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Compacité par compensation et systèmes hyperboliques de lois de conservation. (Compensated compactness and hyperbolic systems of conservation laws). (French) Zbl 0578.35055

The study of N conservation laws \(\partial_ t u_ i+\partial_ x f_ i(u)=0\) for \((x,t)\in \Omega ={\mathbb{R}}\times {\mathbb{R}}^+\), \(u(x,0)=a(x)\) reduces to considering the viscous system \(\partial_ t u_ i^{\epsilon}+\partial_ x f_ i(u^{\epsilon})=\epsilon \partial^ 2_ x u_ i^{\epsilon}\) and showing that \(\lim_{\epsilon \to 0^+}f(u_ i^{\epsilon})=f(\lim_{\epsilon \to 0^+}u_ i^{\epsilon})\) in the weak-star topology of \(L^{\infty}(\Omega)\). When the sequences \(u^{\epsilon}\) and \(\sqrt{\epsilon} u_ x^{\epsilon}\) are bounded in \(L^{\infty}(\Omega)\) and \(L^ 2(\Omega)\), respectively, L. Tartar [NATO ASI Ser., Ser. C 111, 263-285 (1983; Zbl 0536.35003)] shows that then there exists a subsequence \(v^ n=u^{\epsilon_ n}\), \(\epsilon_ n\to 0^+\), and a family of probability measures on \({\mathbb{R}}^ N:\) \(\{v_{x,t}| (x,t)\in \Omega \}\) such that (i) for each function \(F\in ({\mathbb{R}}^ N,{\mathbb{R}})\), \(F\circ v^ n\to Z\) as \(n\to \infty\) in \(L^{\infty}(\Omega)\) weak-star with \(Z(x,t)=<v_{x,t}, F>\) a.e., (x,t)\(\in \Omega\) and \[ (ii)\quad <v_{x,t}, \phi_ 1\psi_ 2-\phi_ 2\psi_ 1>=<v_{x,t}, \phi_ 1>\quad <v_{x,t}, \psi_ 2>- <v_{x,t}, \phi_ 2><v_{x,t}, \psi_ 1>. \] (ii) is a result of compactness by conpensation. In passing to the limit \(\epsilon \to 0^+\) in the viscous system one obtains \(\partial_ t u+\partial_ x w=0\), where \(u(x,t)=<v_{x,t}, id>\) and \(w(x,t)=<v_{x,t}, f>.\) For u to be a solution of the initial problem it suffices that \(w=f(u)\), which is not the case for arbitrary \(v_{x,t}\), at least if f is affine.
Thus L. Tartar (loc. cit.) proposed the following program: if (ii) holds show that \(v_{x,t}\) is not arbitrary and show that (if \(N\leq 2)\) (ii) implies \(v_{x,t}\) is a Dirac mass, which in turn implies \(w=f(u)\). Tartar realized this program for \(N=1\) and R. Di Perna [Arch. Ration. Mech. Anal. 82, 27-70 (1983; Zbl 0519.35054) and Commun. Math. Phys. 91, 1-30 (1983; Zbl 0533.76071)] carried it out for the system of elasticity and isentropic gas dynamics for \(N\leq 2\) for well-behaved functions of state.
In this paper the author gives a number of propositions reproving and generalizing results of Di Perna. These results lead him to conjecture that any probability measure satisfying (ii) is of the form \(v=\exp H(w)\mu_ 1\otimes \mu_ 2,\) where \(\partial^ 2H/\partial w_ 1\partial w_ 2=-(\lambda_ 2-\lambda_ 1)^{-2}\partial (\lambda_ 1,\lambda_ 2)/\partial (w_ 1,w_ 2).\) Here the \(w_ 0\) are the Riemann invariants of the system \(w'_ if'=\lambda_ iw'_ i\) \((i=1,2)\).
Reviewer: N.D.Kazarinoff

MSC:

35L65 Hyperbolic conservation laws
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B20 Perturbations in context of PDEs
35B35 Stability in context of PDEs
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