Existence Theory for the Isentropic Euler Equations

Transcription

1 Arch. Rational Mech. Anal Digital Object Identifier DOI 1.17/s Existence Theory for the Isentropic Euler Equations Gui-Qiang Chen & Philippe G. LeFloch Communicated by C. M. Dafermos Abstract We establish an existence theorem for entropy solutions to the Euler equations modeling isentropic compressible fluids. We develop a new approach for constructing mathematical entropies for the Euler equations, which are singular near the vacuum. In particular, we identify the optimal assumption required on the singular behavior on the pressure law at the vacuum in order to validate the two-term asymptotic expansion of the entropy kernel we proposed earlier. For more general pressure laws, we introduce a new multiple-term expansion based on the Bessel functions with suitable exponents, and we also identify the optimal assumption needed to validate the multiple-term expansion and to establish the existence theory. Our results cover, as a special example, the density-pressure law pρ = κ 1 ρ γ 1 κ 2 ρ γ 2 where γ 1,γ 2 1, 3 and κ 1,κ 2 > are arbitrary constants. 1. Introduction In this paper we establish an existence theory for the Euler equations modeling isentropic compressible fluids: t ρ x ρ v =, t ρv x ρ v 2 pρ =, 1.1 where ρ denotes the density, v the velocity, and pρ the pressure of the fluid. Under the standard assumption p ρ > for ρ>, 1.2 the system 1.1 is strictly hyperbolic, away from the vacuum ρ = at least. Denoting the sound speed by cρ := p ρ, the two wave speeds of the system, v cρ and v cρ, are real and distinct for ρ>. The partial differential equations 1.1 provide an example of particular interest in the mathematical theory of

2 82 Gui-Qiang Chen & Philippe G. LeFloch nonlinear hyperbolic systems of conservation laws. Recall that the main difficulties towards proving the existence of an entropy solution to 1.1 are as follows: 1 Discontinuities may form at a later time even for initially smooth data and, therefore, entropy solutions must be understood in the sense of distributions. 2 Since the equations come from continuum physics modeling, we are interested in solutions that are defined globally in time and start out from arbitrary large data at time t =, and should not impose any smallness condition on the solutions. 3 Additionally, the equations under consideration are degenerate near the vacuum, in the sense that strict hyperbolicity of 1.1 fails at the vacuum: at ρ =, the two wave speeds coincide if p =, which is satisfied by pressure laws arising in continuum physics and will therefore be assumed from now on. The existence of entropy solutions for the Cauchy problem associated with 1.1 was established, in the case of polytropic perfect gases pρ = κρ γ, κ >, γ>1, 1.3 first by DiPerna [6], Ding, Chen &Luo [5], and Chen [1] based on compensated compactness arguments, and then, motivated by a kinetic formulation of hyperbolic conservation laws, by Lions, Perthame & Tadmor [8], and Lions, Perthame & Souganidis [9]. General pressure laws pρ were covered first by Chen &LeFloch [2]; the existence of entropy flux-splittings for 1.1 was established by Chen &LeFloch [3]. The purpose of this paper is to deal with more general pressure functions, including for instance the function pρ = κ 1 ρ γ 1 κ 2 ρ γ 2, γ 1,γ 2 > 1, κ 1,κ 2 >. 1.4 We refer to the statements 2.1 and 3.2 below for our precise assumption on the function pρ. Recall that the results in [2] apply to 1.4 when γ 2 > 1 γ 1. The Green-function approach introduced here turns out to be more accurate than the energy-estimate approach we developed earlier. It allows us to cover more general pressure functions and, in particular, to cover general exponents γ 1 and γ 2 in 1.4. It was pointed out by DiPerna [6] that, in the case 1.3 for polytropic perfect gases, the mathematical entropies associated with the Euler equations admit an explicit formula, based on a fundamental solution of the entropy equation. We refer to this fundamental solution as the entropy kernel. In [6, 5, 1, 8, 9], the explicit formula for the entropies associated with 1.3 is central to establishing the reduction of the Young measures governed by Tartar s commutation equations [1]. The approach developed in [2] for general pressure laws pρ requires less explicit calculations and only certain properties of the entropy kernel singularities and cancellation. Studying the existence of the entropy kernel and its regularity is delicate since the equation for entropies contains singular coefficients near the vacuum. Our general strategy is as follows. We first apply the Fourier transform in the velocity variable so that the linear hyperbolic equation governing the entropy kernel

3 Isentropic Euler Equations 83 is transformed into a family of second-order differential equations with singular coefficients, in which the Fourier variable is as a parameter. To determine the regularity of the entropy kernel, we then determine the asymptotic behavior of the solutions in the Fourier variable, that is, derive a suitable asymptotic expansion which is based on Bessel functions. Our main results are as follows. In Section 2, we determine the optimal assumption on pρ which is required to validate the two-term expansion derived in [2]. For the example 1.4, that condition is γ 2 1 > 3γ 1 1/2. However, for smaller values of γ 2 the two-term expansion does not accurately describe the asymptotic properties of the entropy kernel. A multiple-term expansion is derived in Section 3 for a large class of pressure laws which, in particular, cover the example 1.4 for arbitrary exponents γ 1,γ 2 > 1. Finally, in Section 4, relying on the convergence framework established by the authors in [2], we arrive at the general existence theory for 1.1 with exponents in the physically relevant interval 1.3. The compactness of the solution operator and the decay of periodic entropy solutions follow as in [2]. Note in passing that the genuine nonlinearity assumption ρp ρ 2 p ρ > for ρ> 1.5 is not needed for the proof of existence of the entropy kernel, but solely to establish that it remains non-negative. We refer to [4] for other applications of the techniques and ideas developed here. 2. A two-term asymptotic expansion To begin with, we treat a class of pressure laws that includes, for instance, the example 1.4 for large deviation between γ 1 and γ 2, specifically for γ 2 >3γ 1 2/2. In the present section only, we assume that there exists an exponent γ 1,,a smooth function P = Pρ, and some real ε> such that pρ = κρ γ 1 ρ θ1ε Pρ, Pρ and ρ 3 P ρ are bounded as ρ, 2.1 where κ := γ 1 2 /4γafter normalization. Of course, the function Pρmay exhibit some singularities at ρ =. In fact, 2.1 implies solely that ρp ρ and ρ 2 P ρ remain bounded. The following notation will be used: kρ := cy y dy, θ = γ 1 2, λ = 3 γ 2 γ 1, ν = λ 1 2 = 1 2 θ. Recall that the case γ 1, 3 and ε 1 1/θ is treated in [2]. The entropy kernel solves the following Cauchy problem associated with a linear hyperbolic equation see [2]: a χ ρρ k ρ 2 χ vv =, b χ,v=, c χ ρ,v= δ v=. 2.2

4 84 Gui-Qiang Chen & Philippe G. LeFloch The equation 2.2a is a singular Euler-Poisson-Darboux equation, in which k ρ 2 = θ 2 ρ 2 θ 1 1 oρ with 2 θ 1 2,. Using the Fourier transform in the variable v and denoting by ξ the Fourier variable in 2.2, these yield a ˆχ ρρ = k ρ 2 ξ 2 ˆχ, b ˆχ,ξ=, c ˆχ ρ,ξ= Thus, the problem reduces to a family of second-order differential equations in the density variable ρ, depending on the parameter ξ R. In the particular case 1.3, it is not hard to check that the solution of 2.3 is closely related to the Bessel function J ν = J ν y defined for all y>by 1 J ν y = c,λ y ν cosy x 1 x 2 λ dx =: c,λ y ν fˆ λ y, 1 1 where the constant c,λ is given by c,λ = 1 1 x 2 λ 1. dx See for instance [7]. More precisely, using the fact that J ν is a solution of the second-order differential equation d 2 J ν dy 2 1 y dj ν dy 1 ν2 y 2 J ν =, 2.4 we can check that the solution of 2.3 in the case p ρ := κρ γ is exactly ˆχ ρ, ξ := ξ ν ρ 1/2 J ν ξ ρ θ. Let us review first some properties of Bessel functions which motivate the forthcoming discussion. First of all, we have the asymptotic expansions J ν y = { y ν Oy ν1 when y, c 1,λ y 1/2 cos y λ 1π/2 Oy 3/2 when y, 2.5 for some constant c 1,λ >. On the other hand, the second Bessel function Y ν = Y ν y satisfies after normalization Y ν y = { y ν Oy ν1 when y, c 2,λ y 1/2 sin y λ 1π/2 Oy 3/2 when y, 2.6 for some constant c 2,λ >. Similar formulas are available for the derivatives J ν y and Y ν y. To handle the above expansions, it is convenient to set, for y =, { y ±ν for y 1, Q ±ν y := y 1/2 for y 1.

5 Isentropic Euler Equations 85 From the asymptotic formula 2.5 we see that, for some C 1 >, J ν y C 1 Q ν y, y >. 2.7 Consider also the kernel K associated with the two Bessel functions ρ,s > and ξ R: K ρ,s; ξ = Y ν ξ kρ J ν ξ ks J ν ξ kρ Y ν ξ ks. Based on 2.5 and 2.6, it is easy to check that, for some C 2 >, K ρ,s; ξ C 2 Q ν ξ kρ Q ν ξ ks 1 R ξ ks, s ρ, ξ R, 2.8 where the function R is defined by Ry := { 1 if y 1, 1/ y if y Finally, since J ν and Y ν are two independent solutions of 2.4 with the expansions 2.5 and 2.6 at, we see that their Wronskian is wρ, ξ :=Y ν ξ kρ J ν ξ kρ J ν ξ kρ Y ν ξ kρ 1 = ξθkρ = 2 ν ξ kρ. 2.1 For the general problem 2.3, we now prove the following theorem. Theorem 2.1. Suppose that the function p = pρ satisfies the assumptions 1.2 hyperbolicity and 2.1 behavior near the vacuum for some γ 1, and some arbitrary ε >. Then there exists a solution of the problem 2.3, ˆχ = ˆχρ,ξ, defined for ρ and ξ R. It is smooth for ρ>, continuous when ρ, and is given by the expansion ˆχρ,ξ = where θ kρ 1/2 Jν ξ kρ k ρ ξ ν Nρ J ν1 ξ kρ ξ ν1 Nρ := c 3 k λ 1 k 1/2 k λ1 k 1/2 dτ rρ,ξ, 2.11 for some constant c 3 := 2λ 1c,λ / 4λ 1c,λ1 = 4λ 2 1/8λλ 1. The remainder satisfies rρ,ξ ρ r ρ ρ, ξ C Q ν ξ kρ ξ min2,1ε for some C>. 2.12

6 86 Gui-Qiang Chen & Philippe G. LeFloch Note that, in fact, N = Nρ only depends on second-order derivatives of the function k at most, since N = c 3 k λ 1 k 1/2 k λ1 k 1/2 c 3 k λ 1 k 1/2 k λ1 k 1/2 dτ. Proof. To simplify the notation, we assume ξ > throughout the proof. The functions ˆχ θ kρ 1/2 1 ρ, ξ := J ν ξ kρ k ρ ξ ν, ˆχ θ kρ 1/2 2 ρ, ξ := k ξ ν Y ν ξ kρ ρ are two independent solutions of the modified equation cf in [2]: ˆχ ρρ k ρ 2 ξ 2 βρ ˆχ =, 2.13 where β := k λ 1 k 1/2 k λ1 k 1/2. On the other hand, the remainder θ kρ 1/2 rρ,ξ := k rρ,ξ ρ satisfies a non-homogeneous version of the same equation cf. 3.3 in [2]. Indeed, we have r ρρ k ρ 2 ξ 2 βρ r = Hρ,ξ βρ r 2.14 with cf. Section 3, Step 1, in [2] for an explicit expression of α, α, and A: Hρ,ξ := α 2λ 3 ρ 2λ 1 α ρ fˆ λ1 ξ kρ =: Aρ fˆ λ1 ξ kρ = Aρ J ν1 ξ kρ c,λ1 ξkρ ν1. A tedious but straightforward calculation based on the assumption 2.1 shows that g := β k k = λλ 1 k k 3 4 kk 2 k kk 2 k and = Oρ 1θ1ε 2.15 Bρ := θkρ 1/2 Aρ k ρ c,λ1 kρ ν = Oρ 1θ1ε. 2.16

7 Isentropic Euler Equations 87 Using the fact that ˆχ 1 and ˆχ 2 are two independent solutions of the homogeneous equation 2.13 and relying on the method of variation of parameters, we easily arrive at the following integral formulation for the solution of 2.14: θkρ 1/2 K ρ,s; ξ θks 1/2 rρ,ξ = k ρ Ws,ξ k Hs,ξ βs rs,ξ ds, s 2.17 where the kernel K was defined earlier and the Wronskian W is Wρ,ξ := ˆχ 1 ˆχ 2, ˆχ 1, ˆχ 2 ρ, ξ = ξθkρ J ν ξkρ Y ν ξkρ J ν ξkρ Y νξkρ = ξθkρwρ,ξ= 1, thanks to 2.1. Hence, 2.17 becomes rρ,ξ = θ K ρ,s; ξ Bs J ν1 ξks gs rs, ξ ds ξks Introduce now the notation Gρ, ξ := R ξ ks s 1θ1ε ds. Clearly, for ρ within any given bounded set, we have On the other hand, setting Gρ, ξ C, ρ >, ξ R Sy := { 1 for y 1, 1/y 2 for y 1 and using the assumption 2.1, it is not hard to check that, for some C 3 >, We will also use below the inequality S ξ ks s 1θ1ε ds. 2.2 ξ min2,1ε Q 1 ν y Ry J νy Sy for y> We now apply a fixed-point procedure to 2.18, defining a sequence of functions r n by r = and, for n, r n1 ρ, ξ = θ K ρ,s; ξ Bs J ν1 ξks r n s, ξ gs ds. ξks C 3

8 88 Gui-Qiang Chen & Philippe G. LeFloch In view of 2.8 and, by 2.15, gρ C 3 ρ 1ε for some C 3 >, we have for all n 1 r n1 ρ, ξ r n ρ, ξ 1 θc 2 Q ν ξ kρ Qν ξ ks R ξ ks r n r, ξ r n 1 s, ξ gs ds 1 θc 2 C 3 Q ν ξ kρ Qν ξ ks R ξ ks r n s, ξ r n 1 s, ξ s 1θ1ε ds 1 = θc 2 C 3 Q ν ξ kρ Q ν ξ ks r n s, ξ r n 1 s, ξ G ρ s,ξds. In other words, 1 Q ν ξ kρ r n1 ρ, ξ r n ρ, ξ 1 θc 2 C 3 Q ν ξ ks r n s, ξ r n 1 s, ξ G ρ s,ξds On the other hand, we estimate r 1 as follows. Using 2.8, then 2.7, 2.21, and Bρ C 4 ρ 1θ1ε for some C 4 > thanks to 2.16, yields r 1 ρ, ξ θ K ρ,s; ξ Bs J ν1 ξ ks ξks ds 1 θc 2 Q ν ξ kρ Qν ξ ks R ξ ks J ν1 ξ ks Bs ds ξks θc 1 C 2 C 4 Q ν ξ kρ S ξ ks s 1θ1ε ds Q ν ξ kρ Q ν ξ kρ θc 1 C 2 C 3 C 4 ξ min2,1ε =: C 5, 2.23 ξ min2,1ε where we used 2.2 for the last inequality. By induction on n, it follows from 2.21 and 2.23 that r n1 ρ, ξ r n ρ, ξ Q ν ξ kρ C6 Gρ, ξ n C 5 ξ min2,1ε, 2.24 n! where C 6 := θc 1 C 3. We deduce from 2.24 that {r n } is a Cauchy sequence on the space of uniformly bounded and continuous functions on each compact subset

9 Isentropic Euler Equations 89 in ρ and that rρ,ξ r n1 ρ, ξ r n ρ, ξ n= C 5 Q ν ξ kρ ξ min2,1ε C6 Gρ, ξ n n= n! = C 5 Q ν ξ kρ ξ min2,1ε e C 6 Gρ,ξ. Since, by 2.19, the function G is uniformly bounded and the first part of 2.12 is now established. Finally, differentiating 2.18, we observe that the derivative r ρ satisfies r ρ ρ, ξ = θ K ρ ρ,s; ξ Bs J ν1 ξks rs, ξ gs ds, ξks which has the same form as the formula satisfied by r. It is straightforward to check from 2.5 and 2.6 that, instead of 2.8, we now have the bound k ρ K ρ ρ,s; ξ C kρ Q 1 ν ξ kρ Qν ξ ks Rξ ks for s ρ, ξ R. By a direct estimate, we arrive at 2.12 for r ρ. We now translate the above result into the original velocity variable. Because of the hyperbolic nature of the problem 2.1, the support of the entropy kernel should be the domain of dependence K := { ρ, v : ρ, v kρ }. Furthermore, since the initial data is singular, the entropy kernel is not smooth but its derivative contains some singularities localized on the boundary K = { ρ, v : v ± kρ = }. The expansion derived now for the entropy kernel is based on the two functions ρ, v [ kρ 2 v 2] µ for µ = λ, λ 1. Theorem 2.2. Assume that the pressure function p = pρ satisfies 1.2hyperbolicity and 2.1 behavior near the vacuum for some γ 1, and ε>. Then the problem 2.1 admits a solution χ = χρ,v, supported in the set K and smooth in its interior. For each µ<λ, the function [ kρ 2 v 2] µ χρ,v is Hölder continuous in ρ, v with [kρ v λ µ 2 v 2] µ χ C kρ 2 λ µ More precisely, the entropy kernel admits the expansion χρ,v = a ρ [ kρ 2 v 2] λ a ρ [ kρ 2 v 2] λ1 eρ, v, 2.26

10 9 Gui-Qiang Chen & Philippe G. LeFloch where and a := c λ k λ k 1/2 c λ with 1 2 λ 1 = 1 y 2 λ dy 2.27 c λ 2 λ 1 a := 4λ 1 k λ 1 k 1/2 k λ 1 k 1/2 k λ1 k 1/2 ds The remainder e = eρ, v is such that v µ e is Hölder continuous in ρ, v for ρ>for all µ with 1 min2, 1 ε<µ<λ min2, 1 ε and, for all <β<µ, v β eρ, v Cρ µθβθ1 θ/2 [ kρ 2 v 2] µ β Furthermore, if the genuine nonlinearity condition 1.6 holds, the function χ is positive in the interior of its support. We point out that, when θ, 1, that is, γ 1, 3 which is the interval of interest in Theorem 1.1 and in [2], the derivative v λ1 eρ, v of the remainder is Hölder continuous up to the boundary ρ =. For instance, in the estimate 2.29 with the exponent 1 θ/2 >, we see that the factor ρ µθβθ1 θ/2 vanishes as ρ when β = λ 1, provided we choose µ sufficiently close to λ 1. As a corollary, the family of weak entropies for the Euler equations is described by the formula ηρ, v = χρ,v sψsds, R where ψ : R R is arbitrary. Proof. In Theorem 2.1, we established the existence of the solution ˆχ =ˆχρ,ξof the problem 2.2, given by the expansion According to 2.12, the following estimate holds for the remainder: { rρ,ξ C ξ min2,1ε ξ kρ ν for kρ ξ 1, C ξ min2,1ε ξ kρ 1/2 2.3 for kρ ξ 1. Considering the expansion 2.11, we see that the remainder rρ,ξ decays at infinity like ξ 1/2 min2,1ε and therefore faster than the second term J ν1 ξkρ /ξ which is solely ξ 3/2 according to 2.5. This allows us to treat rρ,ξ as a higher-order term. Using the relation between J ν and fˆ λ and observing that the corresponding function f λ is closely related to the function [ kρ 2 v 2] λ, we now apply the inverse Fourier transform to Referring to [2] for the details, we find exactly 2.26 in which eρ, v is determined from the remainder rρ,ξ by θ kρ 1/2 eρ, v = C k cos ξv rρ,ξ ξ ν dξ, 2.31 ρ R 1

11 Isentropic Euler Equations 91 up to some multiplicative constant C>. Given some µ>, by the definition of fractional derivatives we have v µ eρ, v θ kρ 1/2 C k ξ µ ν rρ,ξ dξ. ρ R Then, a straightforward calculation based on 2.3 yields the Hölder bound v µ eρ, v Cρ 1 θ/2, 2.32 provided 1 min2, ε/θ <µ<λ min2, ε/θ. Similarly, from 2.12, we can also obtain ρ ρ v µ eρ, v Cρ 1 θ/2. By a standard embedding theorem Wρ 1,p Cρ,α, we conclude that µ v e are Hölder continuous in ρ, v. Finally, introducing the variable z := v/kρ, we see that the function [1 z 2 ] is positive on the support of K.Wehave z µ eρ, v = kρ µ v µ eρ, v Cρ µθ1 θ/2. The function e = eρ, v is Hölder continuous with exponent µ and vanishes outside K. Therefore, we have eρ, v C sup µ z eρ, v [1 z 2 ] µ C 1 ρ µθ1 θ/2 [1 z 2 ] µ C 2 ρ µθ1 θ/2 [ kρ 2 v 2] µ, which gives 2.29 for β =. More generally, for β<µ, we obtain v β eρ, v Cρβθ sup µ z eρ, v [1 z 2 ] µ β C 1 ρ βθµθ1 θ/2 [ kρ 2 v 2] µ β, which gives This completes the proof of Theorem 2.2. z z 3. A multiple-term expansion We now turn to the general situation of a pressure function p = pρ satisfying the following: There exists a sequence of exponents 1 <γ := γ 1 <γ 2 < <γ N 3γ 1/2 <γ N1 3.1 and a sufficiently smooth function P = Pρsuch that N pρ = κ n ρ γ n ρ γ N1 Pρ, n=1 Pρ and ρ 3 P ρ are bounded as ρ, 3.2

12 92 Gui-Qiang Chen & Philippe G. LeFloch for some coefficients κ n R, where κ 1 := γ 1 2 /4γ after normalization. From the exponent γ we also define θ, λ, and ν as in Section 2, so that the reminder in 3.2 takes the same form as we had in 2.1. The two-term expansion derived in Section 2 is no longer valid when, besides the singularity with exponent γ, the pressure contains singularities with exponents strictly less than 3γ 1/2. Instead, we must introduce the Bessel functions of exponents ν i, i = 1,...,I, ordered in increasing order and determined in the following way. Set { { } N } µ1,µ 2,...,µ I =[,θ] e n γ n γ 1 : e n =, 1,... n=1 and then ν i 1 2 := λ i := λ µ i θ for i = 1,...,I. 3.3 Clearly, µ 1 = and λ = λ 1, ν = ν 1. But, it is important to notice that the coefficients ν i are not associated with the coefficients γ i directly by the same formulas as in Section 2. Finally, we denote by µ I1 the smallest of n e n γ n γ 1 which is greater than θ, and we set µ I1 = θ1 ε. To motivate the introduction of the exponents µ i, we point out that 3.2 implies that, for instance, I cρ = ρ θ q i ρ µ i ρ µ I1 Qρ, where q i,1 i I, are constants and Q is a more regular function. The function k defined as before by cy kρ := y dy admits a completely similar expansion. We now prove the following theorem. Theorem 3.1. Suppose that the function p = pρ satisfies the assumption 3.2 for some sequence of exponents satisfying 3.1. Then there exists a solution of the problem 2.2, ˆχ =ˆχρ,ξ, defined for ρ and ξ R. It is smooth for ρ>, continuous as ρ, and is given by the expansion θ kρ 1/2 ˆχρ,ξ = k ρ I K i J νi ξ kρ ξ ν i Nρ J ν1 ξ kρ ξ ν1 rρ,ξ, 3.4 where the function N = Nρ was already defined in Theorem 2.1, and the constants K i are determined in the proof below with K 1 = 1. The remainder also satisfies the estimate 2.12 in Theorem 2.1.

13 Isentropic Euler Equations 93 Proof. We follow the same strategy as in the proof of Theorem 2.1. As before, we take ξ>. Now, the remainder satisfies rρ,ξ := k ρ 2 ξ 2 βρ r θ kρ 1/2 k rρ,ξ ρ r ρρ = βρ r H ρρ k ρ 2 ξ 2 I i=2 θ kρ 1/2 J νi ξ kρ K i k ρ ξ ν i θ kρ 1/2 I = βρ r H k K i β i ρ J ν i ξ kρ ρ ξ ν i i=2 =: βρ r H, 3.5 where the functions β i = β i ρ are defined by β i := k λ i 1 k 1/2 k λ i1 k 1/ We now rely on the expression of the function H in the proof of Section 2. Using the assumption 3.2, we have Aρ = I q i ρ µ i 1 ρ µ I1 1 Qρ 3.7 for some q i and Q. Since, by 2.5, all of J δ ξ kρ / kρξ δ are equivalent to 1 as ρ, the behavior of the function Hρ,ξnear the vacuum is given by Eρ := Aρ c,λ1 = Aρ c,λ1 θ 1/2 θ kρ 1/2 I k K i β i ρ kρ ν i ρ i=2 I i=2 K i k λ i1 k 1/2. Now, again using the assumption 3.2, we find k λi1 k 1/2 I = ρ θλ i θ 3/2 κ j ρ µ j ρ µ I1 Qρ 3.8 for some constants q j and some regular function Q, with in particular q 1 = θ 1/2 θλ i θ 1/2 θλ i θ 3/2 =. Furthermore, by our definition 3.3, we have exactly θλ i θ 3/2 = µ i 1. j=1

14 94 Gui-Qiang Chen & Philippe G. LeFloch That is, the exponents in the expansion 3.7 coincide with the principal exponent arising in 3.8 for i = 1,...,I. This implies that, by a suitable choice of the constants K i, we can remove all of the singularities in the function E so that Returning to the function H, we conclude that Eρ = Oρ µ I Hρ,ξ { Cρ µ I1 1 for kρξ 1, C kρξ 3/2 for kρξ The rest of the proof follows as in Section 2, by using the following integral formulation for the solution of 2.14: rρ,ξ = θkρ 1/2 K ρ,s; ξ k ρ Ws,ξ θks 1/2 k Hs,ξ βs rs,ξ ds, s where the kernel K was defined earlier. In other words, we have rρ,ξ = θ K ρ,s; ξ ks k Hs,ξ s rs, ξ gs ds The only new feature is the treatment of the term r 1. We now use 3.1 and, in fact, arrive at exactly the same estimate. This completes the proof of Theorem 3.1. From Theorem 3.1, we deduce the existence of the entropy kernel. Theorem 3.2. Assume that the pressure-function p = pρ satisfies 1.2hyperbolicity and 3.1, 3.2behavior near the vacuum. Then the problem 2.2 admits a solution χ = χρ,v, supported in the set K and smooth in its interior. For each µ<λ, the function [ kρ 2 v 2] µ v λ µ χρ,v is Hölder continuous in ρ, v with [kρ 2 v 2] µ χ C kρ 2 λ µ More precisely, the entropy kernel admits the expansion where χρ,v = I a i ρ [ kρ 2 v 2] λ i a ρ [ kρ 2 v 2] λ1 eρ, v, 3.13 a i ρ := K i kρ λ i k ρ 1/ and a ρ was defined in Theorem 2.2. Furthermore, all of the regularity properties stated in Theorem 2.2 hold. Finally, we can also study the entropy kernel.

15 Isentropic Euler Equations 95 Theorem 3.3. Under the assumption of Theorem 3.2, there exists an entropy-flux kernel σ = σρ,v, supported in the set K and smooth in its interior. For each µ<λ, the function [ kρ 2 v 2] µ σρ,vis Hölder continuous in ρ, v with [kρ v λ µ 2 v 2] µ χ C kρ 2 λ µ More precisely, the entropy kernel admits the expansion where σρ,v v χρ, v = I b i ρ [ kρ 2 v 2] λ i b ρ [ kρ 2 v 2] λ1 b i ρ := a i ρ ρk ρ kρ fρ,v, and b = b ρ is some function of ρ. Furthermore, the regularity results stated in Theorem 2.2 for the entropy kernel carry over to the entropy-flux kernel. 4. Existence theory Our main existence result is the following. Theorem 4.1. Consider the isentropic Euler equations 1.1 under the assumptions 1.2 hyperbolicity, 3.1, 3.2 behavior near the vacuum, and 1.7 genuine nonlinearity with γ 1, 3. Then, given any measurable and bounded initial data ρ,v at time t =, there exists a corresponding entropy solution ρ, m = ρ, ρv of the Cauchy problem associated with 1.1, globally defined in time, such that ρx,t C and mx, t C ρx, t for some constant C> and for almost every x, t. Recall that 2.1 is just a special case of 3.1, 3.2. Note also that the condition γ < 3 and the genuine nonlinearity condition 1.3 are assumed in this section only. Proof. Most of the arguments in [2] carry over with the more general expansion discovered in Section 3. Thus, we only need here to discuss some new observations required in the central part of the proof, i.e., the reduction of the Young measures. Denote by ν = ν x,t ρ, v a Young measure associated with the sequence of approximate solutions to 1.1. It is known that ν satisfies Tartar s commutation relations at almost every point x, t: ν, η1 q 2 η 2 q 1 = ν, η1 ν, q2 ν, η2 ν, q1 4.1 for any two weak entropy pairs η 1,q 1 and η 2,q 2 of 1.1. The objective is to deduce from 4.1 that the support of ν in the ρ, v-plane is either a single point or

16 96 Gui-Qiang Chen & Philippe G. LeFloch a subset of the vacuum line { ρ = }. To this end, the cancellation properties stated in Lemmas 4.2 and 4.3 in [2] play the central role. Using the notation in [2], in particular, G λi ρ, v := [ kρ 2 v 2] λ i, we reconsider the distribution in the variables ρ,v,s 1,s 2,s 3 : E := 2 λ1 χ 2 3 λ1 σ 3 3 λ1 χ 3 2 λ1 σ 2 = 2 λ1 χ 2 3 λ1 σ3 vχ 3 λ1 3 χ 3 2 λ1 σ2 vχ 2 I = λ1 2 λ1 3 λ1 3 λ1 2 a i G λ i,2 a G λ1,2 g 2 I s 3 v b i G λ i,3 b G λ1,3 h 3 I a i G λ i,3 a G λ1,3 g 3 I s 2 v b i G λ i,2 b G λ1,2 h 2, where for instance χ 1 := χρ,v s 1, etc. Thus, we have E = I a i λ1 2 G λi,2 a λ1 2 G λ1,2 λ1 2 g 2 I s 3 v b i λ1 3 G λi,3 b 3 λ1 G λ1,3 I 3 λ1 h 3 λ 1 b i λ 3 G λ i,3 λ 1b 3 λ G λ1,3 I a i λ1 3 G λi,3 a λ1 3 G λ1,3 λ1 3 g 3 I s 2 v b i λ1 2 G λi,2 b 2 λ1 G λ1,2 I 2 λ1 h 2 λ 1 b i λ 2 G λ i,2 λ 1b 2 λ G λ1,2 =:E I E II E III.

17 Isentropic Euler Equations 97 To decompose E, we rely on the crucial observation that, by 3.17, the ratios b i /a i are independent of the index i. We define I I E I := s 3 s 2 a i λ1 2 G λi,2 b i λ1 3 G λi,3, E II := I a i λ1 2 G λi,2 s 3 v b 3 λ1 G λ1,3 λ 1 I a I a i λ1 3 G λi,3 s 2 v b 2 λ1 G λ1,2 λ 1 b i λ1 2 G λi 1,2s 3 v λ1 3 G λ,3 λ1 3 G λ1,3 s 2 v λ i1 2 G λ,2, I b i λ 3 G λ i,3 I b i λ 2 G λ i,2 and E III is the remainder. We must determine the limit of the first two terms as s 2,s 3 s 1 in the sense of distributions. Dealing with E III is easy since it involves only products of Hölder-continuous functions by measures or more regular products. By symmetry, we get E III ass 2,s 3 s weakly in the sense of distributions, uniformly in ρ, v. On the other hand, E I contains the favorable factor s 2 s 3. Therefore, by the arguments developed in [2], we have again E I ass 2,s 3 s weakly in the sense of distributions and uniformly in ρ, v. Finally, the term E II contains products of functions of bounded variation by bounded measures. However, it is not hard to see that only the specific product treated in [2] generates a non-trivial contribution. The reason is that the terms containing G λi for i = 1 are more regular than those associated with the first term G λ1 = G λ. This completes the proof of Lemma 4.3 in [2] for the multiple-term expansion and, in turn, establishes the reduction theorem for Young measures. By relying on the framework developed in [2], this also completes the proof of Theorem 4.1. Acknowledgements. Gui-Qiang Chen was supported in part by NSF grants DMS-24225, DMS , and INT Philippe G. LeFloch was supported in part by NSF grant INT and the Centre National de la Recherche Scientifique CNRS. This work was done in August 2 while P.G.L. was visiting the Department of Mathematics at Northwestern University.

18 98 Gui-Qiang Chen & Philippe G. LeFloch References 1. Chen, G.-Q.: Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics III. Acta Math. Sci. 6, in English; 8, in Chinese 2. Chen, G.-Q., LeFloch, P.G.: Compressible Euler equations with general pressure law. Arch. Rational Mech. Anal. 153, Chen, G.-Q., LeFloch, P.G.: Entropy and entropy-flux splittings for the isentropic Euler equations. Chinese Annals Math. 22B, Chen, G.-Q., LeFloch, P.G.: in preparation 5. Ding, X., Chen, G.-Q., Luo, P.: Convergence of the Lax-Friedrichs scheme for the isentropic gas dynamics I II. Acta Math. Sci. 5, 483 5, in English; 7, ; 8, in Chinese 6. DiPerna, R.J.: Convergence of the viscosity method for isentropic gas dynamics. Commun. Math. Phys. 91, Gelfand, I.M., Shilov, G.E.: Generalized functions. Vol. I, Academic Press Lions, P.-L., Perthame, B., Tadmor, E.: Kinetic formulation for the isentropic gas dynamics and p-system. Commun. Math. Phys. 163, Lions, P.-L., Perthame, B., Souganidis, P.E.: Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Comm. Pure Appl. Math. 49, Tartar, L.: The compensated compactness method applied to systems of conservation laws. In Systems of Nonlinear Partial Differential Equations, J.M. Ball ed., NATO ASI Series, C. Reidel publishing Col., pp Department of Mathematics Northwestern University Evanston, Illinois USA gqchen@math.northwestern.edu and Centre de Mathématiques Appliquées & Centre National de la Recherche Scientifique UA 756, Ecole Polytechnique Palaiseau Cedex France lefloch@cmap.polytechnique.fr Accepted June 1, 22 Published online December 3, 22 Springer-Verlag 22