Compactness Methods and Nonlinear Hyperbolic Conservation Laws

Transcription

1 Compactness Methods and Nonlinear Hyperbolic Conservation Laws Gui-Qiang Chen Department of Mathematics, Northwestern University Evanston, IL , USA Contents 1. Introduction Compactness Methods and Conservation Laws BV Compactness L 1 Compactness Compensated Compactness Compactness and Large-Time Behavior Problems Compensated Compactness and Conservation Laws Scalar Conservation Laws Strictly Hyperbolic Systems Nonstrictly Hyperbolic Conservation Laws Isentropic Euler Equations Hyperbolic Conservation Laws with Umbilic Degeneracy Compressible Euler Equations Non-isentropic Euler Equations Multi-D Compressible Euler Equations with Geometric Structure Hyperbolic Systems of Conservation Laws with Relaxation 63 Acknowledgments 69 References 70 c 0000 (copyright holder)

2 34 Compactness Methods and Nonlinear Hyperbolic Conservation Laws 1. Introduction We are concerned with a hyperbolic system of conservation laws: t u + x f(u) =0, x R d,u R n, (1.1) where f =(f 1,,f d ):R n (R n ) d is a nonlinear mapping. Since the conservation law is a fundamental law of nature, most of partial differential equations arising from physical or engineering science can be formulated into form (1.1) or its variants (with dissipation, relaxation, memory, damping, dispersion, etc.). The hyperbolicity of system (1.1) requires that, for all ξ =(ξ 1,,ξ d ) S d 1, the matrix ( d j=1 ξ j f j (u)) n n have n real eigenvalues λ j (u, ξ),j =1,2,,n,and be diagonalizable. When d = n = 1, it is a scalar equation with λ = f (u), which is alway hyperbolic. A system is called nonstrictly hyperbolic or resonant, if there exist some ξ 0 S d 1 and u 0 R n such that λ i (u 0,ξ 0 )=λ j (u 0,ξ 0 )forsomei j. Such a degeneracy is quite generic. For example, when d = 3,n = 2 (mod4), Lax [La2] showed that any system must be nonstrictly hyperbolic. The same situation occurs for n = ±2, ±3, ±4(mod 8) (see [FRS]). Since f(u) is a nonlinear mapping, solutions of the Cauchy problem (even starting from smooth initial data) generally develop singularities in a finite time: T<+, x u(t, x), when t T, and then the solutions become discontinuous functions. This feature reflects the physical phenomenon of breaking of waves and development of shock waves. For this reason, attention focuses on solutions in the space of discontinuous functions. Therefore, one can not directly use the classical analytic techniques that predominate in the theory of partial differential equations of other types. To overcome this difficulty in constructing global discontinuous solutions of the Cauchy problem of system (1.1) with initial data: u t=0 = u 0 (x), (1.2) one natural strategy, motivated from physics, is to construct approximate solutions u ε (t, x) via the singular perturbation methods as the following. (1). Viscosity method: Add the term ε x (D(u) x u) on the right hand side of (1.1). Usually one chooses D(u) the unit matrix. Then u ε (t, x) are generated by the parabolic equations. (2). Numerical methods: u ε (t, x) are generated by the difference equations: D t u ε + D x f(u ε ) 0, ε = t, or other efficient shock capturing numerical methods, associated with (1.1). Other perturbation methods include the relaxation methods and the kinetic methods to construct approximate solutions u ε (t, x). As an example, we consider u ε (t, x), generated by the following Cauchy problem: t u ε + x f(u ε )=ε x u ε, (1.3) with Cauchy data: u ε t=0 = u ε 0 (x) u 0(x) L, a.e. ε 0, (1.4)

3 Gui-Qiang Chen 35 satisfying u ε L C, C independent of ε. (1.5) Assume that u ε (t, x) are compact, that is, there exists a subsequence {u ε k (t, x)} k=1 such that u ε (t, x) u(t, x), a.e. (1.6) Let (η, q) :R n R R d be an entropy-entropy flux pair (an entropy pair, for short), determined by the linear hyperbolic systems: q k (u) = η(u) f k (u), k =1,2,d. (1.7) Taking any convex entropy η(u), i.e. 2 η(u) 0, and multiplying η(u ε )both sides of (1.3), we have t η(u ε )+ x q(u ε )=ε x η(u ε ) ε( x u ε ) 2 η(u ε ) x u ε ε x η(u ε ). (1.8) Then (η(u ε ) t ϕ +q(u ε ) x ϕ)dxdt + η(u ε 0 (x))ϕ(0,x)dx ε η(u ɛ (1.9) ) x ϕdxdt, for any nonnegative test function ϕ(t, x) C0 (R d+1 + ). Take ε = ε k 0,k,and use (1.6) together with (1.5). We conclude (η(u) t ϕ+q(u) x ϕ)dxdt + η(u 0 (x))ϕ(0,x)dx 0, (1.10) for any convex entropy pair (η, q) and any nonnegative test function ϕ(t, x) C0 (R d+1 + ). This implies that the limit function u(t, x) is an entropy solution of (1.1)-(1.2). From the discussion above, it is clear that the essential problem in this strategy is to study the compactness of approximate solutions with respect to ε 0in order to obtain global entropy solutions. That is, whether one can obtain the strong convergence of a subsequence of u ε (t, x). The compactness problem is also closely related to the large-time behavior problems for entropy solutions (see 2.4 and Chen-Frid [CF1-5]). From our further discussions, we will find that the compactness plays indeed an important role in hyperbolic conservation laws. In these notes, we focus on some basic ideas, methods, and approaches via some prototypical examples for solving nonlinear hyperbolic conservation laws. Many important results in compactness methods and various topics in conservation laws are not covered; and some of them can be found in the references cited here. In Section 2, we describe several important compactness frameworks and discuss their important applications to the existence and asymptotic behavior of entropy solutions. As examples, scalar conservation laws and 2 2 strictly hyperbolic systems are analyzed to show how the compactness of exact and approximate solutions is established. In Section 3, we discuss how the compactness methods can be applied to solving important nonstrictly hyperbolic systems of conservation laws, including the isentropic Euler equations and the quadratic systems with umbilic degeneracy. In Section 4, we analyze some further results on the compressible Euler equations with the aid of the compactness methods, in addition to those in Section 3.

4 36 Compactness Methods and Nonlinear Hyperbolic Conservation Laws In Section 5, we discuss the stability and behavior of singular limits of the zero relaxation time for hyperbolic systems of conservation laws with relaxation.

5 Gui-Qiang Chen Compactness Theories and Conservation Laws In this section, we describe several important compactness frameworks and discuss their immediate applications to nonlinear hyperbolic conservation laws BV Compactness. If a sequence of functions u ε (t, x) satisfies u ε L C u 0 L, TV(u ε ) CTV(u 0 ), where C is independent of ε, then there exists a subsequence u ε k such that u ε k (t, x) u(t, x), a.e., when k, by the Helly theorem. The role of these norms is indicated by Glimm s theorem [Gl] for hyperbolic systems (also see [Ol,Vo,CS]) L 1 Compactness. If a sequence of functions u ε satisfies (i). u ε L 1 loc C, C>0independent of ε; (ii). {u ε (t, x)} ε>0 is equicontinuous in L 1 loc (Rd+1 + ), that is, for any compact subset Ω R d+1 +, u ε (t+ t, x + x) u ε (t, x) dxdt Ω uniformly tends to zero as t, x 0. Then there exists a subsequence u ε k such that u ε k (t, x) u(t, x), L 1 loc (Rd+1 + ). The role of the L 1 norm for the compactness is indicated by Kruzkov s theorem [Kr] for scalar conservation laws Compensated Compactness. Let Ω R d+1 + be a bounded open set. Denote y =(t, x) Ω. Assume that a sequence of functions u ε :Ω R n satisfies and u ε d A 0 t + u ε A j = g ɛ (t, x) x j j=1 u ε u L (Ω), (2.1) is compact in H 1 loc (Ω), (2.2) for some m n matrices A j,j =0,1,,d. Then (2.2) can be regarded as m first order differential equations for the n unknowns. A mathematical question is whether f(u ε ) f(u) subsequentially in the sense of distributions for any smooth nonlinear function f. If (2.2) were elliptic, then the weak convergence would be equivalent to the convergence in L 2 loc. However, this is not the case in general. For example, u ε (t, x) =sin ( ) t+x ε,f(u)=u 2. It is well known that u ε 0and(u ε ) Nevertheless, we have the following theorem.

6 38 Compactness Methods and Nonlinear Hyperbolic Conservation Laws Theorem 2.1 [Ta1]. Let u ε :Ω R 4 be measurable functions satisfying w lim u ε = u, in L 2 4(Ω), u ε 1 t + uε 2 x, u ε 3 t + uε 4 are compact in H 1 loc x (Ω). Then there exists a subsequence (still labeled) u ε such that uε 1 u ε 2 u ε 3 u ε u 1 u 2 4 u 3 u 4 in the sense of distributions. Theorem 2.2 [Ta1]. Let K R n be a bounded open set and u ε :Ω K.Then there exists a family of probability measures {ν y (λ) prob.(r n )} y Ω such that supp ν y K, a.e. y Ω, and, for any continuous function f, there is a subsequence u ε k satisfying f(u ε k ) f(λ)dν y (λ) =<ν y,f >. R n As a corollary, we have Theorem 2.3. A uniformly bounded sequence u ε converges to u a.e. in Ω subsequentially if and only if the corresponding family of Young measures ν y reduces almost all points y to a family of Dirac masses concentrated at u(y), i.e. ν y = δ u(y). Proof. Suppose that the uniformly bounded sequence u ε in L, u ε L M, converges to u a.e. Then, for any f C(R n ), there exists a subsequence {u ε k } k=1 {u ε } ε>0 and a family of Young measures {ν y Prob.(R n )} y Ω, supp ν y {λ λ M}, such that f(u ε k (y)) ν y (λ),f(λ), f(u ε k (y)) f (u(y)), a.e. y Ω. The uniqueness of the weak limits implies ν y (λ),f(λ) =f(u(y)) = δ u(y) (λ),f(λ), a.e. for any f C(R n ). This means that ν y (λ) =δ u(y) (λ), a.e. On the other hand, if the Young measures ν y (λ) =δ u(y) (λ), then there exists a subsequence {u ε k } k=1 such that (u ε k ) 2 u 2, u ε k u. Then (u ε k u) 2 dy = ((u ε k ) 2 2uu ε k + u 2 )dy 0, k. Ω Ω Using the uniform boundedness of u ε k (y), one immediately has u ε k u L p (Ω) 0, k. Remark 1. The Young measure ν y can be also understood as the limiting probability distribution of the value of {u ε (y)} near the point y as ε (see Ball [Ba]). Set <νy,r ε,f 1 > = f(u ε (ξ))dξ, f C c (R n ). B(y, r) B(y,r)

7 Gui-Qiang Chen 39 Then νy,r ε 1. Therefore, there exist a subsequence {ε k} k=1 and a measure ν y,r such that ν ε k y,r ν y,r, that is, 1 <ν y,r,f >= <ν ξ,f >dξ. B(y, r) B(y,r) Assume that y is an Lebesgue point. Then <ν y,r,f > <ν y,f >, r 0. Remark 2. The deviation between the weak and strong convergence is measured by the spreading of the support of ν y.iffis a Lipschitz continuous function, then f(ω lim u ε ) ω lim f(u ε ) L = f(< ν y,λ>) <ν y,f(λ)> L f Lip <ν y,λ <ν y,λ>> L f Lip sup(diam(supp ν y )). y The following compactness interpolation is very useful for obtaining the H 1 loc compactness needed in Theorem 2.1. Theorem 2.4. Let q>1andr (q, ] are constants, and p [q, r). Then (compact set of W 1,q loc (compact set of W 1,p loc (Ω)) (bounded set of W 1,r loc (Ω)). (Ω)) This is a generalization of Murat s lemma [Ta1,Mu1]. The proof is simple and can be found in [DCL1,Ch4]. Some more general results and detailed proofs can be found in Tartar [Ta1-4] and Murat [Mu1-4]. Other related references include Ball [Ba], Chen [Ch4], Coifman-Lions-Meyer-Semmes [CLM], Dacorogna [Da], Evans [Ev1], Hörmander [H1], Morrey [Mo1-2], Struwe [St], Young [Yo], and the references cited therein Compactness and Large-Time Behavior Problems. It is clear from the discussions above that the compactness plays an essential role in constructing global entropy solutions. In fact, it is also very important for the large-time behavior problems. One of the main asymptotic problems is the decay of periodic entropy solutions, which is an important nonlinear phenomenon. For the linear case, initial oscillations propagate, and there is no decay. However, this is not the case for systems with certain nonlinearity. For one-dimensional convex scalar conservation laws (f (u) 0), Lax [La1] first showed the decay of periodic solutions with period P : u(t, x) u C t, with u = 1 u 0 (x)dx. P P Glimm-Lax s theory [GL] indicates the decay of periodic solutions obtained through Glimm s method for 2 2 strictly hyperbolic and genuinely nonlinear systems for initial data of small oscillation. Using the generalized characteristics, Dafermos [Da2] showed for such systems that any periodic entropy solution, with local bounded variation and small oscillation, asymptotically decays.

8 40 Compactness Methods and Nonlinear Hyperbolic Conservation Laws In this section, with the aid of compactness arguments, we discuss a new analytic approach to study the asymptotic decay of L periodic solutions of hyperbolic systems of conservation laws, without the restrictions of either smallness or local bounded variation on the L periodic initial data and specific reference to any particular method for constructing the entropy solutions. Let u(t, x) be a period entropy solution for the Cauchy problem (1.1)-(1.2). Then we have the following decay framework. Theorem 2.5 [CF1-2]. Consider system (1.1) endowed with a strictly convex entropy η (u). Let u(t, x) L (R d+1 + ) be an entropy solution of (1.1)-(1.2), which is periodic in x with period P, and the corresponding self-similar scaling sequence u T (t, x) u(tt,tx) is compact in L 1 loc(r d+1 + ). (2.3) Then the periodic solution u(t, x) asymptotically decays in L p to the average ū of its initial data over the period: esslim u(t, x) u t P p dx, for any 1 p<, (2.4) with ū = 1 P P u 0(x)dx. Sketch of Proof. We assume P =[0,1] d for simplicity. We divide the proof into several steps. Step 1. Since u T (t, x) is compact in L 1 loc (Rd+1 + ), there exists a subsequence ut k (t, x) converging to some function u(t, x) L (R d+1 + )inl1 loc (Rd+1 + ). Then we conclude that u(t, x) =u(t) from the periodicity of u T k (t, x). Notice that t u T k (t, x)+ x f(u T k (t, x)) = 0 in the sense of distributions. Setting k,wehave t u(t)=0, that is, u(t) t ϕdxdt + uϕdx =0, ϕ C 0 (R d+1 + ), whereweusedthat u 0 (Tx) u 0 (x)dx u, P since u 0 (x) is periodic. This implies that u(t) =u= P u 0(x)dx. Since the limit is unique, the whole sequence u T (t, x) strongly converges to u in L 1 loc (Rd+1 + )when T. Therefore, we have 1 T T d+1 u(t, ξt) u p t d dξdt 0 ξ r 1 (2.5) = u T (t, x) u p dxdt 0, when T. 0 x rt Step 2. The periodic entropy solution u(t, x) satisfies the entropy inequality (1.10) for the strictly convex entropy pair (η,q ). In (1.10), we use η (u, u) =η (u) η (u) η (u)(u u) 0, q (u, u) =q (u) q (u) η (u)(f(u) f(u)),

9 which is also a strictly convex entropy pair. a R d, η (u(t 2,x),u)dx a+p Gui-Qiang Chen 41 a+p Then we can deduce that, for any η (u(t 1,x),u)dx, (2.6) for all 0 t 1 <t 2,t 1,t 2 (0, ) T,where meas(t )=0. Step 3. Given T > 0,T (0, ) T,wetakealltherightn-prisms given by x a+p,fora Z d,andt [[rt ]/(2r),T], in the interior of the cone { x rt 0 t T }. The number of such prisms is larger than [rt ] d. Using the periodicity of u(t, x), inequality (2.6) with t 2 = T, which holds for a.e. t 1 = t (0,T)overthe period P, and the strict convexity of the entropy η, we obtain that there exists c 0 > 0, independent of T,suchthat 1 T T d+1 0 x rt η (u(t, x), ū)dxdt [rt]d T T d+1 [rt ] 2r P η (u(t, x), ū)dxdt [rt]d T T d+1 [rt ] 2r P η (u(t,x),ū)dxdt (2.7) c 0 P u(t,x) ū 2 dx, whereweused[a] as the largest integer less than or equal a. Noting the uniform boundedness of the periodic solution, we have 1 T T d+1 0 x rt η (u(t, x), ū)dxdt C1 T T d+1 0 x rt u(t, x) ū 2 dxdt C2 T T d+1 0 ξ r u(t, ξt) ū 2 t d dξdt 0, T +. (2.8) Combining (2.7) with (2.8), we obtain u(t,x) ū 2 dx 0, P when T,T (0, ) T. The boundedness of the periodic solution and the Hölder inequality yield (2.4) for all p [1, ). This completes the proof of Theorem 2.5. Remark 1. In Theorem 2.5, we assume that the self-similar scaling sequence u T (t, x) is compact in L 1 loc (Rd+1 + ). This is a corollary of the compactness of solution operators for hyperbolic conservation laws. Remark 2. In Theorem 2.5, the assumption u(t, x) L (R d+1 + ) can be replaced by u(t, x) L q (R d+1 + ),q >2. Then the asymptotic decay of an Lq periodic entropy solution u(t, x) of (1.1)-(1.2), with period P, in the sense of (2.4) for p =2 implies (2.4) for any p [1,q). Remark 3. The decay in (2.5) implies T d +1 T d+1 u(t, ξt) u p t d dt 0, in L 1 loc (Rd ξ ), when T, (2.9) 0 which can be regarded as a decay definition in a weak sense (see [CF1]). Under this weak definition, the periodic solution decays in the sense of long time-average and (2.9) means <µ T ( ), u(,ξ ) u p > 0, in L 1 loc (Rd ξ ), when T,

10 42 Compactness Methods and Nonlinear Hyperbolic Conservation Laws where µ T (t) = d+1 X T d+1 [0,T ] (t)t d dt, which is a family of probability measures. We notice that limit (2.9) is also equivalent to 1 T T 0 u(t, ξt) u p dt 0, in L 1 loc (Rd ξ ), when T, (2.10) for the L periodic solution. In this case, the family of average measures is µ T (t) = 1 T X [0,T ](t)dt. The compactness is also important in solving other asymptotic problems such as the asymptotic stability of Riemann solutions. For the details, see Chen-Frid [CF3-5] with aid of the Gauss-Green formula for divergence-measure fields in L or BV (see [CF6]) Compensated Compactness and Conservation Laws. The next problem is how the compactness required for constructing entropy solutions and their asymptotic behavior (Theorem 2.5) can be achieved. From Murat s lemma [Mu1] and an idea in [Ch5] (also [Ch2]), we first have Theorem 2.6. Consider a hyperbolic system of conservation laws (1.1) with a convex entropy pair (η,q ). Assume that the uniformly bounded sequence u T (t, x) L (R d+1 + ) satisfies t η(u T )+ x q(u T ) 0 in the sense of distributions, for any convex entropy pair (η, q) Λ, where Λ is a linear space of entropy pairs of (1.1) including (η,q ). Then t η(u T )+ x q(u T ) is compact in W 1,p loc (R d+1 + ), p (1, ), (2.11) for any entropy pair (η, q) Λ satisfying 2 η C 2 η. This theorem is designed to prove the compactness of the self-similar scaling sequence u T (t, x) or the solution operator in L 1 loc (Rd+1 + ) with the aid of the compensated compactness methods. Consider the one-dimensional case. Assume that a sequence of functions u ε satisfies u ε L C, (2.12) and, for any entropy pair (η, q) C 2, t η(u ε )+ x q(u ε ) is compact in H 1 loc (R2 +), (2.13) where q = η f. The question is whether u ε (t, x) u(t, x), a.e., ε 0. From Theorem 2.2, there exist a family of probability measures {ν t,x (λ)} (t,x) R 2 + and a subsequence {ε k } k=1 such that (η j (u ε k (t, x)),q j (u ε k (t, x))) (< ν t,x,η j (λ)>, < ν t,x,q j (λ)>),j=1,2, η 1(u ε k (t, x)) q 1 (u ε k (t, x)) η 2 (u ε k (t, x)) q 2 (u ε k (t, x)) < ν t,x, η 1(λ) q 1 (λ) η 2 (λ) q 2 (λ) >, k. Therefore, we have from Theorem 2.1 that <ν t,x, η 1(λ) q 1 (λ) η 2 (λ) q 2 (λ) >= <ν t,x,η 1 (λ)> < ν t,x,q 1 (λ)> <ν t,x,η 2 (λ)> < ν t,x,q 2 (λ)>, (2.14) for any entropy pairs (η j,q j ) C 2,j =1,2.

11 Gui-Qiang Chen 43 To prove u ε (t, x) u(t, x), a.e., it suffices to reduce ν t,x to the Dirac mass δ u(t,x) (λ), for fixed a.e. (t, x) Scalar Conservation Laws. As the simplest example, we now show how the compactness can be achieved for scalar conservation laws t u + x f(u) =0. (2.15) Theorem 2.7. Assume that a sequence u ε (t, x) satisfies (2.12)-(2.13) only for two entropy pairs (η 1 (u),q 1 (u)) = (u k, f(u) f(k)), (η 2 (u),q 2 (u)) = (f(u) f(k), u k f (ξ) 2 dξ), k R. (2.16) Then (i). f(u) is weakly continuous subsequentially with respect to the sequence u ε. That is, there exists a subsequence {u ε k } k=1 {uε } ε>0 such that u ε k (t, x) u(t, x), f(u ε k (t, x)) f(u(t, x)), k. (ii). Furthermore, if there is no interval in which the flux function f(u) is linear, then the sequence u ε (t, x) is compact in L 1 loc (R2 +). Proof. From the framework discussed in 2.5, it suffices to study the Young measures ν t,x (λ) satisfying (2.14). Since the sequence u ε (t, x) satisfies (2.12), then there exists a subsequence {u ε k } k=1 {uε } ε>0 such that u ε k (t, x) u(t, x), f(u ε k (t, x)) ν t,x,f(λ), k. Then, for fixed (t, x) in (2.14), we take (η 1 (λ),q 1 (λ)) = (λ u(t, x),f(λ) f(u(t, x))), λ (η 2 (λ),q 2 (λ)) = (f(λ) f(u(t, x)), f (ξ) 2 dξ). u(t,x) From (2.14), we have λ u(t, x) f(λ) f(u(t, x)) <ν t,x, λ f(λ) f(u(t, x)) f (ξ) 2 > dξ u(t,x) 0 <ν t,x,f(λ) f(u(t, x)) > = λ <ν t,x,f(λ) f(u(t, x)) > < ν t,x, f (ξ) 2 dξ > u(t,x) Hence <ν t,x, (λ u(t, x)) λ u(t,x) + <ν t,x,f(λ) f(u(t, x)) > 2 =0. f (ξ) 2 dξ (f(λ) f(u(t, x))) 2 >. (2.17)

12 44 Compactness Methods and Nonlinear Hyperbolic Conservation Laws Since the first term of (2.17) is nonnegative from the Jensen inequality, we have <ν t,x,f(λ)>=f(u(t, x)) = f(< ν t,x,λ>), which implies Theorem 2.7(i). Furthermore, if there is no interval in which f(u) is linear, one obviously has from the first term of (2.17) that ν t,x = δ u(t,x), which yields Theorem 2.7(ii). Remark. The simple proof presented here is essentially from Chen-Lu [CLu] by using only two entropy pairs (2.16). It is actually very important for applications to use only the two entropy pairs for the reduction, such as in the convergence proof of high-order difference schemes (see [CL]) and zero relaxation limits (see [CLi,CLL]). A reduction proof by using infinite entropy pairs was first given by Tartar [Ta1]. Combining Theorem 2.7 with Theorems , we conclude Theorem 2.8. Consider scalar conservation laws (2.15) satisfying that there is no interval in which the flux function f(u) is linear. Then (i). The solution operator u(t, ) =S t u 0 ( ):L L exists and is compact in L 1 loc. (ii). Given any periodic initial data, there exists a unique periodic entropy solution. Moreover, any periodic entropy solution asymptotically decays. Proof. We divide the proof into three steps. Step 1. For any u 0 (x) L,set { 1, x m, u m 0 (x)=u 0(x)X m (x), X m (x)= (2.18) 0, x m. Then u m 0 (x) u 0 (x), a.e. Fix m Z +, the set of positive integers. Consider the Cauchy problem for the parabolic equations: { t u + x f(u) =ε xx u, (2.19) u t=0 = u m 0 (x), a.e. Then there exist global smooth solutions u ɛ m (t, x) of (2.19) satisfying u ɛ m L u m 0 L u 0 L, (2.20) by the maximum principle. Multiply both sides of the equations in (2.19) by η (u ε m) for any C 2 function η(u) (as an entropy). Then t η(u ε m )+ xq(u ε m )=ε xxη(u ε m ) εη (u ε m )( xu ε m )2, (2.21) where q(u) = u η (v)f (v)dv is the corresponding entropy flux. Take η(u)= u2 in (2.21) and integrate in [0, ) (, ). We have 2 ε x u 0 ( ε m )2 dxdt = 1 u m 0 2 (x)2 dx C m, (2.22) where C m > 0 is independent of ε. Notice that, for any φ(t, x) H0 1(Ω), Ω R2 +, ε xx η(u ε m)φdxdt Cε x u ε m x φ dxdt C m ε ε x u ε m L 2 (Ω) φ H 1 0 (Ω) C m ε φ H 1 0 (Ω),

13 Gui-Qiang Chen 45 where we used (2.22). This implies that ε xx η(u ε m ) H 1 loc (R2 + ) C m ε 0, ɛ 0. Furthermore, we have εη (u ε m)( x u ɛ m) 2 L 1 loc (R 2 + ) C ε x u ε m L 2 loc (R 2 + ) C m, which implies εη (u ε m )( xu ε m )2 is compact in W 1,p loc (R 2 + ), p (1, 2), from an embedding theorem, for fixed m Z +. Then we conclude that t η(u ε m )+ xq(u ε m ) 1,p is compact in Wloc (R 2 + ), 1 <p<2. (2.23) On the other hand, t η(u ε m )+ xq(u ε m ) 1, is bounded in Wloc (R 2 + ). (2.24) Theorem 2.4 indicates from (2.23)-(2.24) that t η(u ε m )+ xq(u ε m ) is compact in H 1 loc (R2 + ). (2.25) Then the compactness theorem (Theorem 2.7) implies that there exists a subsequence {u ε k m } k=1 {uε m} ε>0 such that, for fixed m Z +, u ε k m (t, x) u m (t, x), a.e. k. Step 2. From Step 1, we conclude that the sequence u m (t, x) satisfies (i). u m L u 0 L ; (ii). For any convex entropy pair (η, q), t η(u m )+ x q(u m ) 0 in the sense of distributions. Theorem 2.6 implies that, for any C 2 entropy pair (η, q), t η(u m )+ x q(u m ) is compact in H 1 loc (R2 + ). Then, from Theorem 2.7, there exists a subsequence {u mk } k=1 {u m} m=1 such that u mk (t, x) u(t, x), a.e. k, and the limit function u(t, x) is an entropy solution. Kruzkov s uniqueness theorem indicates that u(t, x) is the unique entropy solution for the Cauchy problem for (2.15), which uniquely defines the solution operator u(t, ) =S t u 0 ( ). Step 3. If the initial data are periodic, then we can follow the same arguments on each period of the periodic viscous solutions for the compactness estimates in Step 1 (instead of the cutoff of the initial data) to conclude that there is a unique periodic entropy solution with the same period as the initial data. The same arguments as in Step 2 yield the compactness of the solution operator. Then we conclude from Theorem 2.5 that the periodic solution asymptotically decays in the sense (2.4). This completes the proof. Remark: For the convex case f (u) 0, the compactness of the solution operator was first obtained by Lax [La1]. If the reflection points {u f (u) =0}are isolated, Dafermos [Da1] showed the decay of periodic solutions. In Theorem 2.8, the reflection points {u f (u) =0}are allowed to be dense, even a nonzero measure

14 46 Compactness Methods and Nonlinear Hyperbolic Conservation Laws (such as the Cantor set). The similar results hold even for multidimensional scalar conservation laws (see Chen-Frid [CF1-2]).

15 Gui-Qiang Chen Strictly Hyperbolic Systems. System (1.1) (d = 1) is called strictly hyperbolic and genuinely nonlinear if λ 1 (u) <λ 2 (u)and λ j (u) r j (u) 0,j =1,2,where r j (u) is the right eigenvector corresponding to the jth eigenvalue λ j (u). We call w i (u),i=1,2, Riemann invariants if they satisfy w i (u) r j (u) =0, i j. Fora2 2 hyperbolic system, we have t w j + λ j x w j =0,j=1,2, and q r j = λ j η r j, from (1.7). Then q wj = λ j (w 1,w 2 )η wj,j=1,2. Therefore, the entropy η satisfies the following equation: η w1w 2 + λ 2w 1 (w 1,w 2 ) η w2 λ 1w 2 (w 1,w 2 ) η w1 =0. (2.26) λ 2 λ 1 λ 2 λ 1 From (2.26), we have a family of the Lax entropy pairs [La3] in the following form: η k = e kwi {V 0 + V1 k + O( 1 k )}, 2 q k = e kwi {λ i V 0 + H1 k + O( 1 (2.27) k )}, 2 where V 0 > 0, and λ i r i λ i V 1 H 1 = V 0 = V 0 λ iwi. w i r i Then q k = λ i + λ iw i η k k + O( 1 ), k >> 1. (2.28) k2 As in the scalar case, an important step needed for the existence and asymptotic decay of entropy solutions is the following compactness result. Theorem 2.9 [Di1]. If a sequence of functions u ε (t, x) satisfies (i). u ε (t, x) K, a.e., where K R 2 is a bounded set. (ii). For any C 2 entropy pair (η, q), t η(u ε )+ x q(u ε ) is compact in H 1 loc (R2 + ). Then u ε (t, x) is compact in L 1 loc (R2 + ). Proof. We need to prove only that ν t,x, determined by {u ε (t, x)} ε>0, is a family of Dirac masses δ u(t,x) for a.e. (t, x), where u(t, x) is the weak-star limit of u ε (t, x). For simplicity, we drop the index (t, x) ofν t,x in the proof. We first define probability measures µ ± k as follows: <µ ± k,h>= <ν,hη ±k > <ν,η ±k >, h C(R2 ), k >> 1, where η ±k are the Lax entropies in (2.27). Since <µ ± k,h> h C,which implies µ ± k M 1, there are subsequences µ ± k j such that µ ± k j weakly converge to some measures µ ±, i.e., <µ ±,h>= lim j <µ± k j,h>, h C(R 2 ).

16 48 Compactness Methods and Nonlinear Hyperbolic Conservation Laws Consider the smallest rectangle Q containing supp ν in the (w 1,w 2 )-coordinates. We claim that the following (a). supp µ ± Q {w=w 1 ± }, (b). <µ +,q λ 1 η>=<µ,q λ 1 η>, for any entropy pair (η, q), imply that ν is a Dirac mass. In fact, take (η, q) =(η k,q k ) from (2.27) in (b). Then <µ +,q k λ 1 η k > = <µ + 1, e kw1 {λ 1w1 k + O( 1 k 2 )} > C 1 where we used the genuine nonlinearity and C 1 > 0. We also have <µ,q k λ 1 η k > C 2 k ekw 1. + k ekw 1, Hence C 1 e kw+ 1 C2 e kw 1, k>>1. That is, e k(w+ 1 w 1 ) C 2, k>>1, C 1 which yields w1 = w+ 1. Similarly, we have w 2 = w+ 2. Hence, ν t,x must be the Dirac mass δ u(t,x). Now the remainder is to prove (a) and (b). (a). For any h C(R 2 ) satisfying h Q + ε =0,where Q + ε =Q {w+ 1 ε<w 1 <w 1 + }, we have <µ + <ν,hη k >,h> = lim k <ν,η k > lim w 1 w1 w+ 1 ε V hekw1 0 dν k w + 1 e ε kw1 V 0 dν 2 w1 w+ 1 lim k e ε 2 k w 1 w1 w+ 1 ε h V 0dν V 0 dν =0. w + 1 ε 2 w1 w+ 1 Thus supp µ + Q {w=w 1 + }. Using the similar method, we can also prove µ Q {w=w1 }. Then (a) is proved. (b). From (2.14), we have <ν,q> <ν,η> <ν,q ±k > <ν,η ±k > =<ν,η ±kq ηq ±k >. <ν,η ±k > Note that q ±k =(λ 1 +O( 1 k ))η ±k. Let k.then <ν,q> <ν,η><µ +,λ 1 >=<µ +,q λ 1 η>, (2.29) and <ν,q> <ν,η><µ,λ 1 >=<µ,q λ 1 η>. (2.30) It suffices for (b) from (2.29)-(2.30) to show <µ +,λ 1 >=<µ,λ 1 >. Take η 1 = η k,η 2 =η k in (2.14), we have <ν,q k > <ν,η k > <ν,q k > <ν,η k > =<ν,η kq k η k q k > <ν,η k >< ν, η k >. (2.31)

17 Gui-Qiang Chen 49 We note that the right hand side of (2.31) tends to 0 as k. Hence <µ +,λ 1 >=<µ,λ 1 >. Then (b) follows. This completes the proof. Now we consider the p-system: { t τ x v =0, t v+ x p(τ)=0, where p(τ) C 2, p (τ) < 0,τp (τ) > 0, as τ 0. The eigenvalues and Riemann invariants are λ 1 = λ 2 = p (τ), (2.31) and the right eigenvectors are τ w j = v +( 1) j+1 p (τ)dτ, j =1,2, r 1 = ( 1, p (τ)), r2 = 0 ( 1, p (τ)). From τp (τ) > 0, we know that λ j r j 0,j =1,2,except the points of the line τ = 0. Therefore, system (2.34) is genuinely nonlinear except the points of the line τ =0. Wedenote(w 1 +,w+ 2 ),(w 1,w 2 )asp, Q, respectively, in the previous proof. Since (2.30) and (a)-(b) still hold, we can easily prove that w 1 + = w+ 2,w 1 = w2 and supp µ + = {P }, supp µ = {Q}, must be true, provided that wj <w+ j,j =1,2. From (b), we have q(p ) λ j (P )η(p )=q(q) λ j (Q)η(Q). Let (η, q) =(τ, v). Since τ = 0 at the points P and Q, v(p )=v(q). Thus v t,x must be a Dirac mass δ u(t,x). Notice that, for this system, there are bounded invariant regions for the approximate solutions generated from the viscosity method. Then we have Theorem The result of Theorem 2.9 still holds for the p-system (2.31), although the genuine nonlinearity fails at the points of line τ = 0. Furthermore, given any initial data in L, there exists a global entropy solution, and the corresponding solution operator is compact in L 1 loc (R2 + ). For the structure of Young measures for 2 2 strictly hyperbolic systems with linear degeneracy, see Serre [Se] for the important reduction and Chen [Ch2] for the method of quasidecouping by using the initial information.

18 50 Compactness Methods and Nonlinear Hyperbolic Conservation Laws 3. Nonstrictly Hyperbolic Conservation Laws In this section we discuss how the compactness methods can be applied to solving important nonstrictly hyperbolic systems of conservation laws, that is, some eigenvalues coincide at some points in the state space. The study of nonstrictly hyperbolic equations has an extensive history, dating back at least two hundred years to the work of Euler [Eu], who posed a degenerate equation, the Euler-Poisson-Darboux equation. Its close relation with wave theory, fluid dynamics, and geometry has attracted great attention for two centuries from such mathematicians as Poisson [Po], Darboux [Dar], Riemann [Ri], Volterra [Vol], and also Weinstein, Erdélyi, Lions, and others in the 1960 s (see [Ya]). On the other hand, the theory of linear equations with multiple characteristics is also well developed. An important feature of such equations is the loss of differentiability [DeG], which leads to ill-posedness in Sobolev spaces but well-posedness in the Gevrey classes [Ge]. Another feature is that sign conditions on the subprincipal symbol play an important role (cf. [Fr,Oh,H2-3]). Nonlinear nonstrictly hyperbolic systems of conservation laws arise from such disparate areas as gas dynamics, multiphase flow in porous media, elasticity, waterwave problems, and magnetohydrodynamics. Such systems appear naturally in multidimensional systems of conservation laws. This means that, in general, planewave solutions for such multidimensional systems are governed by one-dimensional nonstrictly hyperbolic systems. There are two kinds of degeneracy: parabolic degeneracy and hyperbolic degeneracy, which govern different behavior of solutions near the degenerate points. A system is called to have hyperbolic degeneracy at a degenerate point u 0 if the matrix f(u 0 ) is diagonalizable; otherwise, we call the system has parabolic degeneracy. In this section we discuss two prototypes: the system of isentropic Euler equations, which has parabolic degeneracy near the vacuum, and hyperbolic conservation laws with umbilic degeneracy, which have hyperbolic degeneracy Isentropic Euler Equations. The system of isentropic Euler equations for a compressible fluid reads { t ρ + x m =0, t m+ x ( m2 ρ +p(ρ)) = 0, (3.1) where the pressure p(ρ) is a given nonlinear function, determined by the fluid under consideration, such as real gases and other complex fluids (see Courant-Friedrichs [CFr], Stoker [St], Truesdall [Tr], Pedlosky [Pe], and Lions [Lio]). The density ρ 0 and mass m are physically restricted by m Cρ for some constant C > 0so that the function (ρ, m) m 2 /ρ remains Lipschitz continuous, even at the vacuum ρ = 0. Forρ > 0, the velocity v = m/ρ is uniquely defined. Strict hyperbolicity and genuine nonlinearity in the sense of Lax [La4] away from the vacuum for (3.1) require that the pressure law satisfy p (ρ) > 0, 2p (ρ)+ρp (ρ) > 0, for ρ>0. (3.2) At the vacuum, the two characteristic speeds of (3.1) may coincide and the systems be nonstrictly hyperbolic. A simple calculation shows that the vacuum can not be

19 Gui-Qiang Chen 51 avoided for this system even for some Riemann solutions with large Riemann initial data away from the vacuum, in general. This system is an archetype of nonlinear hyperbolic conservation laws in fluid mechanics (see [CFr,Sm,St,Wh,CM,Lio]). The so-called polytropic perfect gas is described by the equation of state: p (ρ) =κρ γ, γ > 1. (3.3) One may assume κ =(γ 1) 2 /(4γ), which is a convenient normalization. For early results on the existence of global entropy solutions of (3.1), we refer to Riemann [Ri] for the Riemann problem, Zhang-Guo [ZG] and Ding-Chang-Wang-Hsiao-Li [DCW] for a special class of initial data with bounded variation, and Nishida-Smoller [NS] for large total variation with small γ 1 or vice versa by using the Glimm scheme [Gl]. For the limit case γ = 1, Nishida [Ni] first established the existence of global solutions in BV for large BV initial data. For arbitrarily large L initial data, the case γ =1+2/N (N 5 odd) was first treated by DiPerna [Di2]. The case γ (1, 5/3], which is the natural interval of γ for polytropic gases, was completed in Ding-Chen-Luo [DCL1] and Chen [Ch1]. As we discussed in 1-2, for a construction of global entropy solutions for (3.1), one needs the compactness (i.e. the strong convergence) of a sequence of approximate solutions, say (ρ ɛ,m ɛ ). To achieve this, it suffices to prove that the corresponding Young measures ν t,x, governed by (2.14) for a.e. (t, x) andforany two weak entropy pairs (η j,q j ),j =1,2,are Dirac masses in the (ρ, m)-plane for our case. This is implied by that the support of any Young measure in the (ρ, v)- plane, { still denoted by } ν t,x, is either a single point or a subset of the vacuum line (ρ, v) ρ =0,v R. One of the main difficulties to reduce the Young measure is that the commutation relation (2.14) holds only for weak entropy pairs, which is not allowed for any C 2 entropy pairs (different from those in 2.7), because of the degeneracy of the system near the vacuum. As we discussed in 1-2, for a construction of global entropy solutions for (3.1), one needs the compactness (i.e. the strong convergence) of a sequence of approximate solutions, say (ρ ɛ,m ɛ ). To achieve this, it suffices to prove that the corresponding Young measures ν t,x, governed by (2.14) for a.e. (t, x) andforany two weak entropy pairs (η j,q j ),j =1,2,are Dirac masses in the (ρ, m)-plane for our case. This is implied by that the support of any Young measure in the (ρ, v)- plane, { still denoted by } ν t,x, is either a single point or a subset of the vacuum line (ρ, v) ρ =0,v R. One of the main difficulties to reduce the Young measure is that the commutative relation (2.14) holds only for weak entropy pairs, which is not allowed for any C 2 entropy pairs (different from those in 2.7), because of the degeneracy of the system near the vacuum. When (3.3) holds, the so-called weak entropies of (3.1) can be expressed by convolution of an arbitrary smooth function ψ(s) and the entropy kernel, χ (ρ, v, s), defined by χ (ρ, v; s) =M [ρ γ 1 (v s) 2 ] λ +, λ = 3 γ 2(γ 1). (3.4) Here y + =max(y, 0) and M > 0 is the constant of normalization. Namely, one has η(ρ, v) = η(ρ, ρv) := RI χ (ρ, v, s) ψ(s) ds. The entropy flux kernel has also an explicit form: σ := ( v + θ (s v) ) χ.

20 52 Compactness Methods and Nonlinear Hyperbolic Conservation Laws In the proof of DiPerna [Di2], Ding-Chen-Luo [DCL1], and Chen [Ch1], the heart of the matter is to construct special functions ψ in order to exploit the set of constraints (2.14). These test functions are suitable approximations of highorder derivatives of the Dirac mass. In DiPerna [Di2], the case that λ 2isan integer was treated, in which all weak entropies are polynomial functions of the Riemann invariants. The idea of applying the technique of fractional derivatives was introduced in Chen [Ch1] and Ding-Chen-Luo [DCL1] in order to deal with real values of λ. Lions-Perthame-Souganidis [LPS] and Lions-Perthame-Tadmor [LPT], motivated by a kinetic formulation for (3.1) and (3.3), made the observation that the use of ψ could be bypassed and (2.14) may be directly expressed by the entropy kernel χ (s) =χ (ρ, v; s) and the entropy flux kernel σ (s) =σ (ρ, v; s). That is, for all s 1 and s 2, <ν t,x,χ (s 1 )σ (s 2 ) χ (s 2 )σ (s 1 )> =<ν t,x,χ (s 1 )><ν t,x,σ (s 2 )> <ν t,x,χ (s 2 )><ν t,x,σ (s 1 )>. (3.5) A simpler proof of the reduction of the Young measures satisfying (3.5) was given in [LPS] for the case γ (1, 3). The range γ [3, ) was treated in [LPT]. The previous techniques of choosing suitable approximations of the derivatives of the Dirac mass, in [Ch1, DCL1, Di2], corresponds in their approach to computing a suitable number of s-derivatives of the commutation equation (3.5), the number of fractional derivatives being related to the exponent λ which characterizes the singularities of the entropy kernel. This is technically delicate since such derivatives of the kernel generate Dirac masses, due to its limited regularity. It was observed that the average <ν,χ >is smoother (as a function of s) thanχ (ρ, v, s) (asa function of (ρ, v, s)). Recently, we established a general compactness framework for approximate solutions of this system with a general pressure law for large initial data in [CL1-2]. The pressure function p(ρ) C 4 (0, ) satisfies that there exist γ (1, 3) and C>0 such that p(ρ) =κρ γ (1 + P (ρ)), P (k) (ρ) Cρ 1 k, 0 k 4, (3.6) for sufficiently small ρ. The solutions under consideration will remain in a bounded subset of {ρ 0} so that the behavior of p(ρ) for large ρ is irrelevant. This means that the pressure law p(ρ) has the same principal singularity as the γ-law gas, but (3.6) allows additional singularities in the derivatives when ρ 0. Indeed observe that, for k>γ+1, ρ γ P (k) (ρ) is unbounded when ρ 0. Observe that p(0) = p (0) = 0, but, for k>γ, the higher derivative p (k) (ρ) is unbounded near the vacuum. Theorem 3.1 (Chen-LeFloch [CL2]). Consider the compressible Euler system (3.1) with the general pressure law (3.2) and (3.6) with Cauchy data (ρ, m) t=0 =(ρ 0 (x),m 0 (x)). (3.7) Then (i). Given any measurable and bounded initial data (ρ 0,m 0 ) satisfying 0 ρ 0 (x) C 0, m 0 (x) C 0 ρ 0 (x), for a.e. x and some C 0 > 0, (3.8)

21 Gui-Qiang Chen 53 there exists an entropy solution (ρ, m) of the Cauchy problem (3.1) and (3.7), globally defined in time, satisfying 0 ρ(t, x) C, m(t, x) Cρ(t, x), for a.e. (t, x), (3.9) where C>0depends only on C 0 and the pressure function p( ). (ii). Let (ρ ε,m ε ) be a sequence of functions satisfying (3.9) uniformly in ε such that, for any weak entropy pair (η, q), t η(ρ ε,m ε )+ x q(ρ ε,m ε ) is compact in H 1 loc (R2 + ). (3.10) Then the sequence (ρ ε,m ε ) is compact in L 1 loc (R2 + ). Since (3.3) is not assumed, the explicit formula (3.4) are no longer available. In particular, the entropy flux kernel, denoted by σ, cannot be directly expressed from the entropy kernel, denoted by χ. It turns out that several crucial steps of the proofs in the previous references can be no longer carried out. As a first step, we constructed the entropy kernel χ and the entropy flux kernel σ. The entropy kernel is governed by a highly singular equation of Euler-Poisson- Darboux type: Λ(w z) χ wz w z (χ w χ z )=0, (3.10) or, equivalently, ρρ χ a(ρ) vv χ =0, (3.11) with initial data: χ ρ=0 =0, χ ρ ρ=0 = δ v=s, (3.12) involving a Dirac mass, where a(ρ) =p (ρ)ρ 2. Notice that the function Λ(w z) is not smooth in w z in general, and that each of its derivatives produces certain extra power of 1/(w z), which is singular near w z 0, that is, ρ 0. Therefore, the equation (3.10) or (3.11) is much more singular than the classical Euler- Poisson-Darboux equation that is the case for the γ-law (3.3). New techniques were developed, and special care was made to study the singularities of their derivatives. One of our major observations is that the principal singularities of χ and σ can be determined (rather explicitly) from the ones in χ, modulo a nonlinear transformation involving the pressure law p(ρ). In addition, we observed several properties of cancellation of singularities for the function E(s 1,s 2 ):=χ(s 1 )σ(s 2 ) χ(s 2 )σ(s 1 ) and its derivatives of order λ +1,ass 2 s 1. Then we proved that any Young measure satisfying the Tartar commutation relation (2.14) for all weak entropy pairs is a Dirac mass. Then we proved that any Young measure satisfying the Tartar-Murat commutation relation (2.14) for all weak entropy pairs is a Dirac mass. As in [LPS], we can write (2.14) directly in terms of the kernels χ and σ. Our proof relies on the following identity: <ν t,x,χ(s 1 )>< ν t,x, s λ+1 2 s λ+1 3 E(s 2,s 3 )> +<ν t,x, s λ+1 2 χ(s 2 ) ><ν t,x, s λ+1 3 E(s 3,s 1 )> (3.14) +<ν t,x, s λ+1 3 χ(s 3 ) ><ν t,x, s λ+1 2 E(s 1,s 2 )>=0 in the sense of distributions, for all real variables s j,j =1,2,3, where we used the fractional derivative operator s λ+1. Then we made s 2 and s 3 converge to s 1 and justified that the sum of the second and third terms converge to zero in a weak sense.

22 54 Compactness Methods and Nonlinear Hyperbolic Conservation Laws On the other hand, the first term is highly singular: the main term of interest has the form of the product of a function of bounded variation by a bounded measure, which is not defined in a classical sense and may actually have several rigorous meanings (see [DLM]). We took advantage of this observation and showed that the first term generally converges weakly to a limit that is not zero, which involved several careful analyses. Further analysis is also necessary to show that the other singular terms vanish at the limit, in which some delicate techniques were needed since there is no explicit formula available. Then the compactness result follows. With the compactness framework, we can conclude Theorem 3.2. Assume that the initial data (ρ 0 (x),m 0 (x)) satisfy (3.8). Then (i). There exists an entropy solution (ρ, m) of the Cauchy problem (3.1) and (3.7), globally defined in time, such that 0 ρ(t, x) C, m(t, x) Cρ(t, x), a.e. (t, x), where C depends only on C 0 and the pressure function p( ). (ii). The solution operator (ρ, m)(t, ) = S t (ρ 0,m 0 )( ) determined by (i) is compact in L 1 loc (R2 + ). (iii). Let (ρ, m) L (R 2 +) be a periodic entropy solution of (3.1) and (3.7) with period [α, β]. Then (ρ, m) asymptotically decays: β ( esslim ρ(t, x) ρ r + m(t, x) m r) dx =0, for all 1 r<, t α where ( ρ, m) := 1 β β α α (ρ 0(x),m 0 (x))dx. (iv). The approximate solutions, generated from the Lax-Friedrichs scheme or the Godunov scheme, strongly converge (subsequentially) to an entropy solution of (3.1) and (3.7). Remark. The results in Theorem 3.2 are somewhat surprising, since the flux function of (3.1) is only Lipschitz continuous. The example found by Greenberg- Rascle [GR] demonstrates that there exist certain systems with only C 1 (but not C 2 ) flux functions admitting time-periodic and space-periodic solutions. This example indicates that the compactness and asymptotic decay of entropy solutions are very sensitive with respect to the smoothness of flux functions. By the Euler-Lagrange transformation, the results established here for (3.1) can be reformulated to those for the Lagrangian system (see Wagner [Wa] and Chen [Ch2]) Hyperbolic Conservation Laws with Umbilic Degeneracy. We are now concerned with hyperbolic systems of conservation laws with hyperbolic umbilic degeneracy. A point u 0 R n in the state space is called an umbilic point for a 2 2 hyperbolic system of conservation laws: t u + x f(u) =0, u R 2, if λ 1 (u 0 )=λ 2 (u 0 ). Such a umbilic degeneracy allows a degree of interaction, or nonlinear resonance, between distinct modes, and leads to high singularities missing in the strictly hyperbolic case.

23 Gui-Qiang Chen 55 Consider a hyperbolic system of conservation laws: { t u + x ( u C(u)) = 0, (3.15) C(u) = 1 2 (1 3 au3 1 + bu 2 1u 2 + u 1 u 2 2), with a unique umbilic point (0, 0). System (3.15) can be reduced from the following system: t u + x f(u) =0,u R 2, (3.16) with an isolated hyperbolic umbilic point u 0. That is, f(u 0 ) is diagonalizable, and there is a neighborhood N of u 0 such that f(u) has distinct eigenvalues for all u N u 0. Take the Taylor expansion for f(u) aboutu=u 0 : f(u)=f(u 0 )+ f(u 0 )(u u 0 )+ 1 2 (u u 0) 2 f(u 0 )(u u 0 ) + high order terms. (3.17) Since f(u 0 ) is diagonalizable, we can make a coordinate transformation to eliminate the linear term from (3.17) and obtain the the quadratic flux systems: t u + x Q(u) =0, (3.18) omitting the high order terms, where Q(u) = 1 2 u 2 f(u 0 )u. Such quadratic flux functions determine the local behavior of the hyperbolic singularity near the umbilic point. Moreover, for any arbitrary system with umbilic hyperbolic degeneracy, at each umbilic degenerate point, the wave curves of each family of elementary waves are tangential to the corresponding curves of one of the systems in (3.18) or (3.15) under a suitable linear transformation. In fact, any quadratic flux system (3.18) can be transformed into the corresponding system in (3.15) by a linear coordinate transformation by using the normal form theorem in [SS1]. The Riemann problem, a special Cauchy problem, for 2 2 quadratic flux systems was discussed by Isaacson, Marchesin, Paes-Lemes, Plohr, Schaeffer, Shearer, Temple, Zumbrum, and many others ( ). For the simplest nontrivial system with an isolated umbilic point, the Cauchy problem was solved by Kan [Ka], and a different independent proof of this result was given by Lu [Lu]. In Chen-Kan [CK1-2], a compactness framework was established for approximate solutions to such systems by combining compensated compactness ideas with classical methods, and by making a detailed analysis of a highly singular equation of Euler-Poisson-Darboux type. New techniques were developed to carry out this strategy, since the entropy equation has different type of singularities from (3.10), much more singular. Then this framework was successfully applied to proving the convergence of the Lax-Friedrichs scheme, the Godunov scheme, and the viscosity method and the existence of global entropy solutions for the Cauchy problem with large initial data for a canonical class of the quadratic flux systems. As a corollary, the compactness of the solution operator in L 1 loc and the asymptotic decay of periodic entropy solutions were established, important nonlinear phenomena, especially for these nonstrictly hyperbolic systems.

24 56 Compactness Methods and Nonlinear Hyperbolic Conservation Laws 4. Compressible Euler Equations In this section, we discuss some more results on the compressible Euler equations with the aid of the compactness methods, in addition to those in Non-isentropic Euler Equations. We consider the compressible non-isentropic Euler equations in Lagrangian coordinates t τ x v =0, t v+ x p=0, (4.1) t (e+ 1 2 v2 )+ x (vp) =0, where τ,v,p, and e denote respectively the deformation gradient, the velocity, the pressure, and the internal energy. Other relevant fields are the entropy s and the temperature θ. The above system of conservation laws is complemented by the Clausius inequality: t s 0, which expresses the second law of thermodynamics. Selecting (τ,v,s) as the state vector, we have the constitutive relations: (e, p, θ) =(ê(τ,s), ˆp(τ,s), ˆθ(τ,s)) (4.2) satisfying the conditions: p = ê τ, θ =ê s. Under the standard assumptions ˆp v < 0andˆθ>0, system (4.1) is strictly hyperbolic. In this section, we first exhibit a class of constitutive relations: p = h(τ αs), e = βs w h(y)dy, θ = αh(τ αs)+β>0, (4.3) where α, β, w = τ αs, and h(w) is a smooth function with h (w) < 0 satisfying { h (w) 4 αh (w) 2 >0, if w<ŵ, (4.4) αh(w)+β < 0, if w>ŵ. The model (4.3) can be regarded as a first-order correction to the general constitutive equation (see Chen-Dafermos [CD1]). Consider the Cauchy problem for (4.1) with periodic initial data: (w, v, s) t=0 =(w 0 (x),v 0 (x),s 0 (x)), (4.5) satisfying w 0 (x) C 0, v 0 (x) C 0, s 0 (x) M loc (R), and w (w 0 (x),v 0 (x)) Σ C1 {(w, v) C 1 v± h (ω)dω C 1 }, which contains only physical admissible states. In particular, if (w, v) Σ C1,then θ=αh(w)+β>0. ŵ

25 Gui-Qiang Chen 57 Theorem 4.1 [CD1]. (i). There exists a global distributional solution (w(t, x),v(t, x),s(t, x)) for the Cauchy problem (4.1) and (4.3)-(4.5), satisfying (w, v) L (R 2 +), (s, s t ) M loc (R 2 +), θ(w) 0, s {[0,T 0 ] [ ct 0,cT 0 ]} CT0 2, (4.6) for any c>0,t 0 >0, with C>0independent of T 0,where s denotes the variation measure associated with the signed measure s. Moreover, (w(t, x), v(t, x), s(t, x)) satisfies the entropy inequality: t η(w, v)+ x q(w, v) 0, s t 0, (4.7) in the sense of distributions for any C 2 entropy pair (η(w, v),q(w, v)) of the subsystem t w x v =0, t v+ x h(w)=0, (4.8) for which the strong convexity condition holds: θη ww αh (w)η w 0, θη vv + αη w 0, (θη ww αh (w)η w )(θη vv + αη w ) ηwv 2 0. (4.9) Furthermore, if the initial date are periodic, then the global distributional solution (w(t, x),v(t, x),s(t, x)) is also periodic with the same period. (ii). Assume that the sequence (w T (t, x),v T (t, x)) satisfies the following: (a). There exists a constant C>0such that (b). The sequence (w T,v T ) L C; (4.10) t η(w T,v T )+ x q(w T,v T ) 0 in the sense of distributions, (4.11) for any C 2 convex entropy pair (η(w, v),q(w, v)) of subsystem (4.8) satisfying (4.9). Then the sequence (w T (t, x),v T (t, x)) is compact in L 1 loc (R2 + ). This theorem is due to Chen-Dafermos [CD1]. The existence is established by using the vanishing viscosity method via the following equations: t τ x v = ε x (κ x (τ αs)), t v + x p = ε x (κ x v), t (e v2 )+ x (vp) =ε x (κv x v)+ε x (ν θ), and the compactness arguments as in 2. The key point is to study the compactness of (w ε,v ε ) first and then the limit behavior of s ε. The main observation is that (w ε,v ε ) satisfy the following 2 2 subsystem, quasidecoupled from s ε, reduced from (4.1): { t w x v = αεκ θ ( xv) 2 + αεκh θ ( x w) 2 + ε x (κ x w), t v + x h(w) =ε x (κ x v). Combining our careful energy estimates with the previous compactness framework (Theorem 2.10), we concluded the strong compactness of (w ε,u ε )inl 1 loc and the weak compactness of s ε in the measure space M.