱu i ( = q i,i=1,2,3; (1.4

2 Vol. 3 (21 Navier Stokes Equations of Isentropic Compressible Fluids 359 together with the no-slip boundary conditions for the velocity: u i =, i =1,2,3. (1.5 The mathematical theory of viscous fluids is far from being complete. Even if the flow is incompressible, i.e., governed by the classical Navier Stokes system, it is still a major open problem whether or not global classical solutions exist in three space dimensions for general (not necessarily small initial data. In spite of the enormous amount of effort, the question of global regularity and uniqueness remains completely open. The problem becomes even more involved when general compressible fluids are considered. The existence of (weak solutions has been largely settled in the space dimension one (see e.g. Antontsev et al. [2], Serre [2]. The spherically symmetric flows with large data were treated by Hoff [11] and in a recent paper by Jiang and Zhang [13]. The truly multidimensional case seems much more complicated. There is a vast amount of literature originated by the paper of Matsumura and Nishida [19] dealing with small and smooth initial data. The requirement of smoothness, but not smallness, was subsequently relaxed by Hoff [12]. Global existence of weak solutions for large data in two space dimensions was proved by Kazhikov and Vaigant [22] on condition that the viscosity coefficient λ depends on ϱ in a very specific way. The obvious mathematical difficulties of the problem led to the development of new concepts such as that of renormalized solutions introduced by DiPerna and Lions [6] and subsequently adapted by many authors in rather different contexts. The crucial question of compactness of the set of bounded solutions has been positively resolved by means of the recent results of the compensated compactness theory, namely, new weakly convergent quantities called paracommutators were identified by Coifman et al. [5] and Li et al. [14]. Recently, Lions [17] presented a theory giving positive existence results in three space dimensions and for general initial data under the restriction γ 9 5. The main goal of the present paper is to extend the results of Lions [17] to the case when the adiabatic constant γ satisfies (1.3. In particular, the physically relevant case of monoatomic gas γ =5/3 will be included. We now give a precise formulation of our results. Formally, multiplying (1.2 by u i, i =1,2,3, integrating by parts, and making use of the continuity equation (1.1, one obtains the energy inequality where d dt E(t+ E(t =E[ϱ, u](t = µ u 2 +(λ+µ div u 2 dx ( ϱ u 2 + a γ 1 ϱγ dx.

3 36 E. Feireisl, A. Novotný and H. Petzeltová JMFM Similarly, multiplying (1.1 by b (ϱ we deduce b(ϱ t + div(b(ϱ u+(b (ϱϱ b(ϱ div u = (1.7 for any differentiable function b. Motivated by the relations (1.6, (1.7, we introduce the concept of finite energy weak solutions of the problem (1.1, (1.2, (1.5; specifically, the functions ϱ, u will comply with the following conditions: ϱ, ϱ L (,T;L γ (, u i L 2 (,T;W 1,2 (, i =1,2,3; the energy E is locally integrable on (,T and the energy inequality (1.6 holds in D (,T; the equations (1.1, (1.2 are satisfied in D ((,T ; moreover, (1.1 holds in D ((,T R 3 provided ϱ, u were prolonged to be zero on R 3 \ ; the equation (1.1 is satisfied in the sense of renormalized solutions, more specifically, (1.7 holds in D ((,T for any b C 1 (R such that b (z for all z R large enough, say, z M (1.8 where the constant M may vary for different functions b. Remark 1.1. The condition (1.8 seems to restrict considerably the class of functions b in comparison with the standard definition introduced in [6]. However, note that it is possible to use the Lebesgue convergence theorem to deduce that (1.7 will hold for any b C 1 (, C[, satisfying b (zz c(z θ +z γ 2 for all z> and a certain θ (,γ/2 provided ϱ, u is a finite energy weak solution in the sense of the above definition. It follows immediately from (1.1, (1.2 that any finite energy weak solution belongs to the class: ϱ C([,T]; L γ weak (, ϱui C([,T]; L 2γ γ+1 weak (, i =1,2,3 (1.9 and, consequently, the initial conditions (1.4 make sense. Accordingly, the data ϱ, q i, i =1,2,3 are supposed to comply with compatibility conditions of the form: ϱ L γ (, ϱ,q i (x = whenever ϱ (x =, Our main result reads as follows: q i 2 L 1 (, i =1,2,3. ϱ (1.1 Theorem 1.1. Assume R 3 is a bounded domain of the class C 2+ν, ν>. Let the data ϱ, q i, i =1,2,3satisfy the compatibility conditions (1.1 and let γ>3/2. Then given T>arbitrary, there exists a finite energy weak solution ϱ, u of the problem (1.1, (1.2, (1.5 satisfying the initial conditions (1.4.

4 Vol. 3 (21 Navier Stokes Equations of Isentropic Compressible Fluids 361 The proof of Theorem 1.1 will be done by means of a three-level approximation scheme based on a modified system: ϱ t + div(ϱ u =ε ϱ, (1.11 (ϱu i t +div(ϱu i u+a(ϱ γ xi +δ(ϱ β xi +ε u i. ϱ = µ u i +(λ+µ(div u xi,i=1,2,3 (1.12 where ε>, δ> are small, β> sufficiently large, and (1.11 is complemented by the homogeneous Neumann boundary conditions ϱ. n =, where n stands for the outer normal vector. (1.13 (i The first step in the proof of Theorem 1.1 consists in solving the modified problem (1.11, (1.12 by means of a Faedo Galerkin approximation where (1.11, (1.13 is solved directly while (1.12 is replaced by a finite-dimensional system (see Section 2. The classical theory of parabolic systems is used to obtain the desired result. The extra term ε u i. ϱ is necessary to save an energy inequality related to (1.6 (see formula (2.15. (ii In the second step, we let the artificial viscosity terms represented by the ε quantities go to zero. This is already a delicate matter due to the lack of suitable estimates on the density component ϱ. Here, we use the technique developed by Lions [17] based on regularity of the effective viscous flux aϱ γ (λ +2µdiv u (cf. Section 3. More specifically, it can be shown that (aϱ γ ε (λ +2µdiv u ε ϱ ε ( aϱ γ (λ +2µdiv u ϱ as ε (1.14 where ϱ ε, u ε is a sequence of approximate solutions and ϱ, u stand for its limit in suitable weak topologies. Here and always, b(ϱ denotes a weak limit of b(ϱ ε. The remarkable properties of the effective viscous flux have been observed by Hoff [12], Serre [21], and Lions [17] where they form the platform for the existence theory. We give an elementary proof of (1.14 based on Div-Curl Lemma known from the theory of compensated compactness. (iii The final step of the proof consists in getting rid of the artificial pressure term δϱ β (Section 4. Our approach is based on the cut-off operators introduced in [8] and [9]. More specifically, we consider a family of functions where T C (R is chosen so that T k (z =kt( z for z R, k =1,2,... (1.15 k T (z =zfor z 1, T(z = 2 for z 3, Tconcave. The main idea, formulated in [7], is to show that lim sup T k (ϱ δ T k (ϱ L γ+1 ((,T c (1.16 δ + where the constant c is independent of k and ϱ δ is the corresponding sequence of approximate solutions. This is an estimate motivated by the paper of Jiang and

5 362 E. Feireisl, A. Novotný and H. Petzeltová JMFM Zhang [13] where the authors prove a weaker statement ϱ θ ϱ θ L 2/θ (,T;L 2/θ (, θ > small enough, for suitably constructed weak solutions to the problem (1.1, (1.2 with radially symmetric initial data. The relation (1.16 is then used to complete the proof of Theorem 1.1. At this final stage, we follow the line of arguments of [7]. Remark 1.2. The reason for introducing the artificial pressure term is that the L p -estimates of the density carried out in Section 4, Part 4.1 are not compatible with the artificial viscosity term in ( The Faedo Galerkin approximation Our first goal is to solve the problem: ϱ t + div(ϱ u =ε ϱ, (2.1 (ϱu i t +div(ϱu i u+a(ϱ γ xi +δ(ϱ β xi +ε u i. ϱ = µ u i +(λ+µ(div u xi,i=1,2,3 (2.2 complemented by the boundary conditions: and modified initial data: ϱ. n =, (2.3 u =, (2.4 ϱ( = ϱ C 2+ν (, <ϱ ϱ (x ϱ, ϱ. n =, (2.5 (ϱ u( = q, q =[q 1,q 2,q 3 ], q i C 2 (, i=1,2,3. ( The Neumann problem for the density The following auxiliary result is classical and may be found e.g. in Lunardi [18, Theorem ]: Lemma 2.1. Assume u is a given vector function belonging to the class u C([,T]; [C 2 (] 3, u = (2.7 Then the initial-boundary value problem (2.1, (2.3, (2.5 possesses a unique classical solution ϱ on the set [,T] such that ϱ(t C 2+ν ( for any fixed t [,T].

6 Vol. 3 (21 Navier Stokes Equations of Isentropic Compressible Fluids 363 Moreover, by virtue of the comparison principle and (2.5, we have ( t ( t ϱ exp div u(s L ( ds ϱ(t, x ϱ exp div u(s L ( ds (2.8 for all t [,T], x. Multiplying (2.1 by ϱ and integrating by parts, we get d ( ϱ 2 dx +2ε ϱ 2 dx 2 u ϱ ϱ + div u ϱ ϱ dx dt and, consequently, d ( ϱ 2 dx + ε dt ϱ 2 dx 1 ( 2 u L (Q + div u L (Q ϱ 2 + ϱ 2 dx ε for all t [,T]. Thus, by virtue of the Gronwall lemma and (2.8, we deduce the estimate T sup ϱ(t 2 W 1,2 ( + ϱ(t 2 W 2,2 ( (T, dt c ε, ϱ, u L ((,T. (2.9 t [,T ] Now, let ϱ 1, ϱ 2 be two solutions (with the same initial data of (2.1, (2.3, (2.5 corresponding to u = u 1, u = u 2 respectively. Multiplying the difference of the equations by (ϱ 1 ϱ 2 and integrating by parts, we obtain d (ϱ 1 ϱ 2 2 dx +2ε (ϱ 1 ϱ 2 2 dx dt 2 div u 1 ϱ 1 ϱ 2 ϱ 1 ϱ 2 + u 1 ϱ 1 ϱ 2 ϱ 1 ϱ 2 dx+ 2 ϱ 2 div( u 1 u 2 ϱ 1 ϱ 2 + ϱ 2 u 1 u 2 ϱ 1 ϱ 2 dx Taking (2.9 into account, we conclude sup ϱ 1 (t ϱ 2 (t W 1,2 ( t [,T ] ( TcT, ε, ϱ, u 1 L ((,T, u 2 L ((,T sup t [,T ] u 1 (t u 2 (t W 1,2 ( where the quantity c is bounded for bounded values of its arguments. Note that (ϱ1 ϱ 2 dx = for all t which allows for using the Poincaré inequality. The results just obtained can be summarized in the following statement: Lemma 2.2. Let the initial datum ϱ satisfy (2.5. Then there exists a mapping S = S( u, S : C([,T]; [C 2 (] 3 C([,T]; C 2+ν ( enjoying the following properties:

7 364 E. Feireisl, A. Novotný and H. Petzeltová JMFM ϱ = S( u ( is the unique classical solution of (2.1, (2.3, (2.5 ; ϱ exp ( t div u(s t L ( ds S( u(t, x ϱ exp div u(s L ( ds for all t ; S( u 1 S( u 2 C([,T ];W 1,2 ( Tc(κ, T u 1 u 2 C([,T ];W 1,2 ( (2.1 for any u 1, u 2 belonging to the set M κ = { u C([,T]; W 1,2 ( u(t L ( + u(t L ( κ for all t} The Faedo Galerkin approximation scheme In order to solve (2.2, consider an suitable orthogonal system formed by a family of smooth functions ψ n vanishing on. One can take e.g. the eigenfunctions of the Laplacian: ψ n = λ n ψ n on, ψ n =. Now, consider a sequence of finite dimensional spaces X n = [span{ψ j } n j=1] 3,n=1,2,... The approximate solutions u n C([,T]; X n we shall look for are required to satisfy the integral equation ϱ(t u n (t.ψ dx q.ψ dx = (2.11 t [ µ u n div(ϱ u n u n + ((λ+µdiv u n aϱ γ δϱ β ] ε ϱ. u n.ψ dx ds for all t [,T] and any function ψ X n. Next, we introduce a family of operators M[ϱ] :X n Xn,<M[ϱ] v, w >= ϱ v. w dx. These operators are invertible provided ϱ is strictly positive on, and we have M 1 [ϱ] L(X n,x n ( inf x ϱ(x 1. Moreover, making use of the identity ( M 1 [ϱ 1 ] M 1 [ϱ 2 ]=M 1 [ϱ 2 ] M[ϱ 2 ] M[ϱ 1 ] M 1 [ϱ 1 ] one can see that the map ϱ M 1 [ϱ] mapping L 1 ( into L(X n,x n

8 Vol. 3 (21 Navier Stokes Equations of Isentropic Compressible Fluids 365 is well-defined and satisfies M 1 [ϱ 1 ] M 1 [ϱ 2 ] L(X n,x n c(n, η ϱ 1 ϱ 2 L1 ( (2.12 for any ϱ 1, ϱ 2 from the set N η = {ϱ L 1 ( inf ϱ η>}. x Now, the identity (2.11 can be rephrased as follows: ( t u n (t =M 1 [ϱ(t] ( q + N [ϱ(s, u n (s] ds (2.13 where < ( q,ψ >= q.ψ dx and < N [ϱ, u n ],ψ >= [ µ u n div(ϱ u n u n + ((λ+µdiv u n aϱ γ δϱ β ] ε ϱ. u n.ψ dx for all ψ X n The approximate solutions The approximate solutions for the problem (2.1 (2.6 will be found by means of (2.13 where we take ϱ = S( u n, the mapping S being defined in Lemma 2.2. Accordingly, the resulting equation reads: ( t u n (t =M 1 [S( u n (t] ( q + N [S( u n (s, u n (s] ds, u n C([,T]; X n. (2.14 In view of (2.1, (2.12, the integral equation (2.14 can be solved, at least on a short time interval [,T(n], T (n T, by means of a standard fixed point argument on the Banach space C([,T]; X n to obtain a local solution ϱ n, u n of the problem (2.1, (2.11 complemented by the initial condition (2.5 for ϱ n. Now, in order to show that T (n =T for any n, it is enough to prove that u n (t stays bounded in X n on the whole interval [,T(n]. Observe that L 2 norm and C 2 norm are equivalent on X n. To get uniform bounds on u n (t, we derive an energy inequality similar to (1.6. Differentiating (2.11 with respect to t and taking ψ = u n (t, we obtain d dt [ 1 2 ϱ n u n 2 + a γ 1 ϱγ n + δ β 1 ϱβ n dx ( ε aγϱ γ 2 n ] + µ u n 2 +(λ+µ div u n 2 dx+ ( δβϱ β 2 n ϱ n 2 dx = on (,T(n.

9 366 E. Feireisl, A. Novotný and H. Petzeltová JMFM The first consequence of (2.15 is the estimate u n 2 L 2 ( dt E δ[ϱ, q] where E δ [ϱ, q]= T (n 1 q 2 + a 2 ϱ γ 1 ϱγ + δ β 1 ϱβ dx. Consequently, since dim(x n is finite, it follows from (2.8 that there exists a constant η = η(n, ϱ, q such that <η(n ϱ n (t, x 1 for all t (,T(n, x. (2.16 η(n Now, the relation (2.15 implies sup t [,T (n] sup t [,T (n] ϱ n (t u n (t 2 dx E δ [ϱ, q] which, combined with (2.16 and the fact that the L and L 2 norms are equivalent on X n, yields ( u n (t L ( + u n (t L ( κ(n, ϱ, q. (2.17 The relations (2.16, (2.17 furnish the desired estimates which, in combination with (2.1, (2.12, give the possibility to repeat the above fixed point argument to conclude, after a finite number of steps, that T (n =T. Moreover, it can be deduced from (2.15 that εδ (ϱn β 2 is bounded in L 2 (,T;W 1,2 (. Using the embedding W 1,2 ( L 6 (, we obtain ϱ β n L1 (,T ;L 3 ( c(ε, δ independently of n. The relation (2.15 also yields sup ϱ β n(t L 1 ( c(δ. t [,T ] Now, by interpolation ϱ β n L2 ( c( ϱ β n 1 4 L1 ( ϱβ n 3 4 L3 ( and, consequently, ϱ β n is bounded in L 4 3 (,T;L 2 ( independently of n, in particular, ϱ n is bounded in L 4 3 β ((,T. Thus ϱ n L β+1 ((,T c(ε, δ provided β 3. To conclude this part, we sum up our information on the sequence of the approximate solutions ϱ n, u n : Lemma 2.3. Assume β 4. Let ϱ n, u n be the solution of (2.1, (2.11 on (,T constructed above.

10 Vol. 3 (21 Navier Stokes Equations of Isentropic Compressible Fluids 367 Then sup ϱ n (t γ L γ ( ce δ[ϱ, q], (2.18 t [,T ] T δ sup t [,T ] ϱ n (t β L β ( ce δ[ϱ, q], (2.19 sup t [,T ] ϱ n (t u n (t 2 L 2 ( ce δ[ϱ, q], (2.2 u n (t 2 L 2 ( + u n(t 2 L 2 ( dt ce δ[ϱ, q], (2.21 and T ε ϱ n (t 2 L 2 ( dt c(β,δ,ϱ, q, (2.22 ϱ n L β+1 ((,T c(ε, δ, ϱ, q (2.23 All the above estimates hold independently of n. Proof. Observing that (2.18 (2.21, and (2.23 follow immediately from the energy inequality (2.15, we have to prove only (2.22. To this end, multiply (2.1 by ϱ n and integrate by parts to get T ε ϱ n (t 2 L 2 ( dt 1 [ ϱ 2 L 2 2 ( + sup ( T 1 ] ϱ n (t 2 L 4 ( T u n 2 L 2 ( dt 2. t [,T ] Consequently, (2.22 follows from (2.19, (2.21 provided β The existence of the first level approximate solutions The final task of this section is to employ the estimates obtained in Lemma 2.3 to pass to the limit for n in the sequence ϱ n, u n to obtain a solution of the problem (2.1 (2.6. The estimates (2.19, (2.22 along with the fact that t ϱ n satisfies (2.1 make it possible to use the Lions-Aubin lemma (see Lions [15, Chapter 1, Theorem V.1] to infer that there exists a function ϱ such that ϱ n ϱ in, say, L 4 ((,T (if β>4 (2.24 and, by virtue of (2.23, ϱ γ n ϱ γ,ϱ β n ϱ β in, say, L 1 ((,T passing to subsequences as the case may be. Analogously, we can find a subsequence so that u n u weakly in L 2 (,T;W 1,2 (. (2.25

11 368 E. Feireisl, A. Novotný and H. Petzeltová JMFM Now, our task is to pass to the limit in the products ϱ n u n and ϱ n u n u n. The estimates (2.18, (2.2, and (2.21 imply boundedness of the sequence ϱ n u n in the space L (,T;L 2γ γ+1 ( which, combined with (2.24, (2.25 yields ϱ n u n ϱ u weakly star in L (,T;L 2γ 2γ γ+1 ( where γ +1 > 6 ( provided γ> 3 2. Consequently, we can pass to the limit in the continuity equation (2.1. In order to continue, we have to show that (2.1 holds, in fact, in the strong sense: Lemma 2.4. There exist r>1,q>2such that t ϱ n, ϱ n are bounded in L r ((,T, ϱ n is bounded in L q (,T;L 2 ( independently of n. Consequently, the limit function ϱ belongs to the same class and satisfies the equation (2.1 almost everywhere on (,T and the boundary conditions (2.3 in the sense of traces. Proof. The idea of the proof is to use the L p theory of parabolic equations. To begin, observe that the functions ϱ n, u n satisfy (2.1 as well as (2.3, (2.4 in the weak sense, i.e., the identity T ϱ n ϕ t ε ϱ n. ϕ + ϱ n u n. ϕ dx dt = (2.27 holds for any ϕ C ((,T vanishing for t =,T. Next, div(ϱ n u n = ϱ n. u n + ϱ n div u n where, by virtue of (2.19,(2.21,(2.22, the first term on the right-hand side is bounded in L 1 (,T;L 3 2( and the second in L 2 (,T;L 2β β+2 ( independently of n. Consequently, in order to apply the L p theory, we have first to improve the integrability in t of the term ϱ n. To this end, we use that ϱ n u n is bounded in L (,T;L 2β β+1 ( L 2 (,T;L 6β β+6 ( and, by interpolation, ϱ n u n L 2 ( c ϱ n u n θ ϱ L 2β n u n 1 θ β+1 ( L 6β β+6 ( for a certain θ (, 1 and β> 3 2.Thus ϱ n u n is bounded in L q (,T;L 2 ( for a certain q>2. Now, the equation (2.27 may be interpreted as an abstract evolution equation v t + Av = g, A = ε + Id (+ the homogeneous Neumann b.c.

12 Vol. 3 (21 Navier Stokes Equations of Isentropic Compressible Fluids 369 with the right-hand side g L q (,T;D(A 1 2 where D(A 1 2 =W 1,2 (. Using the abstract semigroup theory of parabolic equations, namely the maximal regularity estimates (cf. e.g. Amann [1, Chapter III.4], we obtain ϱ n bounded in L q (,T;D(A 1 2 =W 1,2 (. This is enough for the term ϱ n. u n to belong to the space L r ((,T for a certain r>1, and, consequently, div(ϱ n u n L r ((,T. Now, the classical L p theory for parabolic equations yields the desired conclusion. Since ϱ n u n satisfy (2.11, the relation (2.26 can be strengthened to the following statement: 2γ γ+1 ϱ n u n ϱ u in C([,T]; Lweak (. (2.28 Indeed observe that for any fixed ψ in (2.11 the terms t ϱ n. u n.ψ are uniformly continuous in t by virtue of (2.21 and Lemma 2.4. Using compactness of the embedding L 2γ γ+1 ( W 1,2 (, one deduces from (2.28: ϱ n u n ϱ u in C([,T]; W 1,2 ( yielding, together with (2.25 ϱ n u i nu j n ϱu i u j in, say, D ((,T, i,j =1,2,3. Finally, let us clarify the behaviour of the term ϱ n. u n. To this end, we multiply (2.1 by ϱ n and integrate by parts to obtain t t ϱ n (t 2 L 2 ( +2ε ϱ n 2 L 2 ( dt = div u n ϱ n 2 dx dt + ϱ 2 L 2 (. Performing the same treatment with the limit equation satisfied by ϱ, we get t t ϱ(t 2 L 2 ( +2ε ϱ 2 L 2 ( dt = div u ϱ 2 dx dt + ϱ 2 L 2 (. Thus using (2.24 (2.26 and Lemma 2.4 we conclude ϱ n L2 ((,T ϱ L2 ((,T and ϱ n (t L2 ( ϱ(t L2 ( for any t yielding, in particular, strong convergence of ϱ n and, consequently, ϱ n. u i n ϱ. u i in D ((,T, i=1,2,3. Let us write the results achieved in this section in a more concise form: Proposition 2.1. Suppose β>max{4,γ}.let R 3 be a bounded domain with C 2+ν boundary. Assume the initial data ϱ, q satisfy (2.5, (2.6. Then there exists a weak solution ϱ, u of the problem (2.1 (2.6 such that ϱ L β+1 ((,T and the following estimates hold: sup ϱ(t γ L γ ( ce δ[ϱ, q], (2.29 t [,T ]

13 37 E. Feireisl, A. Novotný and H. Petzeltová JMFM and T δ sup ϱ(t β L β ( ce δ[ϱ, q], (2.3 t [,T ] sup ϱ(t u(t 2 L 2 ( ce δ[ϱ, q], (2.31 t [,T ] u(t 2 L 2 ( + u(t 2 L 2 ( dt ce δ[ϱ, q], (2.32 T ε ϱ(t 2 L 2 ( dt c(β,δ,ϱ, q. (2.33 Moreover, the energy inequality d [ 1 dt 2 ϱ u 2 + a γ 1 ϱγ + δ ] β 1 ϱβ dx + µ u 2 +(λ+µ div u 2 dx holds in D (,T. Finally, there exists r>1such that ϱ t, ϱ L r ((,T and the equation (2.1 is satisfied a.a. on (,T. 3. The vanishing viscosity limit The aim of the present section is to pass to the limit in (2.1, (2.2 letting ε. Accordingly, the solution of the problem (2.1 (2.6 obtained in Proposition 2.1 above will be denoted ϱ ε, u ε On the equation div v = f We consider an auxiliary problem div v = f, v =. It can be shown there exists a linear operator operator B =[B 1,B 2,B 3 ] enjoying the properties: B : { f L p ( is a bounded linear operator, i.e., } f = [W 1,p (] 3 B[f] W 1,p ( c(p f L p ( for any 1 <p< ; (3.1 the function v = B[f] solve the problem div v = f in, v =;

14 Vol. 3 (21 Navier Stokes Equations of Isentropic Compressible Fluids 371 if, moreover, f can be written in the form f = div g for a certain g [L r (] 3, g. n =, then B[f] Lr ( c(r g Lr ( (3.2 for arbitary 1 <r<. The operator B was first constructed by Bogovskii [3]. A complete proof of the above mentioned properties may be found in Galdi [1, Theorem 3.3] or Borchers and Sohr [4, Proof of Theorem 2.4] Estimates of the density independent of viscosity We are going to use the operator B to improve the estimates of the density component. To this end, consider the quantities ψ(tb i [ϱ ε m ],ψ D(,T, ψ 1,m = 1 ϱ(tdx as test functions for (2.2. Note that the total mass m = ϱ(t dx is a constant of motion. After a little bit lengthy but straightforward computation, we obtain: T ψ(aϱ γ+1 ε + δϱ β+1 ε dxdt= T ( T m ψ aϱ γ ε + δϱ β ε dx dt +(λ+µ ψ ϱ ε div u ε dx dt T T ψ t ϱ ε u i ε B i [ϱ ε m ]dxdt+µ ψ xj u i ε xj B i [ϱ ε m ]dxdt T T ψ ϱ ε u i εu j ε xj B i [ϱ ε m ]dxdt ε ψ ϱ ε u i ε B i [ ϱ ε ]dxdt T ψ ϱ ε u i [ ( ] T 7 ε B i div ϱε u ε dx dt+ε ψ xj u i ε xj ϱ ε B i [ϱ ε m ]dxdt= I j. j=1 Here and in what follows, the summation convention is used to simplify notation. (i By virtue of (2.3, (2.32, we get T I 1 = m ψ aϱ γ ε + δϱ β ε dx dt+(λ+µ T ψ ϱ ε div u ε dx dt c(δ, ϱ, q. (ii Similarly, by the Hölder inequality, T I 2 = ψ t ϱ ε u i ε B i [ϱ ε m ]dxdt T c ψ t ϱ ε L 2 ( ϱ ε u ε L 2 ( B i [ϱ ε m ] dt c(δ, ϱ, q L ( T ψ t dt

15 372 E. Feireisl, A. Novotný and H. Petzeltová JMFM where the last inequality follows from (2.3, (2.31 and the smoothing properties of the operator B. (iii Analogously, T I 3 = µ ψ xj u i ε xj B i [ϱ ε m ]dxdt µ u ε L 2 ((,T B i [ϱ ε m ] L 2 (,T ;W 1,2 ( c u ε L2 ((,T ϱ ε L2 ((,T c(δ, ϱ, q. (iv Furthermore, we use the Hölder inequality to deduce T I 4 = ψ ϱ ε u i εu j ε xj B i [ϱ ε m ]dxdt T (v Next, one has I 5 = ε ϱ ε L3 ( u ε 2 L 6 ( ϱ ε L3 ( dt c(δ, ϱ, q. T ψ ϱ ε u i ε B i [ ϱ ε ]dxdt T εc ϱ ε L2 ( u ε L6 ( ϱ ε L3 ( dt c(δ, ϱ, q in view of the estimates (2.3, (2.32, (2.33, and (3.2. (vi By virtue of (2.3, (2.32, and (3.2; we infer I 6 = T T ψ ϱ ε u i ε B i [div(ϱ ε u ε ] dx dt u ε 2 L 6 ( ϱ ε 2 L 3 ( dt c(δ, ϱ, q. (vii Finally, using (2.32, (2.33, and (3.1 together with the fact that β 4, we conclude T I 7 = ε ψ xj u i ε xj ϱ ε B i [ϱ ε m ]dxdt ε ε ϱε L2 ((,T u L2 ((,T B i [ϱ ε m ] L ((,T εc(δ, ϱ, q. Summing up the previous results, we have proved the following statement: Lemma 3.1. Let ϱ ε, u ε be the sequence of solutions of the problem (2.1 (2.6 constructed in Proposition 2.1. Then there exists a constant c = c(δ, ϱ, q, independent of ε, such that ϱ ε L γ+1 ((,T + ϱ ε L β+1 ((,T c(δ, ϱ, q. (3.3

16 Vol. 3 (21 Navier Stokes Equations of Isentropic Compressible Fluids The vanishing viscosity limit passage At this stage, we are ready to pass to the limit for ε to get rid of the diffusion term in the equation (2.1 as well as of the ε quantities occuring in (2.2. Note that the parameter δ is kept fixed throughout this procedure so that we may use the estimates derived above. To begin, it is easy to deduce from (2.32, (2.33 that and, analogously, ε ϱ ε. u i ε inl 1 ((,T, i =1,2,3 ε ϱ ε inl 2 (,T;W 1,2 (. By virtue of the estimates (2.3 and (3.3, we get ϱ ε ϱ in C([,T],L β weak ( and weakly in Lβ+1 ((,T (3.4 passing to subsequences if necessary. Moreover, u ε u weakly in L 2 (,T;W 1,2 ( (3.5 and, combining (3.4, (3.5, with (2.31, we infer ϱ ε u ε ϱ u in C([,T]; L 2γ γ+1 weak (. (3.6 Employing the same argument and seeing that 2γ/(γ +1>6/5, we can use the relations (3.5, (3.6 to infer ϱ ε u i εu j ε ϱu i u j in D ((,T, i,j =1,2,3. (3.7 Thus we have proved that the limits ϱ, u satisfy the following system of equations: ϱ t + div(ϱ u =, (3.8 (ϱu i t + div(ϱu i u+p xi =µ u i +(λ+µ(div u xi,i=1,2,3 (3.9 in D ((,T, where aϱ γ ε + δϱ β ε p weakly in L β+1 β ((,T. (3.1 Moreover, in accordance with (3.4, (3.6, the limit functions ϱ, ϱ u satisfy the initial condition ϱ( = ϱ, (ϱ u( = q where ϱ, q are as in (2.5, (2.6. Thus our ultimate goal is to show that p = aϱ γ + δϱ β (3.11 which is equivalent to strong convergence of ϱ ε in L 1 ((,T. This is the aim of the last part of this section.

17 374 E. Feireisl, A. Novotný and H. Petzeltová JMFM 3.4. The effective viscous flux and its properties We introduce the quantity aϱ γ + δϱ β (λ +2µdiv u called usually the effective viscous flux. This quantity enjoys many remarkable properties for which we refer to Hoff [12], Lions [17], or Serre [21]. Here we shall use the result proved by Lions [17, Theorem 5.1, Appendix B]: Lemma 3.2. Let ϱ ε, u ε be the sequence of approximate solutions, the existence of which is guaranteed by Proposition 2.1, and let ϱ, u, and p be the limits appearing in (3.4, (3.5 and (3.1, respectively. Then T ψ T lim ε + ψ φ (aϱ β ε + δϱ β ε (λ +2µdiv u ε ϱ ε dx dt = ( φ p (λ +2µdiv u ϱ dx dt for any ψ D(,T, φ D(. In the rest of this part, we shall give an elementary proof of Lemma 3.2 based on Div-Curl Lemma of compensated compactness. To this end, consider the operators A i = 1 [ xi v],i=1,2,3 (3.12 where 1 stands for the inverse of the Laplace operator on R 3. To be more specific, the Fourier symbol of A i is A j (ξ = iξ j ξ 2. Note that xi A i [v] =v and, by virtue of the classical Mikhlin multiplier theorem (L p regularity theory for the elliptic problems: A i v W 1,s ( c(s, v L s (R 3, 1 <s<, in particular, A i v L q ( c(q, s, v L s (R 3,qfinite, provided 1 q 1 s 1 3, (3.13 A i v L ( c(s, v Ls (R 3 if s>3. Prolonging ϱ ε to be zero outside we consider functions of the form ϕ i (t, x =ψ(tφ(xa i [ϱ ε ],i=1,2,3 where ψ D(,T, φ D(. Now the regularity properties of ϱ, specifically (2.3, justify the choice of ϕ as test functions for the system (2.2. Similarly as in Part 3.2, we arrive at the following formula: T ψ φ ((aϱ γ ε + δϱ β ε (λ +2µdiv u ε ϱ ε dx dt = (3.14

18 Vol. 3 (21 Navier Stokes Equations of Isentropic Compressible Fluids 375 T T (λ + µ ψ div u ε xi φ A i [ϱ ε ]dxdt ψ (aϱ γ ε + δϱ β ε xi φ A i [ϱ ε ]dxdt+ T T µ ψ xj u i ε xj φ A i [ϱ ε ]dxdt ψ ϱ ε u i εu j ε xj φ A i [ϱ ε ]dxdt T T ψ t φϱ ε u i ε A i [ϱ ε ]dxdt ε ψ φϱ ε u i ε A i [div(1 ϱ ε ] dx dt+ T ε ψ φ xj ϱ ε xj u i εa i [ϱ ε ]dxdt+ T µ ψ u i ε xi φϱ ε dxdt µ T T ψ u i ε xj φ xj A i [ϱ ε ]dxdt+ ( ψ φu i ε ϱ ε R i,j [ϱ ε u j ε] ϱ ε u j εr i,j [ϱ ε ] where we have introduced the operators dx dt R i,j [v] = xj A i [v], or, in terms of the Fourier symbols, R i,j (ξ = ξ iξ j ξ 2. (3.15 Here it is important to observe that the function ϱ ε prolonged by zero outside admits the partial derivative with respect to t and ε ϱ ε div(ϱ ε u ε in, t ϱ ε = inr 3 \. Moreover, since u ε vanishes on, we have div(ϱ ε u ε =onr 3 \ and, applying the same argument to ϱ ε : ϱ ε in div(1 ϱ ε = inr 3 \. Now, we repeat the same procedure with the limit equation (3.9 making use of the test functions ϕ i (t, x =ψ(tφ(xa i [ϱ], i =1,2,3, ϱbeing set zero outside. We shall need the following assertion: Lemma 3.3. Let ϱ, u be a solution of (3.8 in D ((,T and such that ϱ L 2 ((,T and u L 2 (,T;W 1,2 (.

19 376 E. Feireisl, A. Novotný and H. Petzeltová JMFM Then, prolonging ϱ, u to be zero on R 3 \, the equation (3.8 holds in D ((,T R 3. Proof. We have to show T ϱϕ t + ϱ u. ϕ dx dt = R 3 for all ϕ D((,T R 3. To this end, consider a sequence of functions φ m D( such that φ m 1, φ m (x = 1 for all x such that dist[x, ] 1 m, (3.16 φ m (x 2mfor all x. Now, we have T T ϱϕ t dx dt = ϱ(φ m ϕ t + ϱ(1 φ m ϕ t dx dt, R 3 and T T ϱ u. ϕ dx dt = ϱ u. (φ m ϕ+ϱ(1 φ m u. ϕ ϱ u. φ m ϕ dx dt. R 3 Since ϱ, u satisfy (3.8, one has T ϱ(φ m ϕ t + ϱ u. (φ m ϕdxdt=; whence it is enough to show T ϱ u. φ m ϕ dx dt asm. (3.17 The velocity components u i, i =1,2,3 belong to L 2 (,T;W 1,2 ( and, consequently, u dist 1 [x, ] L 2 (,T;L 2 (. On the other hand, by virtue of (3.16, dist[x, ] φ m 2, dist[x, ] φ m a.a. on (,T, which yields (3.17 as ϱ L 2 ((,T. Now, similarly as in (3.14, we can use ϕ as test functions for (3.9 to obtain: T ( ψ φ p (λ +2µ div u ϱ dx dt = (3.18 T (λ + µ ψ div u xi φa i [ϱ]dxdt T ψ p xi φa i [ϱ]dxdt+

20 Vol. 3 (21 Navier Stokes Equations of Isentropic Compressible Fluids 377 T T µ ψ xj u i xj φ A i [ϱ]dxdt ψ ϱu i u j xj φ A i [ϱ]dxdt T ψ t φϱu i A i [ϱ]dxdt+ T µ ψ u i xi φϱdxdt µ T T ψ u i xj φ xj A i [ϱ]dxdt+ ψ φu i( ϱr i,j [ϱu j ] ϱu j R i,j [ϱ] dx dt. Going back to the expression (3.14 we have T ε ψ φϱ ε u i ε A i [div(1 ϱ ε ] dx dt (3.19 ε φψ L ((,T where, by virtue of (3.13, T ϱ ε L3 ( u ε L6 ( A i [div(1 ϱ ε ] L2 ( dt A i [div(1 ϱ ε ] L2 ( c ϱ ε L2 (. Consequently, using (2.3, (2.32, and (2.33, we conclude that the integral on the left-hand side of (3.19 tends to zero with ε. Analogously, we have T ε ψ φ ϱ ε. u i εa i [ϱ ε ]dxdt ε ψφ L ((,T ε ϱ ε L 2 ((,T u ε L 2 ((,T A i [ϱ ε ] L ((,T where the right-hand side tends to zero for δ fixed and ε. Accordingly, following Lions [17], we observe that Lemma 3.2 holds provided the integrals on the right-hand side of (3.14 converge to their counterparts in (3.18. To begin, one observes easily that (3.4 together with (3.13 imply A i [ϱ ε ] A i [ϱ]inc((,t and xj A i [ϱ ε ] xj A i [ϱ] inc([,t]; L β weak (. Consequently, with the relations (3.4 (3.7 in mind, it is easy to see that it is enough to show T T ( ψ φu i ε ϱ ε R i,j [ϱ ε u j ε] ϱ ε u j εr i,j [ϱ ε ] dx dt (3.2 ψ φu i( ϱr i,j [ϱu j ] ϱu j R i,j [ϱ] dx dt as ε.

21 378 E. Feireisl, A. Novotný and H. Petzeltová JMFM Using (3.4, (3.5 together with (3.6, we observe that (3.2 will be a consequence of the following crucial assertion: Lemma 3.4. Suppose where 1/p +1/q =1/r < 1. Then v n v weakly in L p (R 3, w n w weakly in L q (R 3 v n R i,j [w n ] w n R i,j [v n ] vr i,j [w] wr i,j [v] weakly in L r (R 3, i,j =1,2,3. Remark 3.1. If p = q = 2, the above assertion may be viewed as a special case treated by Coifman et al. [5, Theorem 5.1]. Proof. It is easy to see that the conclusion of Lemma 3.4 is a particular case of a more general statement: v i nr i,j [w j n] w j nr i,j [v i n] v i R i,j [w j ] w j R i,j [v i ]ind (R 3. (3.21 provided v n =[v 1 n,v 2 n,v 3 n], w n =[w 1 n,w 2 n,w 3 n] are sequences of vector functions such that v n v weakly in [L p (R 3 ] 3, w n w weakly in [L q (R 3 ] 3. Indeed Lemma 3.4 follows from (3.21 taking v n = v n e i, w n = w n e j where e i, i =1,2,3 is the orthogonal basis of R 3. To show (3.21, one can use the symmetry R i,j = R j,i to deduce vnr i i,j [wn] j wnr j i,j [vn]= i ( ( vn (R i i,k [vn] k (R i,j [wn] j wn j (R k,j [wn] k (R i,j [vn] i = U n. V n X n. Y n where div U n = xi (v n i R i,k [vn] k = div X ( n = xj wn j R j,k [wn] k = and V n = ( 1 [ xj wn], j Yn = ( 1 [ xi vn], i i.e., curl( V n = curl( Y n =. Consequently, it is possible to use the L p L q version of Div-Curl Lemma (see e.g. YI [23] to conclude U n. V n U. V, Xn. Y n X. Y in D (R 3 where U i = v i R i,k [v k ],V i =R i,j [w j ],X j =w j R j,k [w k ],Y j =R j,i [v i ],i,j=1,2,3.

22 Vol. 3 (21 Navier Stokes Equations of Isentropic Compressible Fluids 379 Now, combining Lemma 3.4 with (3.4, (3.6 we get ϱ ε R i,j [ϱ ε u j ε] ϱ ε u j εr i,j [ϱ ε ] ϱr i,j [ϱu j ] ϱu j R i,j [ϱ] strongly in L 2 (,T;W 1,2 ( (3.22 provided β > 6γ/(2γ 3 which, together with (3.5, completes the proof of (3.2. Indeed the quantities in (3.22 converge weakly and are uniformly bounded in L α ( for any t (,T where γ +1 2γ + 1 β = 1 α < 5 6. Consequently, the desired conclusion follows from compactness of the embedding L α W 1,2 and the Lebesgue convergence theorem. We have proved Lemma Strong convergence of the density We conclude this section by showing (3.11 and, consequently, strong convergence of the sequence ϱ ε. By virtue of Lemma 3.3, the functions ϱ L 2 (,T;L 2 (, u L 2 (,T;W 1,2 ( solve the continuity equation (3.8 in D ((,T R 3 provided they were prolonged to be zero on R 3 \. Taking a regularizing sequence ϑ m = ϑ m (x, we obtain t S m [ϱ] + div(s m [ϱ] u =r m on (,T R 3 (3.23 where S m [v] =ϑ m vare the standard smoothing operators. Here, by virtue of [16, Lemma 2.3], r m inl 1 ((,T R 3 asm. Following DiPerna and Lions [6] we can multiply (3.23 by b (S m [ϱ] and pass to the limit for m to deduce that ϱ, u solve (3.8 in the sense of renormalized solutions, i.e., (1.7 holds in D ((,T R 3 for any b satisfying (1.8. Moreover, in accordance with Remark 1.1, we are allowed to take b(z =zlog(z. Consequently, one may integrate (1.7, we obtain T ϱ div u dx dt = ϱ log(ϱ dx ϱ(t log(ϱ(t dx. (3.24 On the other hand, ϱ ε solve (2.1 a.e. on (,T, in particular, b(ϱ ε t + div(b(ϱ ε u ε +(b (ϱ ε ϱ ε b(ϱ ε div u ε ε b(ϱ ε for any b convex and globally Lipschitz on R + ; whence T (b (ϱ ε ϱ ε b(ϱ ε div u ε dx dt b(ϱ dx b(ϱ ε (T dx from which we easily deduce T ϱ ε div u ε dx dt ϱ log(ϱ dx ϱ ε (T log(ϱ ε (T dx. (3.25

23 38 E. Feireisl, A. Novotný and H. Petzeltová JMFM Take two nondecreasing sequences ψ n, φ n of nonnegative functions such that ψ n D(,T, ψ n 1,φ n D(,φ n 1. Combining the conclusion of Lemma 3.2 together with (3.24, (3.25, we obtain T T lim sup ψ m φ m (aϱ γ ε + δϱ β ε ϱ ε dx dt lim sup ψ n φ n (aϱ γ ε + δϱ β ε ϱ ε dx dt ε + ε + T ( lim ψ n φ n aϱ γ ε + δϱ β ε (λ +2µdiv u ε ϱ ε dx dt+ ε + T (λ +2µ lim sup ψ n φ n ϱ ε div u ε dx dt ε + T ψ n φ n (p (λ +2µdiv uϱ dx dt+ T T (λ+2µ lim sup ϱ ε 1 ψ n φ n div u ε dx dt+(λ+2µ lim sup ϱ ε div u ε dx dt ε + ε + T T T pϱ dx dt+(λ+2µ ϱ 1 ψ n φ n div u dx dt (λ+2µ ϱdiv udxdt+ [ ] η 1 (n+(λ+2µ ϱ log(ϱ dx lim inf ϱ ε (T log(ϱ ε (T dx ε + T η 1 (n+η 2 (n+ pϱ dx dt for all m n where η 1 (n,η 2 (n for n. Thus we have proved T T lim sup ψ m φ m (aϱ γ ε + δϱ β ε ϱ ε dx dt pϱ dx dt for all m =1,2,... ε + To conclude the proof of (3.11, we make use of a (slightly modified Minty s trick. Since the nonlinearity P (z =az γ + δz β is monotone, we have T ψ m φ m (P (ϱ ε P (v(ϱ ε vdxdt and, consequently, T pϱ dx dt + T ψ m T φ m P (vv dx dt ψ m φ n (pv + P (vϱdxdt Now, letting m,weget T (p P(v(ϱ vdxdt

24 Vol. 3 (21 Navier Stokes Equations of Isentropic Compressible Fluids 381 and the choice v = ϱ + ηϕ, η, ϕ arbitrary, yields desired conclusion: p = aϱ γ + δϱ β. Let us review the results achieved in this section: Proposition 3.1. Let R 3 be a bounded domain of class C 2+ν and let { 6γ } β>max 2γ 3, 4. Then, given initial data ϱ, q as in (2.5, (2.6, there exists a finite energy weak solution ϱ, u of the problem ϱ t + div(ϱ u = (3.26 (ϱu i t + div(ϱu i u+(aϱ γ + δϱ β xi = µ u i +(λ+µ(div u xi, i =1,2,3, (3.27 u = (3.28 satisfying the initial conditions (1.4. Moreover, ϱ L β+1 ((,T and the equation (3.26 holds in the sense of renormalized solutions on D ((,T R 3 provided ϱ, u were prolonged to be zero on R 3 \. Finally, ϱ, u satisfy the estimates: sup ϱ(t γ L γ ( ce δ[ϱ, q], (3.29 t [,T ] δ sup ϱ(t β L β ( ce δ[ϱ, q], (3.3 t [,T ] sup ϱ u(t 2 L 2 ( ce δ[ϱ, q], (3.31 t [,T ] u L2 (,T ;W 1,2 ( ce δ[ϱ, q] (3.32 where the constant c is independent of δ>. 4. Passing to the limit in the artificial pressure term Our ultimate goal is to let δ in (3.27 and to relax our hypotheses on the initial data ϱ, q, which, up to now, have been supposed to satisfy (2.5, (2.6. To begin, consider general initial data ϱ, q satisfying the compatibility conditions (1.1. It is easy to find a sequence ϱ,δ C 2+ν ( such that <δ ϱ,δ (x δ 1/β, ϱ,δ. n =, (4.1 ϱ,δ ϱ in L γ ( as δ. (4.2

25 382 E. Feireisl, A. Novotný and H. Petzeltová JMFM Set By virtue of (1.1, we have q i ϱ,δ (x (x q δ(x i ϱ (x if ϱ (x >, = i=1,2,3. ifϱ (x=, q δ i 2 bounded in L 1 ( independently of δ>, i =1,2,3 ϱ,δ and we can find h i δ C2 ( such that qi δ h i ϱ,δ δ L 2 ( <δ, i=1,2,3. Taking q i δ = hi δ ϱ,δ, i =1,2,3, one checks easily that and q i δ 2 ϱ,δ are bounded L 1 (, i =1,2,3, independently of δ> (4.3 q i δ q i in L 1 ( as δ for i =1,2,3. (4.4 From now on, we shall deal with the sequence of approximate solutions ϱ δ, u δ of the problem (3.26 (3.28 with the initial data ϱ,δ, q δ =[qδ 1,q2 δ,q3 δ ], the existence of which is guaranteed by Proposition 3.1. Observe that, by virtue of (4.1, (4.3, the value of the modified energy E δ [ϱ,δ, q δ ] is bounded and, consequently, the estimates (3.29 (3.32 hold independently of δ On integrability of the density We first derive estimates of the density ϱ δ independent of δ>. The technique is the same as in Section 3, Part 3.2. Since the continuity equation (3.26 is satisfied in the sense of renormalized solutions in D ((,T R 3, we can regularize the equation (1.7. Similarly as in (3.23, we get ] t S m [b(ϱ] + div(s m [b(ϱ] u+s m [(b (ϱϱ b(ϱdiv u = r m (4.5 where r m inl 2 (,T;L 2 (R 3 as m (4.6 provided b is uniformly bounded. Exactly as in Section 3.2, we use the operator B introduced in Section 3.1 to construct multipliers of the form ] ϕ i (t, x =ψ(tb i [S m [b(ϱ δ ] S m [b(ϱ δ ] dx,i=1,2,3, ψ D(,T.

26 Vol. 3 (21 Navier Stokes Equations of Isentropic Compressible Fluids 383 The functions ϕ i are smooth with respect to the x variable while t ϕ i are bounded in L 2 (,T;W 1,2 ( in view of (4.5, (4.6. Consequently, the quantities ϕ i, i =1,2,3 may be used as test functions for the equations (3.27. Taking (4.5 into account one arrives at the following formula: T ψ(aϱ γ δ + δϱβ δ S m[b(ϱ δ ] dx dt = (4.7 T ( ( T ψ (aϱ γ δ +δϱβ δ dx m [b(ϱ δ ] dx dt+(λ+µ ψs m [b(ϱ δ ]div u δ dx dt S T ] ψ t ϱ δ u i δ B i [S m [b(ϱ δ ] S m [b(ϱ δ ]dx dx dt+ T ( ] ψ µ xj u i δ ϱ δ u i δu j δ xj B i [S m [b(ϱ δ ] S m [b(ϱ δ ] dx dx dt+ T [ [ [ ] ψϱ δ u i δ B i S m (b(ϱ δ b (ϱ δ ϱ δ div u δ ] S m (b(ϱ δ b (ϱ δ ϱ δ div u δ ]dx dx dt+ T ] T ] ψϱ δ u i δ B i [r m r m dx dx ψϱ δ u i δ B i [div (S m [b(ϱ δ ] u δ dx dt. Now, making use of (4.6, we can pass to the limit for m in (4.7. Moreover, we can approximate the function z θ b(z to deduce T ψ(aϱ γ+θ δ + δϱ β+θ δ dx dt c 7 I j where the integrals I 1 - I 7 may be treated as follows: (i In view of (3.29, (3.3, T ( dx( I 1 = ψ (aϱ γ δ + δϱβ δ ϱ θ δ dx dt is bounded independently of δ provided θ γ. (ii Similarly, by virtue of (3.29, (3.32, we have T I 2 = ψϱ θ δ div u δ dx dt c on condition that θ γ 2. (iii As for I 3, we have: T I 3 = ψ t ϱ δ u i δ B i {ϱ θ δ j=1 } ϱ θ δdx dx dt c T ψ t dt if, for instance, θ< γ 3. Here, we have used (3.29, (3.31, and (3.1 together with the embedding W 1,p ( L ( for p>3.

27 384 E. Feireisl, A. Novotný and H. Petzeltová JMFM (iv Furthermore, T } I 4 = ψ xj u i δ xj B i {ϱ θ δ ϱ θ δ dx dx dt c provided θ γ 2, and, similarly, (v T } I 5 = ψϱ δ u i δu j δ x j B i {ϱ θ δ ϱ θ δ dx dx dt is bounded independently of δ>ifθ 2 3 γ 1. (vi Next, by virtue of (3.1 and the Hölder inequality, T } I 6 = (1 θ ψϱ δ u i δ B i {ϱ θ δ div u δ ϱ θ δ div u δ dx dx dt T T c 1 ϱ δ Lγ ( u δ L6 ( B[...] Lp (dt c 2 ϱ δ Lγ ( u δ L6 ( ϱ θ δ div u δ Lq (dt where p = 6γ 5γ 6 and q = 6γ 7γ 6. Thus I 6 is bounded independently of δ provided θ 2 3 γ 1. (vii Finally, by virtue of (3.2, we have T } I 7 = sup lim sup ψϱ δ u i δ B i {div (S m [b(ϱ δ ] u δ dx dt b(z z θ m T c 1 sup lim sup ϱ δ Lγ ( u δ L6 ( S m [b(ϱ δ ] u δ Lp ( dt b(z z θ m T c 2 ϱ δ Lγ ( u δ 2 L 6 ( ϱθ δ Lr ( dt where r = 3γ 2γ 3. Using (3.29, (3.32 we infer I 7 is bounded provided, similarly as above, θ 2 3 γ 1. Consequently, we have proved the following result: Lemma 4.1. Let γ> 3 2. Then there exists θ>, depending only on γ, such that T aϱ γ+θ δ + δϱ β+θ δ dx dt c where the constant c is independent of δ>. Remark 4.1. It can be shown (cf. θ = 2 3 γ 1. Lions [17] that the optimal value of θ is

28 Vol. 3 (21 Navier Stokes Equations of Isentropic Compressible Fluids The limit passage The uniform energy estimates (3.29 (3.32 imply and, by virtue of Lemma 4.1, ϱ δ ϱ in C([,T]; L γ weak (, u δ u weakly in L 2 (,T;[W 1,2 (] 3, (4.8 ϱ δ u δ ϱ u in C([,T]; L 2γ γ+1 weak ϱ γ δ ϱγ weakly in L γ+θ γ ((,T (, (4.9 passing to subsequences as the case may be. Moreover, by virtue of (4.2, (4.4, the limits ϱ, ϱ u satisfy the initial conditions (1.4. Since γ>3/2, one has and (4.8, (4.9 yield 2γ γ+1 > 6 5 ϱ δ u i δu j δ ϱui u j,i,j=1,2,... in, say, D ((,T. Finally, making use of Lemma 4.1 again, we get δϱ β δ in, say, L1 ((,T. (4.1 Consequently, ϱ, u satisfy t ϱ + div(ϱ u = (4.11 in D ((,T R 3, (ϱu i t + div(ϱu i u+a(ϱ γ xi =µ u i +(λ+µ(div u xi,i=1,2,3 (4.12 in D ((,T. Thus the only thing to complete the proof of Theorem 1.1 is to show strong convergence of ϱ δ in L 1 or, equivalently, ϱ γ = ϱ γ. Since ϱ δ, u δ is a renormalized solution of the continuity equation (3.26 in D ((,T R 3 we have T k (ϱ δ t +div(t k (ϱ δ u δ +(T k(ϱ δ ϱ δ T k (ϱ δ div u δ =ind ((,T R 3 (4.13 where T k are the cut-off functions introduced in (1.15. Passing to the limit for δ + we obtain where t T k (ϱ + div(t k (ϱ u+(t k (ϱϱ T k(ϱdiv u =ind ((,T R 3 (4.14 (T k(ϱ δ ϱ δ T k (ϱ δ div u δ (T k (ϱϱ T k(ϱ div u weakly in L 2 ((,T and T k (ϱ δ T k (ϱ inc([,t]; L p weak ( for all 1 p<. (4.15

29 386 E. Feireisl, A. Novotný and H. Petzeltová JMFM 4.3. The effective viscous flux Similarly as in Section 3, we prove the following auxiliary result: Lemma 4.2. Let ϱ δ, u δ be the sequence of approximate solutions constructed by means of Proposition 3.1. Then T ( lim ψ φ aϱ γ δ (λ +2µdiv u δ T k (ϱ δ dx dt = δ + T ( ψ φ aϱ γ (λ +2µdiv u T k (ϱ dx dt for any ψ D(,T, φ D(. Proof. Pursuing step by step the proof of Lemma 3.2, we use the quantities ϕ i (t, x =ψ(tφ(xa i [T k (ϱ n ], ψ D(,T, φ D(, i =1,2,3 as test functions for (3.27 (as always, ϱ δ is prolonged by zero outside : T ( ψφ aϱ γ δ + δϱβ δ (λ +2µ div u δ T k (ϱ δ dxdt= (4.16 T [ ] ψ (λ + µdiv u δ aϱ γ δ δϱβ δ xi φ A i [T k (ϱ δ ] dx dt+ T { } µ ψ xj φ xj u i δa i [T k (ϱ δ ] u i δ xj φ xj A i [T k (ϱ δ ]dx dt + u i δ xi φt k (ϱ δ dxdt T { } φϱ δ u i δ t ψ A i [T k (ϱ δ ] + ψa i [(T k (ϱ δ T k(ϱ δ ϱ δ div u δ ] dx dt T ψϱ δ u i δu j δ x j φa i [T k (ϱ δ ] dx dt+ T { } T k (ϱ δ R i,j [φϱ δ u j δ ] φϱ δu j δ R i,j[t k (ϱ δ ] dx dt ψu i δ where the operators R i,j are defined by (3.15. Analogously, we can repeat the above arguments considering the equations (4.12, (4.14 and the test functions to deduce T ϕ i (t, x =ψφa i [T k (ϱ], i=1,2,3 ( ψφ aϱ γ (λ +2µ div u T k (ϱ dxdt= (4.17

30 Vol. 3 (21 Navier Stokes Equations of Isentropic Compressible Fluids 387 T ] ψ [(λ + µdiv u aϱ γ xi φ A i [T k (ϱ] dx dt+ T { } µ ψ xj φ xj u i A i [T k (ϱ] u i xj φ xj A i [T k (ϱ] + u i xi φ T k (ϱ dx dt T φϱ u i{ } t ψ A i [T k (ϱ] + ψa i [(T k (ϱ T k (ϱϱ div u] dx dt T ψϱu i u j xj φa i [T k (ϱ] dx dt+ T ψu i{ } T k (ϱr i,j [φϱ u j ] φϱ u j R i,j [T k (ϱ] dx dt. Similarly as in the proof of Lemma 3.2, it can be shown that all the terms on the right-hand side of (4.16 converge to their counterparts in (4.17 which, together with (4.1, yields the desired conclusion. Here the hardest part, i.e., the convergence of the quantity T { } ψu i δ T k (ϱ δ R i,j [φϱ δ u j δ ] φϱ δu j δ R i,j[t k (ϱ δ ] dx dt T ψu i{ } T k (ϱr i,j [φϱu j ] φϱu j R i,j [T k (ϱ] dx dt follows from (4.8, (4.9, (4.15, and Lemma The amplitude of oscillations The main result of this part is essentially taken over from [7]: Lemma 4.3. There exists a constant c independent of k such that lim sup T k (ϱ δ T k (ϱ L γ+1 ((,T c δ + for any k 1. Proof. One has T lim ϱ γ δ T k(ϱ δ ϱ γ T k (ϱ dxdt= (4.18 δ + T lim (ϱ γ δ δ + ϱγ (T k (ϱ δ T k (ϱ dx dt+ T T lim (ϱ γ δ δ + ϱγ (T k (ϱ δ T k (ϱ dx dt (ϱ γ ϱ γ (T k (ϱ T k (ϱ dx dt

31 388 E. Feireisl, A. Novotný and H. Petzeltová JMFM T lim sup T k (ϱ δ T k (ϱ γ+1 dx dt δ + as z z γ is convex, T k concave on [,, and (z γ y γ (T k (z T k (y T k (z T k (y γ+1 for all z,y. On the other hand, lim δ + T lim δ + 2 sup δ T div u δ T k (ϱ δ div u T k (ϱ dxdt= (4.19 ( T k (ϱ δ T k (ϱ+t k (ϱ T k (ϱ div u δ dx dt div u δ L 2 ((,T lim sup T k (ϱ δ T k (ϱ L 2 ((,T. δ + The relations (4.18, (4.19 combined with Lemma 4.2 yield the desired conclusion The renormalized solutions We are going to use the conclusion of Lemma 4.3 to show the following crucial assertion: Lemma 4.4. The limit functions ϱ, u solve (4.11 in the sense of renormalized solutions, i.e., t b(ϱ + div(b(ϱ u+(b (ϱϱ b(ϱ div u = (4.2 holds in D ((,T R 3 for any b C 1 (R satisfying (1.8 provided ϱ, u are set zero outside. Proof. Regularizing (4.14 one gets t S m [T k (ϱ] + div(s m [T k (ϱ] u+s m [(T k (ϱϱ T k(ϱ div u] =r m (4.21 where r m inl 2 (,T;L 2 (R 3 for any fixed k. Now, we are allowed to multiply (4.21 by b (S m [T k (ϱ]. Letting m we deduce ( t b(t k (ϱ + div(b(t k (ϱ u+ b (T k (ϱt k (ϱ b(t k (ϱ div u = (4.22 b (T k (ϱ[(t k (ϱ T k (ϱϱdiv u] ind ((,T R 3. At this stage, the main idea is to let k in (4.22. We have T k (ϱ ϱ as k in L p ((,T for any 1 p<γ

32 Vol. 3 (21 Navier Stokes Equations of Isentropic Compressible Fluids 389 since and T k (ϱ ϱ Lp ((,T lim inf δ + T k(ϱ δ ϱ δ Lp ((,T T k (ϱ δ ϱ δ p L p ((,T 2p k p γ ϱ δ γ L γ ((,T. (4.23 Thus (4.22 will imply (4.2 provided we show b (T k (ϱ[(t k (ϱϱ T k(ϱdiv u] inl 1 ((,T as k. (4.24 Denote Q k,m = {(t, x (,T T k (ϱ M}, where M is the constant related to b by (1.8. One has T b (T k (ϱ[(t k (ϱϱ T k(ϱdiv u] dx dt sup z M sup b (z z M Q k,m b (z sup u δ L 2 (,T ;W 1,2 ( δ (T k (ϱϱ T k(ϱdiv u dx dt lim inf δ + T k(ϱ δ ϱ δ T k (ϱ δ L 2 (Q k,m. Now, by interpolation, T k(ϱ δ ϱ δ T k (ϱ δ 2 L 2 (Q k,m (4.25 T k(ϱ δ ϱ δ T k (ϱ δ α L 1 ((,T T k(ϱ δ ϱ δ T k (ϱ δ (1 α(γ+1 L γ+1 (Q k,m,α=γ 1 γ where, similarly as in (4.23, T k(ϱ δ ϱ δ T k (ϱ δ L 1 ((,T 2 γ k 1 γ sup ϱ δ γ L γ ((,T, (4.26 δ and, since T k (zz T k(z, T k(ϱ δ ϱ δ T k (ϱ δ L γ+1 (Q k,m (4.27 ( 2 T k (ϱ δ T k (ϱ L γ+1 ((,T + T k (ϱ L γ+1 (Q k,m ( 2 T k (ϱ δ T k (ϱ L γ+1 ((,T + T k (ϱ T k (ϱ L γ+1 ((,T + T k (ϱ L γ+1 (Q k,m 2 T k (ϱ δ T k (ϱ L γ+1 ((,T +2 T k (ϱ T k (ϱ L γ+1 ((,T +2M. By virtue of Lemma 4.3 and (4.27, one gets lim sup T k(ϱ δ ϱ δ T k (ϱ δ L γ+1 (Q k,m 4c +2M δ + which, together with (4.25, (4.26, completes the proof of (4.24.

33 39 E. Feireisl, A. Novotný and H. Petzeltová JMFM 4.6. Strong convergence of the density We are going to complete the proof of Theorem 1.1. To this end, we introduce a family of functions L k : zlog(z for z<k, L k (z = zlog(k+z z k T k(s/s 2 ds for z k. Seeing that L k can be writen as L k (z =β k z+b k (z (4.28 where b k satisfy (1.8, we can use the fact that ϱ δ, u δ are renormalized solutions of (3.26 to deduce t L k (ϱ δ + div(l k (ϱ δ u δ +T k (ϱ δ div u δ =. (4.29 Similarly, by virtue of (4.11 and Lemma 4.4, t L k (ϱ + div(l k (ϱ u+t k (ϱ div u = (4.3 in D ((,T. In view of (4.29, we can assume L k (ϱ δ L k (ϱ inc([,t]; L γ weak ( (4.31 and, approximating z log(z L k (z, ϱ δ log(ϱ δ ϱ log(ϱ inc([,t]; L α weak( for any 1 α<γ. Taking the difference of (4.29 and (4.3 and integrating with respect to t we get (L k (ϱ δ L k (ϱ(tφ dx = (L k (ϱ,δ L k (ϱ φ dx+ t (L k (ϱ δ u δ L k (ϱ u. φ +(T k (ϱ div u T k (ϱ δ div u δ φ dx dt for any φ D(. Passing to the limit for δ and making use of (4.2 together with (4.31, one obtains (L k (ϱ L k (ϱ(tφ dx = t t (L k (ϱ L k (ϱ u. φ dx dt+ lim (T k (ϱ div u T k (ϱ δ div u δ φ dx dt δ + Taking φ = φ m the sequence approximating the characteristic function of as in (3.16 and making use of the boundary conditions (1.5, one derives (L k (ϱ L k (ϱ(t dx= (4.32

34 Vol. 3 (21 Navier Stokes Equations of Isentropic Compressible Fluids 391 t t T k (ϱ div u dx dt lim T k (ϱ δ div u δ dx dt. δ + Observe that the term L k (ϱ L k (ϱ is bounded in view of (4.28. Finally, making use of Lemma 4.2 and the monotonicity of the pressure (cf. (4.18, we can estimate the right-hand side of (4.32: t t T k (ϱ div u dx dt lim T k (ϱ δ div u δ dx dt (4.33 δ + t (T k (ϱ T k (ϱ div u dx dt. By virtue of Lemma 4.3, the right-hand side of (4.33 tends to zero as k. Accordingly, one can pass to the limit for k in (4.32 to conclude ϱ log(ϱ(t =ϱlog(ϱ(t for all t [,T], which implies strong convergence of the sequence ϱ δ in L 1 ((,T. Theorem 1.1 has been proved. 5. Concluding remarks One can prove the same existence result in two space dimensions for any γ>1. The total mass of the solution constructed above is conserved, i.e., m = ϱ(t dxis independent of t [,T]. Note that this property does not follow directly from (1.1 if the density ϱ is not known to be square integrable, i.e., when γ (3/2, 9/5. The proof of Theorem 1.1 remains basically unchanged if the motion of the fluid is driven by a bounded external force, i.e., when (1.2 contains an additional term ϱ f(t, x with f bounded and measurable function. Acknowledgement. The work of E. F. and H. P. was supported by the Grant 21/98/145 of GA ČR. The first author is also grateful to Université de Toulon et du Var for hospitality and financial support. References [1] H. Amann, Linear and quasilinear parabolic problems, I, Birkhäuser Verlag, Basel, [2] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Krajevyje zadaci mechaniki neodnorodnych zidkostej, Novosibirsk, [3] M. E. Bogovskii, Solution of some vector analysis problems connected with operators div and grad (in Russian, Trudy Sem. S. L. Sobolev 8 (1 (198, 5 4.