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The compensated compactness method applied to systems of conservation laws. (English) Zbl 0536.35003
Systems of nonlinear partial differential equations, Proc. NATO Adv. Study Inst., Oxford/U.K. 1982, NATO ASI Ser. Ser., C 111, 263-285 (1983).
[For the entire collection see Zbl 0514.00014.]
The following problem is one of the typical situation with respect to solving systems of (nonlinear) partial differential equations: Given a functional Q and a sequence \((u^ n)_{n\in {\mathbb{N}}}\) of (weak) solutions of an (approximating) system, with the property \(u^ n\to u^{\infty}\) weak (or weak *) in a suitable space. What is the relation between \(\mu =\lim_{n\to}Q(u^ n)\) (if the lim exists in an appropriate sense) and \(Q(u^{\infty})?\) In the paper under review the author gives the following theorem (compensated compactness): Let the system of partial differential operators \[ \sum_{j,k}a_{ijk}\frac{\partial u_ j}{\partial x_ k},\quad i=1,...,q,\quad u=(u_ 1,...,u_ q),\quad a_{ijk}\in {\mathbb{R}}, \] be given and assume for a sequence \((u^ n)_{n\in {\mathbb{N}}}\), \(u^ n\in(L^ 2(\Omega))^ q\), \(\Omega \subset {\mathbb{R}}^ N\), i) \(u^ n\to u^{\infty}\) weakly in \((L^ 2(\Omega))^ q\) and ii) \(\sum_{j,k}a_{ijk}\partial u_ j^{(n)}/\partial x_ k\in\) compact set of \(H^{-1}_{loc}(\Omega)\), \(i=1,...,q\). Furthermore let Q be a quadratic form in \({\mathbb{R}}^ q\) satisfying \(Q(\lambda)\geq 0\) for all \(\lambda \in \Lambda =\{\lambda \in {\mathbb{R}}^ q| \quad \exists \xi \in {\mathbb{R}}^ N\backslash \{0\}\quad \forall i=1,...,q\sum_{j,k}a_{ijk}\lambda_ j\xi_ k=0\}.\) Moreover suppose that \(Q(u^ n)\) converges weakly (*) t \(\mu\) in \({\mathcal M}(\Omega)\). Then \(\mu \geq Q(u^{\infty})\) holds in the sense of \({\mathcal M}(\Omega)\). (\({\mathcal M}(\Omega)\) is the space of Radon measures and the weak * convergence is defined by using the dual space of the space of all continuous functions on \(\Omega\) with compact support.) If in addition \(Q(\lambda)=0\) holds for all \(\lambda\in \Lambda\), then one can show \(Q(u^ n)\to Q(u^{\infty})\) weakly * in \({\mathcal M}(\Omega)\). These results are applied to a hyperbolic system of conservation laws.
Reviewer: N.Jacob

35A35 Theoretical approximation in context of PDEs
35L65 Hyperbolic conservation laws
35G20 Nonlinear higher-order PDEs
35F20 Nonlinear first-order PDEs