## 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

### MSC:

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

Zbl 0514.00014