Key words. Navier-Stokes, compressible, non constant viscosity coefficients, strong solutions, uniqueness, mono-dimensional

Transcription

1 EXISTENCE AND UNIQUENESS OF GLOBAL STONG SOLUTIONS FO ONE-DIMENSIONAL COMPESSIBLE NAVIE-STOKES EQUATIONS A.MELLET AND A.VASSEU Abstract. We consier Navier-Stokes equations for compressible viscous fluis in one imension. It is a well known fact that if the initial atum are smooth an the initial ensity is boune by below by a positive constant, then a strong solution exists locally in time. In this paper, we show that uner the same hypothesis, the ensity remains boune by below by a positive constant uniformly in time, an that strong solutions therefore exist globally in time. Moreover, while most existence results are obtaine for positive viscosity coefficients, the present result hols even if the viscosity coefficient vanishes with the ensity. Finally, we prove that the solution is unique in the class of weak solutions satisfying the usual entropy inequality. The key point of the paper is a new entropy-like inequality introuce by Bresch an Desjarins for the shallow water system of equations. This inequality gives aitional regularity for the ensity (provie such regularity exists at initial time). Key wors. Navier-Stokes, compressible, non constant viscosity coefficients, strong solutions, uniqueness, mono-imensional AMS subject classifications. 76N10, 35Q30 1. Introuction. This paper is evote to the existence of global strong solutions of the following Navier-Stokes equations for compressible isentropic flow: ρ t + (ρu) x = 0 (1.1) (ρu) t + (ρu 2 ) x + p(ρ) x = (µ(ρ) u x ) x, (x, t) +, (1.2) with possibly egenerate viscosity coefficient. Throughout the paper, we will assume that the pressure p(ρ) obeys a gamma type law p(ρ) = ρ γ, γ > 1, (1.3) (though more general pressure laws coul be taken into account). The viscosity coefficient µ(ρ) is often assume to be a positive constant. However, it is well known that the viscosity of a gas epens on the temperature, an thus on the ensity (in the isentropic case). For example, the Chapman-Enskog viscosity law for har sphere molecules preicts that µ(ρ) is proportional to the square root of the temperature (see [CC70]). In the case of monoatomic gas (γ = 5/3), this leas to µ(ρ) = ρ 1/3. More generally, µ(ρ) is expecte to vanish as a power of the ρ on the vacuum. In this paper, we consier egenerate viscosity coefficients that vanish for ρ = 0 at most like ρ α for some α < 1/2. In particular, the cases µ(ρ) = ν an µ(ρ) = νρ 1/3 (with ν positive constant) are inclue in our result (see conitions (2.3)-(2.4) for etails). One-imensional Navier-Stokes equations have been stuie by many authors when the viscosity coefficient µ is a positive constant. The existence of weak solutions Department of Mathematics, University of Texas at Austin, 1 University station C1200, Austin, TX, A. Mellet was partially supporte by NSF grant DMS A. Vasseur was partially supporte by NSF grant DMS

2 2 A. MELLET AND A. VASSEU was first establishe by A. Kazhikhov an V. Shelukhin [KS77] for smooth enough ata close to the equilibrium (boune away from zero). The case of iscontinuous ata (still boune away from zero) was aresse by V. Shelukhin [She82, She83, She84] an then by D. Serre [Ser86a, Ser86b] an D. Hoff [Hof87]. First results concerning vanishing intial ensity were also obtaine by V. Shelukhin [She86]. In [Hof98], D. Hoff prove the existence of global weak solutions with large iscontinuous initial ata, possibly having ifferent limits at x = ±. He prove moreover that the constructe solutions have strictly positive ensities (vacuum states cannot form in finite time). In imension greater than two, similar results were obtaine by A. Matsumura an T. Nishia [MN79] for smooth ata an D. Hoff [Hof95] for iscontinuous ata close to the equilibrium. The first global existence result for initial ensity that are allowe to vanish was ue to P.-L. Lions (see [Lio98]). The result was later improve by E. Feireisl et al. ([FNP01] an [Fei04]). Another question is that of the regularity an uniqueness of the solutions. This problem was first analyze by V. Solonnikov [Sol76] for smooth initial ata an for small time. However, the regularity may blow-up as the solution gets close to vacuum. This leas to another interesting question of whether vacuum may arise in finite time. D. Hoff an J. Smoller ([HS01]) show that any weak solution of the Navier-Stokes equations in one space imension o not exhibit vacuum states, provie that no vacuum states are present initially. More precisely, they showe that if the initial ata satisfies ρ 0 (x) x > 0 E for all open subsets E, then E ρ(x, t) x > 0 for every open subset E an for every t [0, T ]. The main theorem of this paper states that the strong solutions constructe by V. Solonnikov in [Sol76] remain boune away from zero uniformly in time (i.e. vacuum never arises) an are thus efine globally in time. This result can be seen as the equivalent of the result of D. Hoff in [Hof95] for strong solutions instea of weak solutions. Another interest of this paper is the fact that unlike all the references mentione above, the result presente here is vali with egenerate viscosity coefficients. Note that compressible Navier-Stokes equations with egenerate viscosity coefficients have been stuie before (see for example [LXY98], [OMNM02], [YYZ01] an [YZ02]). All those papers, however, are evote to the case of compactly supporte initial ata an to the escription of the evolution of the free bounary. We are intereste here in the opposite situation in which vacuum never arises. The new tool that allows us to obtain those results is an entropy inequality that was erive by D. Bresch an B. Desjarins in [BD02] for the multi-imensional Korteweg system of equations (which correspons to the case µ(ρ) = ρ an with an aitional capillary term) an later generalize by the same authors (see [BD04]) to inclue other ensity-epenent viscosity coefficients. In the one imensional case, a similar inequality was introuce earlier by V. A. Vaĭgant [Vaĭ90] for flows with constant viscosity (see also V. V. Shelukhin [She98]).

3 COMPESSIBLE NAVIE-STOKES EQUATION 3 The main interest of this inequality is to provie further regularity for the ensity. When µ(ρ) = ρ, for instance, it implies that the graient of ρ remains boune for all time provie it was boune at time t = 0. This has very interesting consequences for many hyroynamic equations. In [She98], V. V. Shelukhin establishes the existence of a unique weak solution for one imensional flows with constant viscosity coefficient. In higher imension, D. Bresch, B. Desjarin an C.K. Lin use this inequality to establish the stability of weak solutions for the Korteweg system of equations in [BDL03] an for the compressible Navier-Stokes equations with an aitional quaratic friction term in [BD03]. In [MV06], we establish the stability of weak solutions for the compressible isentropic Navier-Stokes equations in imension 2 an 3 (without any aitional terms). We also refer to [BD05] for recent evelopments concerning the full system of compressible Navier-Stokes equations (for heat conucting fluis). At this point, we want to stress out the fact that in imension 2 an higher, this inequality hols only when the two viscosity coefficients satisfy a relation that consierably restricts the range of amissible coefficients (an in particular implies that one must have µ(0) = 0). This necessary conition isappear in imension 1, as the two viscosity coefficients become one (the erivation of the inequality is also much simpler in one imension). Another particularity of the imension 1, is that the inequality gives control on some negative powers of the ensity (this is not true in higher imensions). This will allow us to show that vacuum cannot arise if it was not present at time t = 0. Finally, we point out that the present result is very ifferent from that of [MV06] where the ensity was allowe to vanish (an the ifficulty was to control the velocity u on the vacuum). Naturally, a result similar to that of [MV06] hols in imension one, though it is not the topic of this paper. Our main result is mae precise in the next section. Section 3 is evote to the erivation of the funamental entropy inequalities an a priori estimates. The existence part of Theorem 2.1 is prove in Section 4. The uniqueness is aresse in Section The result. Following D. Hoff in [Hof98], we work with positive initial ata having (possibly ifferent) positive limits at x = ± : We fix constant velocities u + an u an constant positive ensity ρ + > 0 an ρ > 0, an we let u(x) an ρ(x) be two smooth monotone functions satisfying an ρ(x) = ρ ± when ± x 1, ρ(x) > 0 x, (2.1) u(x) = u ± when ± x 1. (2.2) We recall that the pressure satisfies p(ρ) = ρ γ for some γ > 1 an we assume that there exists a constant ν > 0 such that the viscosity coefficient µ(ρ) satisfies µ(ρ) νρ α ρ 1 for some α [0, 1/2), µ(ρ) ν ρ 1, (2.3) an µ(ρ) C + Cp(ρ) ρ 0. (2.4)

4 4 A. MELLET AND A. VASSEU Note that (2.4) is only a restriction on the growth of µ for large ρ. amissible viscosity coefficients inclue µ(ρ) = ν an µ(ρ) = ρ 1/3. Our main theorem is the following: Theorem 2.1. Assume that the initial ata ρ 0 (x) an u 0 (x) satisfy 0 < κ 0 ρ 0 (x) κ 0 <, ρ 0 ρ H 1 (), u 0 u H 1 (), Examples of (2.5) for some constants κ 0 an κ 0. Assume also that µ(ρ) verifies (2.3) an (2.4). Then there exists a global strong solution (ρ, u) of (1.1)-(1.2)-(1.3) on + such that for every T > 0: ρ ρ L (0, T ; H 1 ()), u u L (0, T ; H 1 ()) L 2 (0, T ; H 2 ()). Moreover, for every T > 0, there exist constants κ(t ) an κ(t ) such that 0 < κ(t ) ρ(x, t) κ(t ) < (t, x) (0, T ). Finally, if µ(ρ) ν > 0 for all ρ 0, if µ is uniformly Lipschitz an if γ 2 then this solution is unique in the class of weak solutions satisfying the usual entropy inequality (3.7). Note that he assumption (2.5) on the initial ata implies, in particular that the initial entropy (or relative entropy) is finite. When the viscosity coefficient µ(ρ) satisfies µ(ρ) ν > 0 ρ 0, (2.6) the existence of a smooth solution for small time is a well-known result. More precisely, we have: Proposition 2.2 ([Sol76]). Let (ρ 0, u 0 ) satisfy (2.5) an assume that µ satisfies (2.6), then there exists T 0 > 0 epening on κ 0, κ 0, ρ 0 ρ H 1 an u 0 u H 1 such that (1.1)-(1.2)-(1.3) has a unique solution (ρ, u) on (0, T 0 ) satisfying ρ ρ L (0, T 1, H 1 ()), t ρ L 2 ((0, T 1 ) ), u u L 2 (0, T 1 ; H 2 ()), t u L 2 ((0, T 1 ) ) for all T 1 < T 0. Moreover, there exist some κ(t) > 0 an κ(t) < such that κ(t) ρ(x, t) κ(t) for all t (0, T 0 ). In view of this proposition, we see that if we introuce a truncate viscosity coefficient µ n (ρ): µ n (ρ) = max(µ(ρ), 1/n), then there exist approximate solutions (ρ n, u n ) efine for small time (0, T 0 ) (T 0 possibly epening on n). To prove Theorem 2.1, we only have to show that (ρ n, u n ) satisfies the following bouns uniformly with respect to n an T large: κ(t ) ρ n κ(t ) t [0, T ], ρ n ρ L (0, T ; H 1 ()), u n u L (0, T ; H 1 ()). In the next section, we erive the entropy inequalites that will be use to obtain the necessary bouns on ρ n an u n.

5 COMPESSIBLE NAVIE-STOKES EQUATION 5 3. Entropy inequalities. In its conservative form, (1.1)-(1.2)-(1.3) can be written as [ ] 0 t U + x [A(U)] = (µ(ρ)u x ) x with the state vector U = [ ρ ρ u ] [ ρ = m ] an the flux [ A(U) = ρ u ρ u 2 + ρ γ ] = m m 2 ρ + ργ. It is well known that H (U) = ρ u γ 1 ργ = m2 2ρ + 1 γ 1 ργ, is an entropy for the system of equations (1.1)-(1.2)-(1.3). More precisely, if (ρ, u) is a smooth solution, then we have with t H (U) + x [ F (U) µ(ρ)uux ] + µ(ρ)u 2 x = 0 (3.1) F (U) = ρu u2 2 + γ γ 1 uργ. In particular, integrating (3.1) with respect to x, we immeiately see that [ρ u2 t ] γ 1 ργ x + µ(ρ) u x 2 x 0. (3.2) However, since we are looking for solutions ρ(x, t) an u(x, t) that converges to ρ ± an u ± at ±, we o not expect the entropy to be integrable. It is thus natural to work with the relative entropy instea of the entropy. The relative entropy is efine for any functions U an Ũ by H (U Ũ) = H (U) H (Ũ) DH (Ũ)(U Ũ) = ρ(u ũ) 2 + p(ρ ρ), where p(ρ ρ) is the relative entropy associate to 1 γ ργ : p(ρ ρ) = 1 γ 1 ργ 1 γ 1 ργ γ γ 1 ργ (ρ ρ). Note that, since p is strictly convex, p(ρ ρ) is nonnegative for every ρ an p(ρ ρ) = 0 if an only if ρ = ρ.

6 6 A. MELLET AND A. VASSEU We recall that ρ(x) an u(x) are smooth functions satisfying (2.1) an (2.2), an we enote [ ] ρ U =. ρ u It is easy to check that there exists a positive constant C (epening on inf ρ) such that for every ρ an for every x, we have ρ + ρ γ C[1 + p(ρ ρ)], (3.3) lim inf ρ 0 p(ρ ρ) C. (3.4) The first inequality we will use in the proof of Theorem 2.1 is the usual relative entropy inequality for compressible Navier-Stokes equations: Lemma 3.1. Let ρ, u be a solution of (1.1)-(1.2)-(1.3) satisfying the entropy inequality t H (U) + x [F (U) µ(ρ)uu x ] + µ(ρ) u x 2 0, (3.5) Assume that the initial ata (ρ 0, u 0 ) satisfies H (U 0 U) x = [ρ 0 (u 0 u) 2 2 ] + p(ρ 0 ρ) x < +. (3.6) Then, for every T > 0, there exists a positive constant C(T ) such that ] (u T u)2 sup [ρ + p(ρ ρ) x + µ(ρ) u x 2 x t C(T ). (3.7) [0,T ] 2 0 The constant C(T ) epens only on T > 0, U, the initial value U 0, γ, an on the constant C appearing in (2.4). Note that when both ρ an ρ 0 are boune above an below away from zero, it is easy to check that p(ρ 0 ρ) C(ρ 0 ρ) 2 an thus (3.6) hols uner the assumptions of Theorem 2.1. Proof of Lemma 3.1. [Daf79]): First, we have (by a classical but teious computation, see t H (U U) = [ t H (U) + x (F (U) µ(ρ)u x u) ] t H (U) x [F (U) µ(ρ)u x u] + x [DF (U)(U U)] D 2 H (U)[ t U + x A(U)](U U) DH (U)[ t U + x A(U)] +DH (U)[ t U + x A(U)] +DH (U) x [A(U U)],

7 where the relative flux is efine by COMPESSIBLE NAVIE-STOKES EQUATION 7 A(U U) = A(U) A(U) DA(U) (U U) [ ] 0 = ρ(u u) 2. + (γ 1)p(ρ ρ) Since U is a solution of (1.1)-(1.2)-(1.3) an satisfies the entropy inequality, an using the fact that U = (ρ, ρu) satisfies (2.1) an (2.2) (an in particular t U = 0), we euce t H (U U) µ(ρ) x u 2 D 2 H (U)[ x A(U)](U U) D 2 H (U)[ x (µ(ρ) x u)] +DH (U) x [A(U U)] +DH (U)[ x A(U)] x [F (U) µ(ρ)uu x ] + x [DF (U)(U U)], where D 2 H (U) = u. We now integrate with respect to x, using the fact that supp ( x U) [, 1], an we get H (U U) x + µ(ρ) x u 2 t Writing D 2 H (U)[ x A(U)](U U) x ( x u)µ(ρ) x u x x [DH (U)]A(U U) x x [DH (U)]A(U) x. 1 ( x u)µ(ρ) x u x xu 2 L µ(ρ) x µ(ρ) x u 2 x, it follows that there exists a constant C epening on U W 1, such that H (U U) x + 1 µ(ρ) x u 2 t 2 C + C 1 U U x + C 1 A(U U) x 1 +C µ(ρ) x. (3.8) To conclue, we nee to show that the right han sie can be controlle by H (U U). First, we note that A(U U) max(1, (γ 1))H (U U),

8 8 A. MELLET AND A. VASSEU an that (3.3) an (2.4) yiel 1 Next, using (3.3) we get: 1 U U x 1 C 1 µ(ρ) x C + p(ρ ρ) x. (3.9) 1 ρ ρ x + (1 + p(ρ ρ)) x 1 ρ u u x + u(ρ ρ) x ( 1 ) 1/2 ( 1 + ρ x ρ(u u) 2 x C 1 (1 + p(ρ ρ)) x ) 1/2 ( 1 ) 1/2 ( 1 + (1 + p(ρ ρ)) x H (U U) x C 1 So (3.8) becomes H (U U) x + 1 t 2 H (U U) x + C. ) 1/2 1 µ(ρ) x u 2 C + C H (U U) x, an Gronwall s lemma gives Lemma 3.1. For further reference, we note that this also implies H (U U) x C(T ). (3.10) t Unfortunately, it is a well-known fact that the estimates provie by Lemma 3.1 are not enough to prove the stability of the solutions of (1.1)-(1.2)-(1.3). The key tool of this paper is thus the following lemma: Lemma 3.2. Assume that µ(ρ) is a C 2 function, an let (ρ, u) be a solution of (1.1)-(1.2)-(1.3) such that u u L 2 ((0, T ); H 2 ()), ρ ρ L ((0, T ); H 1 ()), 0 < m ρ M. (3.11) Then there exists C(T ) such that the following inequality hols: [ 1 2 ρ (u u) + x (ϕ(ρ)) ] 2 + p(ρ ρ) x sup [0,T ] with ϕ such that + T 0 x (ϕ(ρ)) x (ρ γ ) x t C(T ), (3.12) ϕ (ρ) = µ(ρ) ρ 2. (3.13)

9 COMPESSIBLE NAVIE-STOKES EQUATION 9 The constant C(T ) epens only on T > 0, (ρ, u), the initial value U 0, γ, an on the constant C appearing in (2.4). Since the viscosity coefficient µ(ρ) is non-negative, (3.13) implies that ϕ(ρ) is increasing. The lemma thus implies that sup [0,T ] [ 1 2 ρ (u u) + x (ϕ(ρ)) 2 + p(ρ ρ) which, together with Lemma 3.1, yiels ] x C(T ), ρ x (ϕ(ρ)) L (0,T ;L 2 (Ω)) = 2 µ(ρ)(ρ /2 ) x L (0,T ;L 2 (Ω)) C(T ). This inequality will be the corner stone of the proof of Theorem 2.1 which is etaile in the next section. As mentione in the introuction, Lemma 3.2 relies on a new mathematical entropy inequality that was first erive by Bresch an Desjarins in [BD02] an [BD04] in imension 2 an higher. Of course, the computations are much simpler in imension 1. We stress out the fact that it is important to know exactly what regularity is neee on ρ an u to establish this inequality. Inee, unlike inequality (3.7) which is quite classical, there is no obvious way to regularize the system of equations (1.1)- (1.2)-(1.3) while preserving the structure necessary to erive (3.12). Fortunately, it turns out that (3.11), which is the natural regularity for strong solutions, is enough to justify the computations, as we will see in the proof. Proof. We have to show that [ 1 t 2 ρ u u 2 + ρ(u u)(ϕ(ρ)) x + 1 ] 2 ρ(ϕ(ρ))2 x x + t p(ρ ρ) x is boune Step 1. From the proof of the previous lemma (see (3.10)), we alreay know that: [ ] 1 ρ u u 2 x + p(ρ ρ) x C(T ). (3.14) t 2 t Step 2. Next we show that: ρ (ϕ(ρ))2 x x = t 2 ρ 2 ϕ (ρ)(ϕ(ρ)) x u xx x (2ρϕ (ρ) + ρ 2 ϕ (ρ))ρ x (ϕ(ρ)) x u x x. (3.15) This follows straightforwarly from (1.1) when the secon erivatives of the ensity are boune in L 2 ((0, T ) ). In that case, it is worth mentionning that the right han sie can be rewritten as ρ 2 ϕ (ρ)(ϕ(ρ)) xx u x x. However, we o not have any bouns on ρ xx. erivation of (3.15): It is thus important to justify the

10 10 A. MELLET AND A. VASSEU First, we point out that (3.15) makes sense when (ρ, u) satisfies only (3.11) (we recall that since ρ an u are constant outie (, 1), (3.11) implies that ρ x L ((0, T ); L 2 ()) an u x L 2 ((0, T ); H 1 ())). Moreover, we note that the computation only makes use of the continuity equation. The rigorous erivation of (3.15) (uner assumption (3.11)) can thus be achieve by carefully regularizing the continuity equation. The etails are presente in the appenix (see Lemma A.1). Step 3. Next, we evaluate the erivative of the cross-prouct: ρ(u u) x (ϕ(ρ)) x t = x (ϕ(ρ)) t (ρ(u u)) x + ρ(u u) t x (ϕ(ρ)) x = x (ϕ(ρ)) t (ρ(u u)) x (ρ(u u)) x ϕ (ρ) t ρ x. (3.16) Multiplying (1.2) by x ϕ(ρ), we get: x (ϕ(ρ)) t (ρ(u u)) x = x (ϕ(ρ)) t (ρu) x = (ϕ(ρ)) x (µ(ρ)u x ) x x x (ϕ(ρ)) x (ρ γ ) x x (ϕ(ρ)) x (ρu 2 ) x + ϕ (ρ)(ρu) x ρ x u x. x (ϕ(ρ))( t ρ)u x The continuity equation easily yiels: (ρ(u u)) x ϕ (ρ) t ρ x = ((ρu) x ) 2 ϕ (ρ) x + (ρu) x (ρu) x ϕ (ρ) x. Note that those equalities hol as soon as ρ an u satisfy (3.11). Step 4. If ϕ an µ satisfy (3.13) then we have (ϕ(ρ)) x (µ(ρ)u x ) x x = ρ 2 ϕ (ρ)(ϕ(ρ)) x u xx x (2ρϕ (ρ) + ρ 2 ϕ (ρ))ρ x (ϕ(ρ)) x u x x, so (3.15) an (3.16) yiels { ρ(u u) x ϕ(ρ) + ρ xϕ(ρ) 2 } x + x ϕ(ρ) x (ρ γ ) x t 2 = x (ϕ(ρ)) x (ρu 2 ) x + ((ρu) x ) 2 ϕ (ρ) x + (ρu) x ϕ (ρ)[ρ x u (ρu) x ] x = ϕ (ρ)[ ρ x (ρu 2 ) x + ((ρu) x ) 2 ] x (ρu) x ϕ (ρ)ρu x x = ρ 2 ϕ (ρ)u 2 x x (ρu) x ϕ (ρ)ρu x x,

11 an using (3.13), we euce { t COMPESSIBLE NAVIE-STOKES EQUATION 11 ρ(u u) x ϕ(ρ) + ρ xϕ(ρ) 2 } x + x ϕ(ρ) x (ρ γ ) x 2 = µ(ρ)(u x ) 2 x µ(ρ)u x u x x ρ x (ϕ(ρ))u u x x. (3.17) Moreover, since u x has support in (, 1) an using the bouns given by Lemma 3.1 an inequality (3.9), it is reaily seen that the right han sie in this equality is boune by: C C 1 µ(ρ) u x 2 x + C µ(ρ) u x 2 x + C µ(ρ) x + C p(ρ, ρ) x + C 1 ρ x ϕ(ρ) 2 x + C ρu 2 x ρ (u u) + x ϕ(ρ) 2 x + C(T ). Putting (3.17) an (3.14) together, we euce [ 1 t 2 ρ (u u) + x (ϕ(ρ)) ] 2 + p(ρ ρ) x + x (ϕ(ρ)) x (ρ γ ) x [ ] 1 C µ(ρ) u x 2 x + C 2 ρ (u u) + xϕ(ρ) 2 + p(ρ, ρ) x + C(T ). Finally, using the bouns on the viscosity from Lemma 3.1 an Gronwall s inequality we easily euce (3.12). 4. Proof of Theorem 2.1. In this section, we prove the existence part of Theorem 2.1. The proof relies on the following proposition: Proposition 4.1. Assume that the viscosity coefficient µ satisfies (2.3)-(2.4) an consier initial ata (ρ 0, u 0 ) satisfying (2.5). Then for all T > 0, there exist some constants C(T ), κ(t ) an κ(t ) such that for any strong solution (ρ, u) of (1.1)-(1.2)- (1.3) with initial ata (ρ 0, u 0 ), efine on (0, T ) an satisfying ρ ρ L (0, T, H 1 ()), t ρ L 2 ((0, T ) ), u u L 2 (0, T ; H 2 ()), with ρ an ρ boune, the following bouns hol t u L 2 ((0, T ) ), 0 < κ(t ) ρ(t) κ(t ) t [0, T ], ρ ρ L (0,T ;H 1 ()) C(T ), u u L (0,T ;H 1 ()) C(T ). Moreover the constants C(T ), κ(t ) an κ(t ) epen on µ only through the constant C arising in (2.3) an (2.4). Proof of Theorem 2.1. We efine µ n (ρ) to be the following positive approximation of the viscosity coefficient: µ n (s) = max(µ(s), 1/n).

12 12 A. MELLET AND A. VASSEU Notice that µ n verifies µ µ n µ + 1. In particular µ n satisfies (2.3) an (2.4) with some constants that are inepenent on n. Next, for all n > 0, we let (ρ n, u n ) be the strong solution of (1.1)-(1.2)-(1.3) with µ = µ n : ρ t + (ρu) x = 0 (ρu) t + (ρu 2 ) x + p(ρ) x = (µ n (ρ) u x ) x. This solution exists at least for small time (0, T 0 ) thanks to Proposition 2.2 (note that T 0 may epen on n). Proposition 4.1 then implies that for all T > 0 there exists C(T ), κ(t ), an κ(t ) > 0, inepenent on n, such that κ(t ) ρ n (t) κ(t ) t [0, T ], ρ n ρ L (0,T ;H 1 ()) C(T ), u n u L (0,T ;H 1 ()) C(T ). In particular we can take T 0 = in Proposition 2.2 (for all n). Moreover, since the boun from below for the ensity is uniform in n for any T > 0, by taking n large enough (namely n 1/κ(T )), it is reaily seen that (ρ n, u n ) is a solution of (1.1)-(1.2)-(1.3) on [0, T ] with the non-truncate viscosity coefficient µ(ρ). From the uniqueness of the solution of Proposition 2.2, we see that, passing to the limit in n, we get the esire global solution of (1.1)-(1.2)-(1.3). The rest of this section is thus evote to the proof of Proposition 4.1. First, we will show that ρ is boune from above an from below uniformly by some positive constants. Then we will investigate the regularity of the velocity by some stanar arguments for parabolic equations A priori estimates. Since the initial atum (ρ 0, u 0 ) satisfies (2.5), we have ρ 0 (u 0 u) 2 x < an ρ 0 x (ϕ(ρ 0 )) 2 x < +. Moreover, (ρ, u) satisfies (3.11), so we can use the inequalities state in Lemmas 3.1 an 3.2. We euce the following estimates, which we shall use throughout the proof of Proposition (4.1): Ω ρ(u u) L (0,T );L 2 (Ω)) C(T ), ρ L (0,T ;L 1 loc Lγ loc (Ω)) C(T ), ρ ρ L (0,T ;L 1 (Ω)) C(T ), µ(ρ)(u) x L 2 (0,T ;L 2 (Ω)) C(T ). (4.1) an µ(ρ) x (ρ /2 ) L (0,T ;L 2 (Ω)) C(T ), µ(ρ) x (ρ γ/2/2 ) L 2 (0,T ;L 2 (Ω)) C(T ). (4.2)

13 COMPESSIBLE NAVIE-STOKES EQUATION Uniform bouns for the ensity.. The first proposition shows that no vacuum states can arise: Proposition 4.2. For every T > 0, there exists a constant κ(t ) > 0 such that ρ(x, t) κ(t ) (x, t) [0, T ]. The proof of this proposition will follow from two lemmas. First we have: Lemma 4.3. For every T > 0, There exists δ > 0 an (T ) such that for every x 0 an t 0 > 0, there exists x 1 [x 0 (T ), x 0 + (T )] with ρ(x 1, t 0 ) > δ. This nice result can be foun in [Hof98]. We give a proof of it for the sake of completeness. Proof. Let δ > 0 be such that p(ρ ρ) C 2 ρ < δ (such a δ exists thanks to (3.4)). Then, if sup ρ(x, t 0 ) < δ x [x 0,x 0 +] we have p(ρ ρ) x C an since the integral in the left han sie is boune by a constant (see Lemma 3.1), a suitable choice of leas to a contraiction. Lemma 4.4. Let w(x, t) = inf(ρ(x, t), 1) = 1 (1 ρ(x, t)) +. Then there exists ε > 0 an a constant C(T ) such that x w ε L (0,T ;L 2 ()) C(T ). Proof. We have In particular (4.2) gives so using (2.3) we euce: x w = x ρ1 {ρ 1}. µ(w) w 3/2 w x L (0,T ;L 2 (Ω)) C w α 3/2 x w L (0,T ;L 2 (Ω)) = x w α/2 L (0,T ;L 2 (Ω)) C,

14 14 A. MELLET AND A. VASSEU an the result follows with ε = 1/2 α > 0. Proof of Proposition 4.2. Together with Sobolev-Poincaré inequality, Lemma 4.3 an 4.4 yiel that w ε is boune in L ((0, T ) ): w ε (x, t) C(T ) (x, t) (0, T ). This yiels Proposition 4.2 with κ(t ) = C(T ) /ε. Next, we fin a boun for the ensity in L : Proposition 4.5. For every T > 0, there exist a constant κ(t ) such that ρ(x, t) κ(t ) (x, t) (0, T ). Proof. Let s = (γ 1)/2, then (3.12) with (2.3) an (3.13) yiels x (ρ s ) boune in L 2 ((0, T ) ). Moreover, for every compact subset K of, we have x ρ s x = ρ s x ρ x K K ( ( an so using (3.3) we get ( x ρ s x C K + Since K we euce that K K ) 1/2 ( ) 1/2 ρ 1+2s 1 x K ρ 3 ( xρ) 2 x ) 1/2 ( ρ γ x ρϕ (ρ) 2 ( x ρ) 2 x K K ) 1/2 ) 1/2 ( 1/2 p(ρ ρ) x ρϕ (ρ) 2 ( x ρ) x) 2. K ρ s 1 + ρ γ ρ s is boune in L (0, T ; W 1,1 loc ()), an the W 1,1 (K) norm of ρ s (t, ) only epens on K. Sobolev imbeing thus yiels Proposition 4.5. Proposition 4.6. There exists a constant C(T ) such that ρ(x, t) ρ(x) L (0,T ;H 1 ()) C(T ). Proof. Proposition 4.5 yiels 1 ( x ρ) 2 x κ 3 ρ 3 ( xρ) 2 x κ 3 ρ (µ(ρ)) 2 (φ (ρ)) 2 ( x ρ) 2 x νκ 3 inf (1, κ 2α ρ( x φ(ρ)) 2 x ) C(T ). An the result follows.

15 COMPESSIBLE NAVIE-STOKES EQUATION Uniform bouns for the velocity. Proposition 4.7. There exists a constant C(T ) such that an u u L 2 (0,T ;H 2 ()) C(T ) t u L 2 (0,T ;L 2 ()) C(T ). In particular, u u C 0 (0, T ; H 1 ()). Proof. First, we show that u u is boune in L 2 (0, T ; H 1 ()). Since ρ κ > 0, an using (2.3), it is reaily seen that there exists a constant ν > 0 such that an so (3.7) gives an µ(ρ(x, t)) ν (x, t) [0, T ], x u is boune in L 2 ((0, T ) ) u u is boune in L (0, T ; L 2 ()). Therefore u u is boune in L 2 (0, T ; H 1 ()). Note that this implies that t ρ is boune in L 2 ((0, T ) ). Since ρ ρ is boune in L (0, T ; H 1 ), it follows (see [Ama00] for example) that for some s 0 (0, 1). ρ C s0 ((0, T ) ) Next, we rewrite (1.2) as follows: ( ) µ(ρ) t u ρ u x = γρ γ 2 ρ x + uu x + ( x (ϕ(ρ)) (u u))u x (4.3) x where we recall that ϕ, which is efine by ϕ (ρ) = µ(ρ)/ρ 2, is the function arising in the new entropy inequality (see Lemma 3.2). In orer to euce some bouns on u, we nee to control the right han sie of (4.3). The first term, ρ γ 2 ρ x, is boune in L (0, T ; L 2 ()) (thanks to Proposition 4.6). The secon term is boune in L 2 ((0, T ) ) since u is in L. For the last part, we write (using Höler inequality an interpolation inequality): ( x (ϕ(ρ)) (u u)u x L 2 (0,T ;L 4/3 ()) x (ϕ(ρ)) (u u) L (0,T ;L 2 ()) u x L 2 (0,T ;L 4 ()) x (ϕ(ρ)) (u u) L (L 2 ) u x 2/3 L 2 (L 2 ) u x 1/3 L 2 (0,T ;W 1,4/3 ()) C u x 1/3 L 2 (0,T ;W 1,4/3 ()) (here we make use of (3.12) an Proposition 4.2). So regularity results for parabolic equation of the form (4.3) (note that the iffusion coefficient is in C s0 ((0, T ) )) yiel u x L 2 (0,T ;W 1,4/3 ()) C u x 1/3 L 2 (0,T ;W 1,4/3 ()) + C,

16 16 A. MELLET AND A. VASSEU an so u x L 2 (0,T ;W 1,4/3 ()) C. Using Sobolev inequalities, it follows that u x is boune in L 2 (0, T ; L ()). Finally, we can now see that the right han sie in (4.3) is boune in L 2 (0, T ; L 2 ()), an classical regularity results for parabolic equations give an which conclues the proof. u u is boune in L 2 (0, T ; H 2 ()) t u is boune in L 2 (0, T ; L 2 ()), It is now reaily seen that Proposition 4.1 follows from Propositions 4.2, 4.5, 4.6 an Uniqueness. In this last section, we establish the uniqueness of the global strong solution in a large class of weak solutions satisfying the usual entropy inequality. This result can be rewritten as follows: Proposition 5.1. Assume that µ(ρ) ν > 0 for all ρ 0, an that there exists a constant C such that µ(ρ) µ( ρ) C ρ ρ for all ρ, ρ 0. Assume moreover that γ 2, an let (ρ, u) be the solution of (1.1)-(1.2)-(1.3) given by Theorem 2.1. If ( ρ, ũ) is a weak solution of (1.1)-(1.2)-(1.3) with initial ata (ρ 0, u 0 ) an satisfying the entropy inequality (3.5) an relative entropy boun (3.7), an if lim ( ρ ρ ±) = 0, x ± lim (ũ u ±) = 0, x ± then ( ρ, ũ) = (ρ, u). Notice that we o not nee to assume that ρ oes not vanish. This Proposition will be a consequence of the following Lemma: Lemma 5.2. Let Ũ = ( ρ, ρũ) be a weak solution of (1.1)-(1.2)-(1.3) satisfying the inequality (3.5) an let U = (ρ, ρu) be a strong solution of (1.1)-(1.2)-(1.3) satisfying the equality (3.1). Assume moreover that Ũ an U are such that lim ( ρ ρ) = 0, lim x ± (ũ u) = 0. (5.1) x ±

17 COMPESSIBLE NAVIE-STOKES EQUATION 17 Then we have: H t (Ũ U) x + µ( ρ) [ x (ũ u)] 2 C x u H (Ũ U) x x u[µ( ρ) µ(ρ)] [ x (ũ u)] x x (µ(ρ) x u) + ( ρ ρ) (u ũ) x. (5.2) ρ The proof of this lemma relies only on the structure of the equation an not on the properties of the solutions. We postpone it to the en of this section. Proof of Proposition 5.1. In orer to prove Proposition 5.1, we have to show that the last two terms in (5.2) can be controle by the relative entropy H (Ũ U) an the viscosity. Since γ 2 an ρ κ > 0, we note that there exists C such that Then, we can write p( ρ ρ) C ρ ρ 2 for all ρ 0. x u[µ( ρ) µ(ρ)] [ x (ũ u)] x ( ) 1/2 ( C x u L () ρ ρ 2 x x (ũ u) 2 x C x u 2 L () ρ ρ 2 x + 1 µ( ρ) x (ũ u) 2 x 4 C x u 2 L () H (Ũ U) x + 1 µ( ρ) x (ũ u) 2 x 4 ) 1/2 which oes the trick for the first of the last two terms in (5.2). For the last term, we see that if we ha x (µ(ρ) x u) boune in L ((0, T ) ), a similar computation woul apply. However, writing x (µ(ρ) x u) = µ (ρ)( x ρ) ( x u) + µ(ρ) xx u it is reaily seen that x (µ(ρ) x u) is only boune in L 2 ((0, T ) ). For that reason, we nee to control ũ u in L, which is mae possible by the following Lemma: Lemma 5.3. Let ρ 0 be such that p( ρ ρ) x < +. Then there exists a constant C (epening on p( ρ ρ) x) such that for any regular function h: ( 1/2 ( 1/2 h L () C ρ h x) 2 + C h x x) 2.

18 18 A. MELLET AND A. VASSEU Using Lemma 5.3 with h = ũ u, we euce: x (µ(ρ) x u) ( ρ ρ) (u ũ) x ρ x (µ(ρ) x u) ρ ρ ρ L 2 () u ũ L () L 2 () ( C x (µ(ρ) x u) ρ H (Ũ U) 1 2 (H (Ũ U)12 + L 2 () 2 C x (µ(ρ) x u) ρ H (Ũ U) + 1 µ( ρ) x (u ũ) 2 x. L 2 () 4 x (u ũ) 2 x ) 1 ) 2 So (5.2) becomes H t (Ũ U) x µ( ρ) [ x (ũ u)] 2 C(t) H (Ũ U) x where C(t) L 1 (0, T ). Gronwall Lemma, together with the fact that yiels Proposition 5.1. H (Ũ U)(t = 0) = 0 Proof of Lemma 5.3. Using (3.4), we see that there exists some δ > 0 an C such that {x ; ρ δ} C p( ρ ρ) x. We take = C p( ρ ρ) x + 1. Then, for every x 0 in, we know that in the interval (x 0 /2, x 0 + /2), ρ is larger than δ is a set of measure at least 1 we enote by ω this set: ω = (x 0 /2, x 0 + /2) { ρ δ}. Then, for all x ω, we have x ( 1/2 h(x 0 ) h(x) + h x (y) y h(x) + 1/2 h x (y) y) 2. x 0 Integrating with respect to x in ω, we euce: h(x 0 ) 1 ω ω 1 ω 1/2 h x + 1/2 ( ( h 2 x ω ) 1/2 h x 2 x ) 1/2 + 1/2 ( 1/2 h x x) 2. Finally, since ρ δ in ω, we have ( ) 1/2 ( ) 1/2 1 h(x 0 ) ρ h 2 x + 1/2 h δ 1/2 ω 1/2 x 2 x ω

19 an since ω 1, the result follows. COMPESSIBLE NAVIE-STOKES EQUATION 19 Proof of Lemma 5.2. To prove the lemma, it is convenient to note that the system ( )-(1.3) can be rewritten in the form t U i + x A i (U) = x [ Bij (U) x (D j H (U)) ], where B(U) is a positive symmetric matrix an DH enotes the erivative (with respect to U) of the entropy H (U) associate with the flux A(U). The existence of such an entropy is equivalent to the existence of an entropy flux function F such that D j F (U) = i D i H (U)D j A i (U) (5.3) for all U. Then strong solutions of (1.1)-(1.2)-(1.3) satisfy th (U) + x F (U) x (B(U) x DH (U))DH (U) = 0. In our case, we have A(U) = m m 2 ρ + ργ [ = ρu ρu 2 + ρ γ ] an [ 0 0 B(U) = µ(ρ) 0 1 ] Then, a carefull computation (using (5.3)) yiels t H (Ũ U) = = [ t H (Ũ) + x(f (Ũ)) x(b(ũ) xdh (Ũ))DH (Ũ)] [ t H (U) + x F (U) x (B(U) x DH (U))DH (U) ] x [F (Ũ) F (U)] D 2 H (U)[ t U + x A(U) x (B(U) x (DH (U)))](Ũ ( )) U) DH (U)[ t Ũ + x A(Ũ) x B(Ũ) x (DH (Ũ) ] +DH (U)[ t U + x A(U) x (B(U) x (DH (U)))] + x [DF (U)(Ũ U)] +DH [B(Ũ) (U) x [A(Ũ U)] ] ] + x xdh (Ũ) B(U) xdh (U) [DH (Ũ) DH (U) + x (B(U) x DH (U))DH (Ũ U), where the relative flux is efine by A(Ũ U) = A(Ũ) A(U) DA(U) (Ũ U)

20 20 A. MELLET AND A. VASSEU Using the fact that Ũ an U are solutions satisfying the natural entropy inequality an equality, we euce t H (Ũ U) x [F (Ũ) F (U)] + x[df (U)(Ũ U)] +DH [B(Ũ) (U) x [A(Ũ U)] ] ] + x xdh (Ũ) B(U) xdh (U) [DH (Ũ) DH (U) + x (B(U) x DH (U))DH (Ũ U), Integrating with respect to x an using (5.1), we euce H (Ũ U) x t x [DH (U)]A(Ũ U) x [ ] ] B(Ũ) xdh (Ũ) B(U) xdh (U) x [DH (Ũ) DH (U) x + x [B(U) x DH (U)]DH (Ũ U) x. Finally, we check that x [DH (U)]A(Ũ U) = ( xu)[ρ(u ũ) 2 + (γ 1)p(ρ ρ)], x [B(U) x DH (U)]DH (Ũ U) = x(µ(ρ) x u) ρ ( ρ ρ) (u ũ), an [ ] ] B(Ũ) xdh (Ũ) B(U) xdh (U) x [DH (Ũ) DH (U) = [µ( ρ) x ũ µ(ρ) x u] x [ũ u] = µ( ρ) [ x ũ x u] 2 + x u[µ( ρ) µ(ρ)] x [ũ u]. It follows that which gives the Lemma. H t (Ũ U) x + µ( ρ) [ x (ũ u)] 2 C x u H (Ũ U) x x u[µ( ρ) µ(ρ)] [ x (ũ u)] x x (µ(ρ) x u) + ( ρ ρ) (u ũ) x. ρ

21 COMPESSIBLE NAVIE-STOKES EQUATION 21 Appenix A. Proof of equality Lemma A.1. Let (ρ, u) satisfy (3.11) an { t ρ + x (ρ u) = 0 ρ(x, 0) = ρ 0 (x). (A.1) Then (ρ, u) satisfies (3.15). Proof. We enote by h ε the convolution of any function h by a mollifier. Convoluting (A.1) by the mollifier, we get where t ρ ε + x (ρ ε u) = r ε r ε = x (ρ ε u) x (ρu) ε. Since ρ ε is now a smooth function, a straightforwar computation yiels: (ϕ(ρ ε ) x ) 2 ρ ε x t 2 (ρ ε ) 2 ϕ (ρ ε )ϕ(ρ ε ) x u xx x (2ρ ε ϕ (ρ ε ) + (ρ ε ) 2 ϕ (ρ ε ))(ρ ε ) x ϕ(ρ ε ) x u x x = ρ ε ϕ(ρ ε ) x (ϕ (ρ ε )r ε ) x x. (A.2) In orer to pass to the limit ε 0, we note that ρ ε ρ ρ ρ in L (0, T ; H 1 ()) strong, which is enough to take the limit in the left han sie of (A.2) (note that it implies the strong convergence in L (0, T ; L ())). To show that the right han sie goes to zero, we only nee to show that r ε goes to zero in L 2 (0, T ; H 1 ()) strong (an thus in L 2 (0, T ; L ())). We write x r ε = 2[ x ρ ε x u ( x ρ x u) ε ] +ρ ε xx u (ρ xx u) ε + xx ρ ε u ( xx ρ u) ε. The first two terms converge to zero thanks to the strong convergence of ρ ε L (0, T ; H 1 ()). For the last term, we note that in x ρ L (0, T ; L 2 ()) an u L 2 (0, T ; W 1, ()) so the strong convergence to zero in L 2 ((0, T ) ) follows from Lemma II.1 in DiPerna- Lions [DL89].

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