Compensated compactness for 2D conservation laws

We introduce a new framework for studying two-dimensional conservation laws by compensated compactness arguments. Our main result deals with 2D conservation laws which are nonlinear in the sense that their velocity fields are a.e. not co-linear. We prove that if $u^epsilon$ is a family of uniformly bounded approximate solutions of such equations with $H^{-1}$-compact entropy production and with (a minimal amount of) uniform time regularity, then (a subsequence of) $u^epsilon$ convergences strongly to a weak solution. We note that no translation invariance in space — and in particular, no spatial regularity of $u(cdot, t)$ is required. Our new approach avoids the use of a large family of entropies; by a judicious choice of entropies, we show that only two entropy production bounds will suffice. We demonstrate these convergence results in the context of vanishing viscosity, kinetic BGK and finite volume approximations. Finally, the intimate connection between our 2D compensated compactness arguments and the notion of multi-dimensional nonlinearity based on kinetic formulation is clarified.