Didier Bresch 1, Pascal Noble 2 and Jean-Paul Vila Introduction

Transcription

1 ESAIM: PROCEEDINGS AND SURVEYS, Decembe 17, Vol. 58, p Stéphane DELLACHERIE, Gloia FACCANONI, Béénice GREC, Fédéic LAGOUTIERE, Yohan PENEL RELATIVE ENTROPY FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY DEPENDENT VISCOSITIES AND VARIOUS APPLICATIONS Didie Besch 1, Pascal Noble and Jean-Paul Vila 3 Abstact. This pape povides the full poof of the esults announced by the authos in C. R. Acad. Sciences (16. We intoduce an oiginal elative entopy fo compessible Navie-Stokes equations with density dependent viscosities and discuss some possible applications such as inviscid limit o low Mach numbe limit. We fist conside the case µ( = µ and λ( = and a pessue law unde the fom p( = a γ with γ > 1, which coesponds in paticula to the fomulation of the viscous shallow wate equations. We pesent some mathematical esults elated to the weak-stong uniqueness, the convegence to a dissipative solution of compessible o incompessible Eule equations. Moeove, we show the convegence of the viscous shallow wate equations to the inviscid shallow wate equations in the vanishing viscosity limit and futhe pove convegence to the incompessible Eule system in the low Mach limit. This extends esults with constant viscosities ecently initiated by E. Feieisl, B.J. Jin and A. Novotny in J. Math. Fluid Mech. (1. 1. Intoduction Since the pioneeing woks of C. Dafemos 8 and of H.-T. Yau 3, elative entopy methods have become a cucial and widely used tool in the study of asymptotic limits and long-time behavio fo nonlinea PDEs. In a ecent pape E. Feieisl, B.J. Jin, A. Novotny (see 1 have intoduced elative entopies, suitable weak solutions and weak-stong uniqueness fo the compessible Navie-Stokes equations with constant viscosities. The inteested eade is efeed to 18, 9 and efeences cited theein. Based on such elative entopies, vaious papes have been dedicated to singula petubations, see the inteesting book 11, the aticles 1, fo instance. See also the ecent inteesting wok by Th. Gallouët, R. Hebin, D. Maltese, A. Novotny in 1 whee elative entopy technics ae developed to obtain eo estimates fo a numeical appoximation of the compessible Navie Stokes equation with constant viscosities. Hee we focus on the extension of the esults by 1, 1 and to the compessible Navie-Stokes equations with degeneate viscosities depending on the density. This extension is not staightfowad: elative entopy fo the one-dimensional compessible Navie-Stokes equations with degeneate density dependent viscosity has been, fo instance, ecently studied by B. Haspot in 14 unde the athe estictive assumption that the viscosity function µ( equal to the pessue law p( up to a multiplicative constant. The main objective is to get id of this hypothesis and to extend the esult to the multi-dimensional in space case. Fo that 1 Laboatoie de Mathématiques UMR517 CNRS Bâtiment le Chablais Univesité de Savoie Mont-Blanc Le Bouget du lac, Didie.Besch@univ-savoie.f Institut de Mathématiques de Toulouse, UMR519, Univesité de Toulouse, CNRS, INSA, F-3177 Toulouse, Fance, e- mail: Pascal.Noble@math.univ-toulouse.f 3 Institut de Mathématiques de Toulouse, UMR519, Univesité de Toulouse, CNRS, INSA, F-3177 Toulouse, Fance, e- mail: vila@insa-toulouse.f c EDP Sciences, SMAI 17 Aticle published online by EDP Sciences and available at o

2 ESAIM: PROCEEDINGS AND SURVEYS 41 pupose, we will take advantage of the -entopy intoduced ecently by the fist autho, B. Desjadins and E. Zatoska in 5. We intoduce a new elative entopy based on a monotonicity popety which allows to elax the elation between the viscosity and the pessue asked in 14. In paticula, we ae able to handle the case µ( = µ and a geneal pessue p( stictly monotone like p( = a γ with γ > 1. This coesponds to the compessible Navie Stokes system consideed by A. Vasseu and C. Yu in. By a combination of estimates based on the so called B-D (Besch-Desjadin entopy 4, Mellet-Vasseu estimates 17 and some oiginal enomalization techniques, they ecently obtained the fist existence esult of global weak solutions without additional egulaizing/damping tems (fiction, suface tension. The eades ae also efeed to an othe appoach by J. Li and Z. Xin in 15. Note that such elative entopy will be used to design appopiate schemes fo the compessible Navie Stokes equation with degeneate viscosities in 6 and pove convegence. As an application of ou elative entopy estimate, we pesent some mathematical esults elated to the weakstong uniqueness, the convegence to a dissipative solution of compessible o incompessible Eule equations. In paticula, this mathematically justify the convegence of the solutions of the viscous shallow-wate system to the solutions of inviscid shallow-wate system o to the incompessible Eule system in the vanishing viscosity and/o low Mach limit. In contast to constant viscosities, we pove an exponential ate of decay fo density dependent viscosities. Finally, we discuss moe geneal viscosities µ( and λ( assuming an algebaic elation on the coefficients that was intoduced by the fist autho and B. Desjadins in 3. Fo the eade convenience, let us ecall the compessible Navie-Stokes equation with density dependent viscosities: t div (u =, (1 t (u div(u u p(ρ div(µ(d(u (λ(divu =. In this pape, we assume an additional algebaic elation intoduced by the fist autho and B. Desjadins in 3: λ( = (µ ( µ(. It has been obseved in 5 that this system may be efomulated though an augmented system. Intoducing the intemediate velocity u ϕ( with ϕ (s = µ (s/s, a dift velocity (1 ϕ( and a mixtue coefficient, the augmented vesions ae: i Case µ(ρ = µρ, λ(ρ =. The augmented system eads t div ( (v µ log =, t (v div(v (v µ log p(ρ = µdiv((1 D(v µdiv(a(v ( µdiv (1 w, ( t (w div(w (v log = µdiv( w µdiv( (1 v T. with v = u µ log and w = (1 µ log. The associated enegy estimate (named -entopy eads fo all t, T : sup τ,t µ ( v w (τ dx F ((τ dx ( A(v D( 1 v ( w µ p ( ( v dx dx w ( dx F (( dx (3

3 4 ESAIM: PROCEEDINGS AND SURVEYS ii Case µ(ρ and λ(ρ = (µ ( µ(. By intoducing ϕ such that ϕ (s = µ (s/s, one finds t div ( (v ϕ =, t (v div(v (v ϕ p(ρ = div(µ(ρ(1 D(v div(µ(ρa(v div t (w div(w (v ϕ(ρ ( (1 µ(ρ w ((λ( (µ ( µ(divu, = div(µ(ρ w div( (1 µ(ρ v T ((µ ( µ(divu (4 with v = u ϕ( and w = (1 ϕ(. The associated enegy estimate (named -entopy eads fo all t, T sup τ,t ( v w (τ dx µ( A(v dx ds F ((τ dx µ (p ( dx ds µ( D( 1 v ( w dx (µ ( µ( div ( 1 v ( w dx ds ( v w ( dx F (( dx The pape will be divided in six sections. In the fist section, we establish the elative entopy fo system, then we pove a weak-stong uniqueness esult. The thid section coesponds to the convegence of the global weak solution sequences of the viscous compessible Navie-Stokes system to dissipative solution of the compessible Eule system. The fouth section concens the convegence to the incompessible Eule equations. In the last section we discuss possible genealization to moe geneal viscosities satisfying the algebaic elation intoduced by the fist autho and B. Desjadins. (5. Definition of -entopy solution. In what follows, we set R d an open subset o = T d a peiodic box. Let us ecall hee the definition of -entopy solution fo the compessible Navie Stokes equation with degeneate viscosities as intoduced ecently in 5 Definition 1. Let T > and be such that < < 1, the couple of functions (, u is called a global entopy solution to compessible Navie Stokes system with degeneate viscosities if the following popeties ae satisfied: The mass equation is satisfied in the following sense fo all ξ Cc (, T. T T t ξ dxdτ u ξ dxdτ = ξ( dx (6

4 ESAIM: PROCEEDINGS AND SURVEYS 43 The momentum equation is satisfied in the following sense T T T u t φ dxdτ λ(div u div φ dxdτ T (u u : φ dxdτ T p(div φ dxdτ = µ(d(u : φ dxdτ u φ( dx (7 fo all φ (C c (, T 3. Moeove (, u satisfies, fo all t, T, the following -entopy estimates ( u ϕ( sup t,t T µ( A(u dx ds (1 T (1 ϕ( T µ (p ( dx ds (t dx e((t dx µ( D(u dx (µ ( µ( div u dx ds ( u ϕ( (1 ϕ( ( dx e( dx (8 with ϕ (s = µ (s/s, A(u = 1 ( u t u and the intenal enegy e( defined by de( d = p(. 3. Relative entopy. In this section, we assume µ(ρ = µ ρ and we deive a elative entopy between a weak solution (ρ, v, w of the augmented system ( and any othe state (, V, W of the fluid. Let us conside the elative enegy functional, denoted E(ρ, v, w, V, W, defined by E(ρ, v, w 1, V, W = ( w W v V (F ( F ( F (( which measues the distance between a -entopic weak solution (, v, w to any smooth enough test function (, V, W. We can pove that any weak solution (ρ, v, w of the augmented system satisfies the following so-called

5 44 ESAIM: PROCEEDINGS AND SURVEYS elative entopy inequality E(ρ, v, w, V, W (τ E(ρ, v, w, V, W ( τ τ µ A(v V µ D( (1 (v V (w W τ µ p ( log p ( log log log τ ( ((v w W (W w ((v (1 τ τ τ τ ( t W (W w t V (V v τ t F (( (p( p( div(v F ( (v (1 W (1 (1 w V (V v w (V (1 W p ( µ 1 W (9 (1 τ ( µ D ( (1 V ( W : (D ( (1 (V v ( (W w τ τ µ A(V : A(V v µ p ( ( τ (1 µ A(W : A(v V A(w W : A(V fo all τ, T and fo any pai of test functions C 1 (, T, >, V, W C 1 (, T. Note that the ight hand-side is well defined using the global weak egulaity of (, v, w. This is the analogue fo the density dependent Navie-Stokes equations of what has been poven fo the constant viscosity baotopic Navie-Stokes equations in 1. Note also the pesence of the additional new tem τ p ( log p ( log log log which has a pioi no sign but that we will ewite late in an appopiate manne. This is indeed the cone-stone of ou study. The intoduction of this tem allows to elax the estictive assumptions between the viscosity µ and the pessue law p made in 14. Moe pecisely, we pove the following esult: Theoem 1. Let be a peiodic box. Suppose that the pessue law satisfies p( = a γ with γ > 1. Let (, u be a finite -entopy solution to the compessible Navie Stokes system with degeneate viscosity µ( = µ and λ( = in the sense of definition 1. Then (, u satisfy the elative entopy inequality (9 fo any C 1 (, T with > and V, W C 1 (, T. We explain afte the poof how to elax by a density agument the equied egulaity fo the test functions. Remak that existence of -entopic solution fo the compessible Navie-Stokes equations with µ( = µ with µ constant and λ( = without exta tems (capillay, dag, singula pessue has been ecently poved in the nice pape.

6 ESAIM: PROCEEDINGS AND SURVEYS 45 Poof of the elative entopy estimate. Let us wite E(, v, w, V, W (τ E(, v, w, V, W ( 1 = ( w v F ( (τ 1 1 ( V W (F ( F (( 1 ( V W (F ( F (( ( w v F ( ( ( w W v V (τ ( w W v V ( (1 We also calculate that τ µ µ = µ µ µ µ τ τ τ τ µ µ τ A(v V µ τ p ( log p ( log τ τ A(v µ τ D( (1 (v V (w W log log D( (1 v ( w τ p ( (11 A(V : A(V v (1 ( D( (1 V ( W : ( D( (1 (V v ( (W w p ( ( A(v : A(V µ p (. τ ( D( (1 v ( w : ( D( (1 V ( W Recall the enegy estimate satisfied by (, v, w 1 µ ( w v τ F ( (τ 1 ( w v τ A(v µ D( (1 v ( w F ( ( τ p ( (13 Let us now test the equations satisfied by v and w espectively by V and W. This computation is mathematically justified due to the egulaity assumptions on V and W. Note that we can wite the weak fomulation satisfied by w = (1 log ρ playing with the egulaity satisfied by the entopy solutions and with

7 46 ESAIM: PROCEEDINGS AND SURVEYS the weak fomulation fo the mass equation. We get the following elation (v V w W (τ τ µ τ τ τ ((v (1 A(v : A(V µ p(div(v τ (v V w W ( w V v ((v τ 1 W (1 µ A(v : A(W A(w : A(V = τ v t V w t W (1 w W w ( D( (1 v ( w : ( D( (1 V ( W τ p ( W 1 (14 Let us now test the mass equation by V / and by W / and add the two, we get the identity 1 ( V W (τ 1 ( V W ( (15 τ τ = (V t V W t W ( ((v w V V ((v (1 (1 w W W. Remak now that t F ( F ( = t F ( and thus F ( F ((τ F ( F (( = t F ( (16 Let us now multiply the mass equation by F ( and integate in space and time τ τ (F ((τ (F (( = t F ( (v (1 w F ( (17 Recalling that F ( F ( = p(, we obseve that = = div ( (F ( F ((V p( div(v Using (1 (18, we get the desied elative entopy. (1 W (V (1 W (18 (1 W F ( 3.1. Relaxation of the egulaity on the test function (, V, W : Let us ecall the egulaity on the global weak-solution of the degeneate compessible Navie-Stokes equations with linea degeneate viscosity µ( = µ and λ( = and assuming the pessue law to be p( = a γ : w L (, T ; L (, u L (, T ; L (, v L (, T : L (, F ( L (, T ; L 1 (, p ( L (, T ; L (

8 ESAIM: PROCEEDINGS AND SURVEYS 47 with the elation F ( F ( = p( assuming that initially F ( L 1 (, m L (, M L (, whee m = w and M = v if and if =. Moeove we have the following estimates on the velocity due to the Mellet-Vasseu estimate ln(1 u (1 u L (, T ; L ( if initially it is the case. We need to impose at least the weak egulaity on the taget (, V, W and < c < to define the left-hand side namely U L (, T ; L (, F ( L (, T ; L 1 (, W L (, T ; L (, V L (, T : L p ( (, L (, T ; L (. Concening the ight-hand side to be well defined, we need the exta egulaity t F ( L 1 (, T ; L 3/ ( L γ/(γ1 (, F ( L 1 (, T ; L 3 ( L γ/(γ1 ( V, W L 1 (, T ; L ( L (, T ; L 3 (. 4. Weak-Stong Uniqueness. Let us conside a entopy solution (, u of ( and define the modified velocities v = u µ log and w = µ (1 log. Moeove, we assume that (, W, V satisfies the augmented system togethe with the egulaity assumptions of the pevious section. We futhe suppose that W = µ (1 log. Then let us pove that (, v, w = (, V, W that means weak-stong uniqueness popety. This gives (, u = (, U with U = V W/ (1. Moe pecisely let us pove the following esult Theoem. Let be a peiodic box. Suppose that the pessue law satisfies p( = a γ namely the powe law pessue. Let (, u be a -entopy solution to the compessible Navie Stokes system in the sense of the definition 1. Assume that thee exists a stong solution of the compessible Navie-Stokes equations satisfying the positivity and egulaity popeties descibed peviously in (3.1 and that log( L (, T ; W 1, ( L 1 (, T ; W, (. Then we have the weak-stong uniqueness esult: (, u = (, U. Remak that we cannot test the equations satisfied by V and W by V v and W w because v and w ae not contolled close to vacuum but v and w ae well defined. Thus, let us take the equations satisfied by

9 48 ESAIM: PROCEEDINGS AND SURVEYS V and W and test them espectively against (W w/ and (V v/, we get τ τ t V (V v ( ((V τ τ τ τ τ (1 W V (V v µ(1 D(V : D(V v τ µ (1 W : D(V v τ µa(v : A(V v µ (1 W : A(V v p( (V v (19 µ(1 D(V : ( τ (V v µa(v : ( (V v µ (1 W : ( (V v = ( and τ τ τ τ τ t W (W w µ (W : (W w µ (1 D(V : (W w µ (W : ( τ (W w ( ((V (1 W W (W w τ µ (1 A(V : (W w µ (1 ( V t : ( (W w =. Collecting the two pevious equality, we get τ τ τ t V (V v t W (W w µ D( (1 V ( W : D( (1 (V v ( (W w τ µ A(V : A(V v τ (1 µ A(W : A(v V A(w W : A(V τ ( τ ( = ((V (1 W V (V v ((V (1 W W (W w τ τ p( (V v µ(1 D(V : ( (V v τ µ(1 A(V : ( τ (V v µ(1 (W : ( V v τ µ (W : ( τ (W w µ (1 ( V t : ( (W w

10 ESAIM: PROCEEDINGS AND SURVEYS 49 Now we can use this esult to efomulate (9. We fist emak that F ( = p (/ and theefoe we can wite τ = τ p( (V v F ( (V v Thus, using the following identity (1 (W w τ p ( (1 (W w p ( µ 1 W µ (1 p ( p ( (W w = (1 which is tue if W = µ (1 log and w = µ (1 log, we get Note now that E(ρ, v, w, V, W (τ E(ρ, v, w, V, W ( τ τ µ A(v V µ D( (1 (v V (w W τ p ( p ( τ ( ((v w (V (1 (1 W V (V v τ ( ((v w (V (1 (1 W W (W w τ ( t F ( (V (1 W F ( ( τ (p( p( div(v (1 W (1 τ µ(1 D(V : ( (V v τ µ(1 A(V : ( τ (V v µ(1 (W : ( (V v τ µ (W : ( τ (W w µ (1 ( V t : ( (W w ( t F ( (V (1 W F ( ( = div(v (1 W ( p ( because F ( = p ( and ( t div (V (1 W =.

11 5 ESAIM: PROCEEDINGS AND SURVEYS Thus we conclude that E(ρ, v, w, V, W (τ E(ρ, v, w, V, W ( τ τ µ A(v V µ D( (1 (v V (w W τ p ( log p ( log log log τ ( ((v w (V (1 (1 W V (V v τ ( ((v w (V (1 (1 W W (W w τ ( p( p( p (( div(v (1 W ( τ µ(1 D(V : ( (V v τ µ(1 A(V : ( τ (V v µ(1 (W : ( (V v τ µ (W : ( τ (W w µ (1 ( V t : ( (W w. Note that ( ((v w (V (1 (1 W V (V v ( ((v w (V (1 (1 W W (W w ( p( p( p (( div(v (1 W c ( V L ( W L ( E(, v, w, V, W (3 Finally ecalling that w = (1 /, we emak that ( = log log = 1 (1 (w W Then we wite µ(1 D(V : ( (V v µ(1 A(V : ( (V v µ (W : ( (W w µ(1 (W : ( (V v µ (1 ( V t : ( (W w c( V L ( W L (E(, v, w, V, W (4

12 ESAIM: PROCEEDINGS AND SURVEYS 51 Let us now study the tem in the left hand-side I 1 = τ which has a pioi no sign. We can wite it as I = τ p ( log p ( log ( log log p ( τ log log (p ( p ( log ( log log = I 1 I The fist tem I 1 is positive, thee it emains to bound the second tem I. Let us ecall that we conside the case p( = a γ. Then, we have to conside the quantity which may be witten I = τ I = τ γ1 log γ1 log ( log log τ γ log log γ γ1 log log log = I 1 I The fist quantity I 1 is positive. It emains to contol I. It is impotant now to obseve the following identity Let us notice that p ( p ( log log log = p( p( p (( log (5 (p ( p ( p (( log Thus I 1 γ τ J 1 J. (p( p( p (( log (6 τ ((p ( p ( p (( log The key-point is now to ecall that p( p( p (( F ( whee F ( = F ( F ( F (( by definition and that it is possible to pove (see Lemma.. 1 that Thus I C τ (p ( p ( p (( F ( ( log L ( log L ( E(, v, w, V, W. This ends the poof of weak-stong uniqueness using Gonwall Lemma asking log to be bounded in L (, T ; W 1, ( L 1 (, T ; W, (.

13 5 ESAIM: PROCEEDINGS AND SURVEYS 5. Convegence to a dissipative solution of compessible Eule equations. Let us ecall the definition of a dissipative solution of compessible Eule equations. Such concept has been intoduced by P. L. Lions in the incompessible setting: see fo instance 16. The eade is efeed to 9,, 1 fo the extension to the compessible famewok. Definition. The pai (, u is a dissipative solution of the compessible Eule equations if and only if (, u satisfies the elative enegy inequality E(, u,, U, (t E(, u,, U, ( exp c ( divu(τ L (dτ exp c ( s divu(τ L ( E(, U (U u dxds fo all smooth test functions (, U defined on, T so that is bounded above and below away fom zeo and (, U solves t div(u =, t U U U F ( = E(, U fo some esidual E(, U. We pove the following esult. Theoem 3. Let ( ε, u ε be any finite -entopy solution to the viscous compessible Navie-Stokes equations in the peiodic setting namely µ( = ε, λ( = and p( = a γ. Then, any weak limit (, u of ( ε, u ε, when ε, in the sense ε weakly in L (, T ; L γ (, ε v ε u weakly in L (, T ; L ( ε w ε weakly in L (, T ; L ( with v ε = u ε ε log ε and w ε = ε (1 log ε as ε tends to zeo, is a dissipative solution to the compessible compessible Eule equations.

14 ESAIM: PROCEEDINGS AND SURVEYS 53 Poof. Let us conside the entopy solution ( ε, u ε of system ( with µ( = ε and λ( =. It has been shown that it satisfies the so called elative entopy inequality E( ε, v ε, w ε, V, W (τ E(ε, v ε, w ε, V, W ( τ τ ε ε A(v ε V ε ε D( (1 (v ε V (w ε W τ ε ε p ( ε log ε p ( log log ε log τ ε (((v ε (1 w ε W (W w ε ((v ε (1 w V (V v ε τ ε ( t W (W w ε t V (V v ε τ τ τ τ t F (( ε τ (p( p( ε div(v F ( ε (v ε (1 W (1 w ε (V (1 W p ( ε ε ε 1 W (7 (1 (D ( (1 (V v ε ( (W w ε ε ε (D ( (1 V ( W : τ τ ε ε ε A(V : A(V v ε ε p ( ( ε ε τ (1 ε ε A(W : A(v ε V A(w ε W : A(V Let us now conside (, U as in the definition of dissipative solution fo compessible Eule equations and define (, V ε, W ε though the elations W ε = ε (1 / and V ε = U ε /. Theefoe t div(u =, t V ε U V ε F ( = E(, U ε div(( Ut = E ε 1(, U (8 t W ε U W ε = (1 ε div(( U t = E ε (, U Obseve now that p( ε p( = (F ( ε F ( F ( ε whee F ( ε = F ( ε F ( F (( ε.

15 54 ESAIM: PROCEEDINGS AND SURVEYS Then afte some calculations we can pove that the elative entopy may be witten as E( ε, v ε, w ε, Vε, W ε (τ E( ε, v ε, w ε, Vε, W ε ( τ τ ε ε A(v ε V ε ε ε D( (1 (v ε V ε (w ε W ε τ ε ε p ( ε log ε p ( log log ε log τ ε (E1(, ε U (V ε v ε E(, ε U (W ε w ε τ ( ε (F ( ε F ( ( ε F ( F ( ε divu (9 τ ε ε (D ( (1 V ε ( W ε : (D ( (1 (V ε v ε ( (W ε w ε τ ε ε A(V ε : A(V ε v ε Now we use as in 1 that thus afte some Cauchy-Schwaz inequalities, we get ε ( F ( ε F ( ( ε F ( F ( ε E(ρ ε, v ε, w ε, V ε, W ε (τ E(ρ ε, v ε, w ε, V ε, W ε ( τ τ ε ε A(v ε V ε ε ε D( (1 (v ε V ε (w ε W ε τ ε ε p ( ε log ε p ( log log ε log τ ε (E1(, ε U (V ε v ε E(, ε U (W ε w ε τ c( divu L (F ( ε τ ε ε (D ( (1 V ε ( W ε : (D ( (1 (V ε v ε ( (W ε w ε τ ε ε A(V ε : A(V ε v ε (3. It emains now to deals with the pessue tem in the left-hand side I = τ ε p ( ε log ε p ( log log ε log which as a-pioi no sign. We can ecall what as been done in the weak-stong uniqueness pat namely I = τ ε p ( ε τ log ε log ε (p ( ε p ( log ( log ε log = I 1 I The fist tem I 1 is positive, thee it emains to bound the second tem I. Eveything wok as fo the weakstong uniqueness fo p( = a γ and µ( = µ, λ( = to contol the tem linked to the pessue namely I

16 ESAIM: PROCEEDINGS AND SURVEYS 55. Moe pecisely we can pove that E( ε, v ε, w ε, V ε, W ε (τ E( ε, v ε, w ε, Vε, W ε ( ( exp c( divu(τ L ( ε( log (τ L ( log (τ L ( exp ( ( c( divu(τ L ( ε( log (τ L ( log (τ L ( dτ s ( ε( E1(, ε U (V ε v ε E(, ε U (W ε w ε C ε ε U (31 This ends the poof letting ε tend to zeo. 6. Convegence to the stong solution of incompessible Eule equations in the well-pepaed case. In this pat, we want to pove that ou augmented fomulation is well adapted to the low Mach numbe and inviscid limit togethe. This will help in a fothcoming pape, see 6, to justify the intuitive scheme fo incompessible Eule System aleady defined in 13: Remak that may play the ole of h in 13 which is the mesh size. We will choose as pessue state p(ρ = aρ γ. This extends to the density dependent case, the method and esult obtained by E. Feieisl et. al. in the constant viscosity case, see 9 and 18 and efeences cited theein. It would emain to conside the ill pepaed case to end up the complete asymptotic analysis but it is not the objective of this poceeding. We also assume that the initial density convege to a constant taken as 1 namely (ρ ε 1/ε in L (, the initial velocity conveging to an incompessible velocity U in L (. We stat with the augmented system t div ( (v µ log =, t (v div(v (v µ log 1 p(ρ = div(µρ(1 D(v div(µρa(v ε ( div (1 µρ w, (3 t (w div(w (v µ log = div(µρ w div( (1 µρ v T. We want to pass to the limit with espect to ε and µ fo some elation between the two. The entopy eads sup τ,t µ ( v w (τ dx F ((τ dx ( A(v D( 1 v ( w µ ε p ( ( v dx dx w ( dx F (( dx Let us choose = 1, V = U the stong solution of the incompessible Eule Equation t U U U Π =, div U = (33

17 56 ESAIM: PROCEEDINGS AND SURVEYS and W =. Recalling that E(ρ, v, w 1 1, U, = ( w v U a ε ( γ 1 γ( 1 (γ 1 the elative entopy eads E(ρ, v, w 1, U, (τ E(ρ, v, w 1, U, ( τ τ µ A(v U µ D( (1 (v U w µ τ ε p ( log τ ( (((v U (1 w U (U v τ ( t U (U v τ ( µ D ( (1 U : (D ( (1 (U v ( w τ µ A(U : A(U v. Playing with the egulaity of (U, Π solution of the incompessible Eule equations, we can get as in the othe pats a diffeential inequality allowing to conclude of the convegence when ε. Note that if we choose µ = O(ε, we get though the pessue tem a convegence in L (, T ; H 1 ( nom of ρp (ρ ρ ε to 1. Refeences 1 C. Bados, T. Nguyen. Remaks on the inviscid limit fo the compessible flows. Contempoay Mathematics, AMS, To appea (16. S. Benzoni-Gavage, R. Danchin, S. Descombes On the well-posedness fo the Eule-Koteweg model in seveal space dimensions Indiana Univesity Mathematics Jounal, 56, no 4, (7, D. Besch, B. Desjadins. Quelques modèles diffusifs capillaies de type Koteweg, C. R. Acad. Sci. Pais, section mécanique, 33, no. 11, (4, D. Besch, B. Desjadins, C.K. Lin. On some compessible fluid models: Koteweg, lubication and shallow wate systems. Commun. Pat. Diff. Eqs., Vol. 8, (3, D. Besch, B. Desjadins, E. Zatoska. Two-velocity hydodynamics in fluid mechanics: Pat II. Existence of global - entopy solutions to compessible Navie-Stokes systems with degeneate viscosities. J. Math. Pues Appl., 14, Issue 4, (15, D. Besch, P. Noble, J. P. Vila, P. Villedieu. Numeical Schemes fo some extended fomulations of compessible Navie- Stokes equations. In pepaation (15. 7 D. Besch, P. Noble, J. P. Vila. Relative entopy fo compessible Navie Stokes equations with density-dependent viscosities and applications. C. R. Acad. Sciences: Section Mathématiques. Volume 354, Issue 1, (16, C. M. Dafemos. The second law of themodynamics and stability Ach. Rational Mech. Anal. 7(: , ( E. Feieisl. Relative entopies, dissipative solutions, and singula limits of complete fluid systems. Hypebolic Poblems: Theoy, Numeics, Applications,, edited by: Fabio Ancona, Albeto Bessan, Pieangelo Macati, Andea Mason, AIMS on Applied Mathematics, vol. 8, AIMS, Singfield, 14, E. Feieisl, B.J. Jin, A. Novotny. Relative entopies, suitable weak solutions and weak-stong uniqueness fo the compessible Navie Stokes system. J. Math. Fluid Mech, 14 (4:717 73, E. Feieisl, A. Novotny. Singula Limits in Themodynamics of Viscous Fluids. Bikhäuse Velag, (9. 1 Th. Gallouët, R. Hebin, D. Maltese, A. Novotny. Eo estimates fo a numeical appoximation to the compessible baotopic Navie Stokes equations. 13 N. Genie, J. P. Vila, P. Villedieu. An accuate low-mach scheme fo a compessible two-fluid model applied to fee-suface flows. J. Comput. Phys. 5, (13, 1 19.

18 ESAIM: PROCEEDINGS AND SURVEYS B. Haspot. Weak-Stong uniqueness fo compessible Navie-Stokes system with degeneate viscosity coefficient and vacuum in one dimension. Submitted ( J. Li, Z. Xin. Global Existence of Weak Solutions to the Baotopic Compessible Navie-Stokes Flows with Degeneate Viscosities. Axiv (15: axiv: v 16 P. L. Lions. Mathematical topics in fluid mechanics. Vol. 1 incompessible models. Oxfod Science publication, (6. 17 A. Mellet, A. Vasseu. Existence and Uniqueness of Global Stong Solutions fo One-Dimensional Compessible Navie?Stokes Equations. SIAM J. Math. Anal. 39 (4 ( A. Novotny. Lectue Notes on Navie-Stokes-Fouie system. Panoama et synthèses, to appea SMF ( B. Pethame, P.E. Souganidis. Dissipative and entopy solutions to non-isotopic degeneate paabolic balance laws. Ach. Rational Mech Anal., (3, F. Sueu. On the inviscid limit fo the compessible Navie Stokes system in an impemeable bounded domain. J. Math. Fluid Mech. 16, (14, no.1, C. Chistofoou, A.E. Tzavaas. Relative entopy fo hypebolic-paabolic systems and application to the constitutive theoy of themoviscoelasticity: axiv: A. Vasseu, C. Yu. Existence of global weak solutions fo 3D degeneate compessible Navie-Stokes equations. Inventiones 1 4, (16. 3 H.-T. Yau. Relative entopy and hydodynamics of Ginzbug-Landau models. Lett. Math. Phys. (1:63 8, (1991.