ON ISENTROPIC APPROXIMATIONS FOR COMPRESSIBLE EULER EQUATIONS

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1 ON ISENTROPIC APPROXIMATIONS FOR COMPRESSILE EULER EQUATIONS JUNXIONG JIA AND RONGHUA PAN Abstract. In this paper, we first generalize the classical results on Cauchy problem for positive symmetric quasilinear systems to more general esov space. Through this generalization, we obtain the local well-poseness with initial ata in the space,1 (R ) which has critical regularity inex. We then apply these results to give an explicit characterization on the Isentropic approximation for full compressible Euler equations in R 3. This characterization tells us that Isentropic Compressible Euler equations is a reasonable approximation to Non-isentropic Compressible Euler equations in the regime of classical solutions. The failure of such characterization was illustrate when singularities occur in the solutions. 1. Introuction This note is evote to the explicit characterization an mathematical justification on the isentropic approximation for the compressible invisci flui flow. For this purpose, we consier the following Cauchy problem of compressible Euler equations t ρ + u ρ + ρ ivu =, x R, ρ ( t u + u u) + p =, t s + u s =, ρ(x, ) = ρ (x), u(x, ) = u (x), s(x, ) = s (x), (1.1) where ρ is the ensity, u is the velocity fiel, s stans for the specific entropy an p(ρ, s) = ρ γ e s is the pressure law for polytropic gas, with the aiabatic exponent γ > 1. In many applications, if in the thermoynamical process the specific entropy has only very small changes near a constant equilibrium s, an isentropic approximation is applie by assuming s(x, t) = s which reuces (1.1) to the isentropic Euler equations t ρ + u ρ + ρ ivu =, x R ρ ( t u + u u) + p =, (1.) ρ(x, ) = ρ (x), u(x, ) = u (x), where p(ρ) = ρ γ e s. Now, if one assumes s (x) = s 1 Mathematics Subject Classification. 35L65. Key wors an phrases. Compressible Euler equation, Isentropic approximation, critical regularity, esov spaces. 1

2 J. JIA AND R. PAN in (1.1), the solutions of (1.1) are expecte to equal to the corresponing one of (1.) formally. The main goal of this paper is to aress this issue. More precisely, we will stuy the limiting process from solutions of (1.1) to corresponing solutions of (1.) when (s (x) s). The main results of the current paper show that, when the solutions of (1.1) an (1.) are classical, then such a picture can be justifie with sharp error estimates. However, when the solutions of Euler equations blow up an singularities are evelope, the expectation above is not true at least by the measurement of Sobolev norms. The main iea of this paper is the following observation. When both (1.1) an (1.) amit smooth solutions in R n [, T ] for some positive T, the justification of isentropic limit as (s (x) s) can be obtaine by means of the continuity epenence of initial ata for solutions of (1.1) near the initial ata (ρ, u, s). This comes along with the local well-poseness theory for smooth initial ata. Therefore, the justification of isentropic limit will be achieve by a careful energy metho for the symmetric hyperbolic systems. One of the new ingreients of this paper is that the results will be establishe for solutions in the critical esov space,1 (R ). On the other han, when singularities, say shock waves, occur in the solution, such picture breaks own. We will show this by an explicit example. Therefore, our results are somehow optimal for initial ata with lowest possible regularity, which is,1 (R ). In section, we will list some basic information on esov spaces. In Section 3, we will iscuss the local well-poseness theory for symmetric hyperbolic systems, an we will establish the corresponing theory in the critical esov space,1 (R ) which is applicable to compressible Euler equations (1.1). We will then justify the isentropic limit in Section 4 within the regime of classical solutions. Finally, we will iscuss the failure of isentropic approximation when singularity occurs in the solutions in the last section. For compressible Euler equations in one space imension, such a problem was investigate by Saint-Raymon [1] for V solutions, where the ifference between solutions of isentropic an full Euler equations measure by V-norm was shown to grow at most linearly in time. For the steay Euler flows, similar results were obtaine in [] an [8]. It remains an interesting open problem on how to offer a (physically an mathematically) soun explanation on the isentropic approximation for physically amissible weak solutions for compressible Euler equations.. Preliminaries In this section, we collect some basic facts on esov spaces that will be use in this paper. The following notations will be use throughout this paper. For any tempere istribution u, both û an F(u) enote the Fourier transform of u. For every p [1, ], L p enotes the norm of the Lebesgue space L p. The norm in the mixe space-time Lebesgue space L p ([, T ]; L r (R )) is enote by L p (with the obvious generalization to T Lr L p T X for any norme space X). For any pair of operators P an Q on some anach space X, the commutator [P, Q] is given by P Q QP.

3 COMPRESSILE EULER EQUATIONS 3 For any function u, i u stans for xi u with i = 1,,,. Then, we give a short introuction to the esov type spaces. Details about esov type space can be foun in [1] or [11]. There exist two raial positive functions χ D(R ) an ϕ D(R \{}) such that χ(ξ) + q ϕ( q ξ) = 1; q 1, suppχ suppϕ( q ) = φ, suppϕ( j ) suppϕ( k ) = φ, if j k, For every v S (R ) we set 1 v = χ(d)v, q N, j v = ϕ( j D)v an S j = With our choice of ϕ, one can easily verify that 1 m j 1 m. (.1) j k f = if j k (.) j (S k 1 f k f) = if j k 5. (.3) Like in ony s ecomposition, we split the prouct uv into three parts with uv = T u v + T v u + R(u, v), (.4) T u v = j S j 1 u j v, R(u, v) = j j u j v, where j = j 1 + j + j. We now efine inhomogeneous esov spaces. For p, r [1, + ] an s R we efine the inhomogeneous esov space s p,q as the set of tempere istributions u such that u s p,q := ( js j u L p) l q < +. Notice that the usual Sobolev spaces H s coincie with, s for every s R. For simplicity, we will use s to stan for the esov space,1 s. The following two Lemmas will be use frequently in this paper. Lemma.1. [4] Let s > an 1 p, q. Then s p,q is an algebra an we have where C is a positive constant. uv s p,q C( u v s p,q + v u s p,q ), Lemma.. [4] Let I be an open interval of R. Let s > an σ be the smallest integer such that σ s. Let F : I R satisfy F () = an F W σ, (I; R). Assume that v s p,q has values in J I. Then F (v) s p,q an there exists a constant C epening only on s, I, J an N, an such that F (v) s p,q C (1 + v ) σ F W σ, (I) v s p,q.

4 4 J. JIA AND R. PAN 3. Positive symmetric hyperbolic systems In this section, we concentrate on the following quasilinear system A (U, x, t) t U + A k (U, x, t) k U = (U, x, t), U(x, ) = U (x), (3.1) where the n n matrices A (U, x, t), A k (U, x, t) an the source term (U, x, t) epen on U R n, x R, an t [, ) smoothly. For τ 1 < τ, an for subsets O of R n, an Ω of R, (3.1) is calle positive symmetric hyperbolic on O Ω [τ 1, τ ], if, for any U O, x Ω, an t [τ 1, τ ], A k (k =,, ) are symmetic an A is positive efinite. The following Theorem is ue to [5], [7], [9], [3], an [1]. Theorem 3.1. Assume that (3.1) is positive symmetric hyperbolic for its arguments, an U belongs to H s (R ) for some s >. There exits a positive time T such that (3.1) has a unique solution U(x, t) C([, T ]; H s ) C 1 ([, T ]; H s 1 ). Moreover, T can be boune from below by c 1 U (x) 1 H, where c s 1 epens only on A k (k =,, ). The maximal time of existence T of such a solution is inepenent of s an satisfies T < = T U(, t) t =. The positive symmetric hyperbolic systems cover an important class of quasi-linear hyperbolic systems. Although many physical systems are not exactly in positive symmetric form, like (1.1) an (1.), they are symmetrizable, in the sense that there exists a transformation of unknown functions renering the system into postive symmertic hyperbolic systems in the new variables. In particular, when a quasi-linear hyperbolic system enowe with a convex entropy, it is symmetrizable by Herssian matrix of this entropy. Therefore, Theorem 3.1 fins applications for many physical systems incluing isentropic Euler (1.) an full Euler systems (1.1). One of our goals is to establish the analog theory of Theorem 3.1 with intial ata of crticital regularity, i.e., in the esov space,1 (R ) for (3.1). In [1], this was achieve when A = I n, an the result there fins application to the isentropic Euler equations (1.). In fact, if one introuces the quantity σ = (4γe s ) 1 γ 1 ρ γ 1, (3.) then the isentropic Euler system turns into { σ t + u σ + γ 1 σiv u =, u t + u u + γ 1 σ σ =. (3.3) This, however, is not the case for the full Euler equations (1.1). In orer to generalize this theory to cover full Euler equations, we will exten the results of [1] for A = I n to (3.1) for positive efinite A = A (U).

5 COMPRESSILE EULER EQUATIONS Linear positive symmetric system. We first consier the following positive symmetric linear system A (x, t) t U + A k (x, t) k U = (x, t)u + F (x, t). (3.4) where the n n matrices A k (k =,, ) an are smooth functions. The following classical results (c.f. [6]) is often very useful. Theorem 3.. Assume that (3.4) is positive symmetric for all x R an t, such that A (x, t) satisfying βi n A β 1 I n, for some positive constant β in the sense of quaratic forms. Assume that A, A k, an their first orer erivatives are boune. Then, for all T > an U C([, T ]; H 1 ) C 1 ([, T ]; L ), it hols that β U(t) L e γt U() L + t for all t [, T ]. γ is chosen to be large enough, so that β(γ 1) We now prove the following result. t A + e γ(t τ) F (τ) L τ, (3.5) k (A k ) +. (3.6) Theorem 3.3. Assume that (3.4) is positive symmetric for all x R an t, such that A (x, t) satisfying βi n A β 1 I n, for some positive constant β in the sense of quaratic forms. Suppose that T ( (R )), an that A, A = {A k } 1 k have the following form A k (x, t) = Āk + Ãk(x, t), A (x, t) = Ā + Ã(x, t), (3.7) where Āk, Ā are constant matrices, an Ãk T ( ), à T ( ), t à T (L ). Then, for all T > an U C([, T ]; ) C 1 ([, T ]; ), it hols that β U(t) U() ecβ Mt + CM t where M = t A T ( ) + T ( ) + A T ( ) e Cβ M(t τ) A 1 F (τ) τ, (3.8) + A T ( ) A 1 T ( ) + A T ( ) (A 1 A) T ( ), M = 1 + à T ( ).

6 6 J. JIA AND R. PAN Proof. Taking the operator j on both sies of (3.4), we obtain j (A t U) + Following [1], we efine j (A k k U) = j (U) + j F. (3.9) S j = S j, if j ; an S j = 1, otherwise, T k U (A k) = S j + k U j A k, j an therefore, (3.9) is reuce to 3 A t j U + S j 1 (A k ) k j U = j (U) + j (F ) + Rj l [ j, A ] t U, (3.1) l=1 where Rj 1 = [ j, S j 1(A k )] j k U, j j N 1 Rj = ( Sj 1(A k ) S j 1 (A k ) ) j j k U, j j 1 Rj 3 = j T k U (A k), where N 1 is the fixe integer efine as in the proof Lemma 4.14 in [1] page 184 such that j S j 1A k j k U = j j j j N 1 A k j k U Hence, we can easily give the estimates about R 1 j, R j an R3 j as in [1] j( ) R 1 j L Cc j A U, j( ) R j L Cc j A U, (3.11) j( ) R 3 j L Cc j A U. The first two terms on the right han sie of (3.1) can be estimate as follows j( ) j (U) L Cc j U, j( ) j (F ) L = j( ) j (A A 1 F ) L ) Cc j (1 + Ã A 1 F. (3.1)

7 COMPRESSILE EULER EQUATIONS 7 Now, we consier the term [ j, A ] t U. Since { } [ j, A ] t U = [ j, A ] A 1 A k k U + A 1 U + A 1 F For I, we have For I 1, we know that = [ j, A ]A 1 A k k U [ j, A ]A 1 U [ j, A ]A 1 = I + II + III. I = = ( j (A k k U) A j ( A 1 A k k U )) [ j, A k ] k U = I 1 + I. j( ) I 1 L A [ j, A 1 A k] k U j( ) [ j, A k ] k U L F (3.13) Cc j A U, where we use Lemma.1 in [1]. For I, we have j( ) I L j( ) A [ j, A 1 A k] k U L Cc j A ( (3.14) A 1 A) U, where we also use Lemma.1 in [1]. The term II will be treate as follows j( ) II L j( ) [ j, A ]A 1 U L For III, we have j( ) { j (U) L + A j (A 1 U) L } Cc j / U + Cc j A A 1 U. (3.15) j( ) III L j( ) [ j, A ]A 1 F L Cc j à A 1 F (3.16). Due to the following two equalities ( Sj 1 (A k ) k j U, j U ) = 1 ( k ( S j 1 A k ) j U, j U ), 1 t (A j U, j U) = (A t j U, j U) + 1 ( ta j U, j U),

8 8 J. JIA AND R. PAN we have 1 t (A j U, j U) = 1 ( k ( S j 1 A k ) j U, j U ) + 1 ( ta j U, j U) + ( j (U), j U) + ( j (F ), j U) 3 ( ) + Rj, l j U + (I, j U) (II, j U) l=1 (III, j U). Since A is positive efinite an symmetric, we know that there exist symmetric positive efinite matrix A satisfies t (A j U, j U) = ( A j U, A j U). t Combining this with equation (3.1) an the estimates (3.11), (3.1), (3.13), (3.14), (3.15), (3.16), we have where β 1 j U(t) L β 1 j U() L t ( ) + β 3 t A + A j U(τ) L τ + Cβ 1 cj j( ) t M 1 U(τ) τ + Cβ 1 cj j( ) M t A 1 F (τ) τ, M 1 = T ( ) + A T ( ) + A T ( ) (A 1 A) T ( ) + A T ( ) A 1 T ( ). M = 1 + Ã T ( ). (3.17) Multiplying j( ) on both sies of (3.17), an summing up for j, we obtain β 1 U(t) / β 1 U() / + Cβ M + Cβ 1 M t t A 1 F (τ) / τ. β 1 U(τ) / τ Finally, by Gronwall s inequality, we complete the proof of this Lemma. Remark 3.1. Here, we actually prove the following stronger result β j(/) j U t (L ) U() /1e Cβ Mt j 1 + CM t e Cβ M(t τ) A 1 F (τ) / τ. (3.18)

9 COMPRESSILE EULER EQUATIONS Quasilinear positive symmetric system. We now consier the following quasilinear system A (U) t U + A k (U) k U = (U), (3.19) U(x, t) = U (x). For this problem, we make the following assumptions. Conition 3.1. Assume the n n symmetric matrices A k (U) an the source term (U) smoothly epen on U R n an () =. A k (U) has the following form A k (U) = Āk + Ãk(U) for k =, 1,...,, where Āk are constant matrices, Ãk(U) smoothly epen on U R n, Ãk() = an A (U) is positive efinite. We now prove the following local theory. Theorem 3.4. Assume Conition 3.1. Let O be any open subset of R n. For the Cauchy problem (3.19) with initial ata U (R ) taking values in O, there exists T > such that the system has a unique solution U C 1 (R [, T ]). Furthermore, U belongs to C([, T ]; (R )) C 1 ([, T ]; (R )). Proof. The proof is base on the following iteration scheme A (U n ) t U n + A k (U n ) k U n = (U n ), U n t= = S N +nu. Since S N U tens to U in (R ) when N goes to +, by the embeing (R ) (R ), (3.) there are N, δ > an a relatively compact open subset V of O such that any smooth function U satisfying the estimate U S N U δ (3.1) takes values in V. The relatively compact subset V will serve as a reference for energy estimates. We take β > so that βi n A (U) β 1 I n, (3.) for all U V. We procee by inuction. Initially, we have U (t) = S N U an thus U S N U T ( ) = δ, for any T an δ. We assume that for all l k, U l is efine inuctively by (3.) an satisfies the estimate U l S N U T ( δ. (3.3) )

10 1 J. JIA AND R. PAN We are going to show that the same estimate (3.3) hols for l = k + 1, provie that T is suitably chosen. We introuce the notations V k := U k S N U. y efinition, V k must solve the following Cauchy problem A (U k ) t V k + A j (U k ) j V k = (U k ) A j (U k ) j S N U, (3.4) j=1 j=1 V k () = S N +ku S N U. Applying Theorem 3.3 to the above iterative system (3.4), we have where β V k (t) V k () ecβ Mt + CM e Cβ Mt G(t) = A 1 (U k )(U k ) + j=1 A 1 t (U k )A j (U k ) j (S N U ) G(τ) τ. (3.5), M = t A (U k ) T ( ) + (U k ) T ( ) + A(U k ) T ( ) + A (U k ) T ( ) A 1 (U k )(U k ) T ( ) + A (U k ) T ( ) (A 1 (U k )A(U k )) T ( ). M = 1 + Ã(U k ) T ( ), With he help of the inuctive assumption, Lemma.1 an Lemma., we know that there exists a constant M > such that { } max Hence, the inequality (3.5) implies M, M, sup G(t) t [,T ] M < +. β V k (t) V k () ecβ M T + CT M e Cβ M T. (3.6) Taking N large enough, T small enough, we finally obtain sup V k (t) t [,T ] δ. (3.7) Using the similar metho as in [6], we can prove U n is a Cauchy sequence in T (L ) for small enough T >. y interpolation, U n is also a Cauchy sequence in T (s ) for any s < + 1. The limit U of U n is obvious a solution of (3.19). Using the Fatou property for the esov space, we conclue that U belongs to ([, T ]; ) C([, T ]; s ) C 1 ([, T ]; s 1 ),

11 COMPRESSILE EULER EQUATIONS 11 for any s < + 1. In orer to prove that U belongs to C([, T ]; ), we observe from (3.7) an Remark 3.1 that j( ) j U T (L ) δ + S N U, j 1 which implies that for any positive ɛ, there is some integer j such that j( ) j U T (L ) ɛ 4. j j Therefore, for t, t [, T ], we have U(, t) U(, t ) j<j j( ) j (U(, t) U(, t )) L + j j j( ) j U T (L ) C j ( ) U(, t) U(, t ) L + ɛ. ecause U is in C([, T ]; L ), the first term on the right-han sie tens to when t t. This implies that U is continuous in time with values in. Now, one can follow the stanar continuity argument to exten the existence time T of the solution to a maximal one; c.f. [1]. Therefore, the following theorem can be prove in the same manner as in [1]. Theorem 3.5. Uner the Conition 3.1, assume that (3.19) is positive symmetric hyperbolic for its arguments, with the initial ata U (x) (R ). There exits a positive time T such that (3.19) has a unique solution U(x, t) C([, T ]; ) C 1 ([, T ]; ). Moreover, T can be boune from below by c 1 U (x) 1, where c 1 epens only on A k (k =,, ). The maximal time of existence T of such a solution satisfies T < = T U(, t) t =. Remark 3.. It is clear that if the initial ata U Ū for some constant Ū. Replacing () = an Ãk() = with (Ū) = an Ãk(Ū) = in Conition 3.1, Theorem 3.5 hols for U Ū. The following theorem is about the continuous epenence of the solutions, which will be useful in the next section for the escription of isentropic approximations for compressible Euler equations. Theorem 3.6. Assume that A is positive efinite such that βi n A β 1 I n

12 1 J. JIA AND R. PAN for β > in the sense of quaratic forms. Let U ɛ an U be solutions of (3.19) on R [, T ], with initial ata U ɛ, U (R ) respectively. Then sup U ɛ (t) U(t) L e 1 (λ+cβ )T U ɛ U t [,T ] β L. where C epens on the -norms of U an U ɛ, an λ is a large enough constant satisfing ta (U) + k A k (U) β(λ 1). T ( ) Proof. From Theorem 3.5, it is clear that T > is smaller than the maximimal existence times of U ɛ or U. Therefore, we have that sup ( (U t, U)(, t) + (Ut ɛ, U ɛ )(, t) ) C, t T for some positive constant C. Denote δu = U ɛ U, we know that A (U) t δu + A k (U) k δu = F, (3.8) where F = (U ɛ ) (U) (A (U ɛ ) A (U)) [A (U ɛ )] 1 (U ɛ ) + (A (U ɛ ) A (U)) [A (U ɛ )] 1 A k (U ɛ ) k U ɛ (A k (U ɛ ) A k (U)) k U ɛ. Using mean-value theorem, it hols that F (, t) L C( (U, U ɛ ) T ( ) + U ɛ T ( )) U ɛ (, t) U(, t) L C U ɛ (, t) U(, t) L. (3.9) Multiplying both sies of (3.8) with (δu) T, an then integrating over R, integrating by parts, we obtain t (A (U)δU, δu) β(λ 1)(δU, δu) + β(δu, δu) + β 1 (F, F ) Therefore, we have Hence, the proof is complete. λ(a (U)δU, δu) + β 1 F (, t) L λ(a (U)δU, δu) + Cβ 1 (δu, δu) (λ + Cβ )(A (U)δU, δu). sup U ɛ (t) U(t) L e(λ+cβ )T t [,T ] β U ɛ U L.

13 COMPRESSILE EULER EQUATIONS Explicit characterization about isentropic approximation In this section, we apply the general results obtaine in Section 3 to the compressible Euler equations, an then give an explicit characterization about isentropic approximation. Firstly as in [1], we use w = p (γ 1) γ to transform Euler equations (1.1) into A (U) t U + = ρ (γ 1) e (γ 1)s γ (4.1) A j (U) j U =, j=1 U(x, ) = U (x), for U = (w, u 1, u, u 3, s), an the matrices 1 A (U) = (γ 1) 4γ e s γ I 3, 1 A j (U) = γ 1 we j (γ 1) 4γ e s γ u j I 3 u j u j γ 1 wet j (j = 1,, 3). (4.) Here, e j is the j th row of I 3. Theorem 3.4 an Therem 3.5 give the following Theorem. Theorem 4.1. If for some constants ρ, s an w = ρ (γ 1) e (γ 1) s γ, the initial ata U = (w, u, s ) satisfies that U ( w,, s) 5 (R 3 ), there exists a constant T > an a unique solution U = (w, u, s) to the problem (4.) such that U ( w,, s) C([, T ]; 5 (R 3 )) C 1 ([, T ]; 3 (R 3 )). If in aition w (x) for all x R 3, then w(x, t) for all (x, t) R 3 [, T ]. Equivalently, if 1 < γ 3, ρ C 1 (R 3 ), ρ an U ( w,, s) 5 (R 3 ), then there exists a positive number T an a unique solution (ρ, u, s)(x, t) C 1 ([, T ] R 3 ) to problem (1.1) such that ρ(x, t) for all (x, t) R 3 [, T ] an ( ) U(x, t) = ρ (γ 1) e (γ 1)s γ, u, s (x, t) is the solution of (4.) such that U ( w,, s) C([, T ]; 5 (R 3 ) C 1 ([, T ]; 3 (R 3 )). Remark 4.1. This Theorem inclues the cases of initial ata with or without vacuum. For the H s theory with s > 5, the case with initial ata incluing vacuum was given in [1], an the case with initial ata away from vacuum was given in [9].

14 14 J. JIA AND R. PAN In orer to give a precise escription on the isentropic approximation for compressible Euler equations, we now assume that s (x) s = εφ(x) (4.3) for φ(x) 5 (R 3 ), an ε > is the controlling parameter. As explaine in the introuction, given an initial ata (ρ, u, s ) or (w, u, s ) as in Theorem 4.1, the Cauchy problem (1.1) has a smooth solution (w, u, s)(x, t) efine on R 3 [, T ] for some positive T. We enote this solution as U ε (x, t) = (w, u, s)(x, t). If one assigns an initial ata (ρ, u, s) or ( w, u, s) with w = w(ρ, s) to (1.1), the unique solution of (1.1) on R 3 [, T ] is the same one to (1.) with initial ata (ρ, u ) or ( w, u ), an we enote this solution by U I (x, t) = (w I, u I, s). We can now apply Theorem 3.6 to obtain the following theorem. Theorem 4.. Suppose 1 < γ 3, ε (, 1], ρ C 1 (R 3 ), ρ an for some ρ, an ρ (γ 1) ρ (γ 1) 5 (R 3 ), u 5 (R 3 ), φ 5 (R 3 ), Then, (1.) has a unique solution an (1.1) has a unique solution ρ (γ 1) e (γ 1)( s+εφ) γ ρ (γ 1) e (γ 1) s γ 5 (R 3 ). U I (x, t) = (w I, u I, s)(x, t), U ε (x, t) = (w, u, s)(x, t), both efine on R 3 [, T ]. Furthermore, the following estimate hols sup U ε (, t) U I (, t) L Cε φ L, t [,T ] where C is a positive constant epening on C 1 norms of U (x) an T, but not on ε. Proof. y Theorem 3.6, we only nee to compute U ɛ U L to complete the proof of this Theorem, since A was boune by s. From (4.3) an (4.1), we have ( ( ) ) U ε U L ε φ L + ρ (γ 1) e (γ 1) s γ e (γ 1)εφ 1/ γ 1 x R 3 ε φ L + C( ρ, γ, s) Since φ 5 (R 3 ), it is clear that e (γ 1)εφ γ the following calculations. e (γ 1)εφ γ 1 e (γ 1)εφ γ ( R 3 ( e (γ 1)εφ γ 1) x) 1/. (4.4) 1 tens to zero as x. We now make 1 C( φ, γ)(ε φ + ε φ ). (γ 1)εφ (γ 1)εφ + γ γ (4.5)

15 Therefore, we conclue that which conclues the proof of this Theorem. COMPRESSILE EULER EQUATIONS 15 U ε U L ε φ L + Cε φ L Cε φ L, Remark 4.. This theorem gives a precise justification with an explicit error estimate on isentropic approximation for compressible Euler equations in the regime of smooth solutions. 5. Failure of isentropic limit In the previous sections, the justification for isentropic limit has been prove for classical solutions of compressible Euler equations. It is well-known that shock waves may evelope in finite time even for generic small smooth initial ata. When shock forms, the justification of isentropic limit in previous sections breaks own. This can be seen easily by the following example. Consier the full compressible Euler equations in one space imension ρ t + (ρu) x =, x R, (ρu) t + (ρu + p(ρ, s)) x =, (5.1) (ρe) t + (ρeu + pu) x = ρ(x, ) = ρ (x), u(x, ) = u (x), s(x, ) = s, where E = 1 u + e an ρe = cv R p with two positive constants c v an R, an its isentropic reuction ρ t + (ρu) x =, x R, (ρu) t + (ρu + p(ρ, s)) x =, (5.) ρ(x, ) = ρ (x), u(x, ) = u (x). We remark here that unlike (1.1), we replace the entropy equation by the energy conservation law in (5.1), since the entropy equation is no longer vali if singularity occurs in the solutions. It is clear that for any C 1 functions ρ an u, both (5.1) an (5.) share exactly the same C 1 solution (ρ(x, t), u(x, t), s) up to a maximal existence time T 1 >. However, when this solution blows up at (x 1, T 1 ) for some x 1 R, an shocks appear in the solution, the shock solution for (5.1) is ifferent from that of (5.). Inee, the Riemann problems of (5.1) an (5.) with the same Riemann ata lim (ρ, u, s)(x, T 1) = (ρ, u, s); x x 1 lim (ρ, u, s)(x, T 1) = (ρ +, u +, s), (5.3) x x 1 are ifferent since the former one has a variable s (entropy s must increase across a shock wave, see [13]), while the latter has a constant s in the solution. Therefore, the framework we use in justification the isentropic limit process in the previous sections is no longer vali when singularity occurs in the solutions of Euler equations. New insights an techniques are require to offer possible escription of isentropic approximation for entropy weak solutions. Some research activities have been carrie out in this irection, see [1], [] an [8], where the ifference between solutions of isentropic

16 16 J. JIA AND R. PAN an full Euler equations measure by V-norm was shown to grow at most linearly in time. Acknowlegments The authors are very grateful to valuable suggestions of the anonymous referees improving the presentations of this paper. J. Jia woul like to thank China Scholarship Council that has provie a scholarship for supporting his research in the Unite States. J. Jia s research is supporte partially by National Natural Science Founation of China uner the grant no , an by the National asic Research Program of China uner the grant no. 13C3944. The research of R. Pan is supporte by the National Science Founation through the grant DMS References [1] H. ahouri, J. Chemin, R. Danchin, Fourier Analysis an Nonlinear Partial Differential Equations, Grunlehren er mathematischen Wissenschaften, Springer, (11). [] S. Chen, J. Geng, Y. Zhang, Isentropic approximation of quasi-one-imensional unsteay nozzle flow, SIAM J. Math. Anal. 41 (9), no. 4, [3] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag (). [4] R. Danchin, Fourier Analysis Methos for PDE s, Lecture Notes (5) November 14. [5] K. O. Frierichs, Symmetric hyperbolic linear ifferential equations, Comm. Pure Appl. Math. 7 (1954), [6] S.. Gavage, D. Serre, Multi-imensional hyperbolic partial ifferential equations: First-orer Systems an Applications. Oxfor Mathematical Monographs. Oxfor Science Publications, (7). [7] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Rational Mech. Anal. 58 (1975), no. 3, [8] C. Liu, Y. Zhang, Isentropic approximation of the steay Euler system in two space imensions. Commun. Pure Appl. Anal. 7 (8), no., [9] A. Maja, Compressible Flui Flow an Systems of Conservation Laws in Several Space Variables, Springer Appl. Math. Sci., (1984) [1] T. Makino, S. Ukai, S. Kawashima, Sur la Solution à Support Compact e l Equation Euler Compressible, Japan J. Appl. Math., 3(1986), [11] J. Peetre, New thoughts on esov spaces, Duke Univ. Math. Ser. I, Durham, N.C., (1976). [1] L. Saint-Raymon, Isentropic approximation of the compressible Euler system in one space imension, Arch. Ration. Mech. Anal. 155 (), no. 3, [13] J. A. Smoller, Shock Waves an Reaction-Diffusion Equations, Springer-Verlag, n Eition, (1994). Department of Mathematics, Xi an Jiaotong University, Xi an 7149, China; aress: jiajunxiong@163.com Department of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Skiles uiling Atlanta, GA aress: panrh@math.gatech.eu