Preprint 2005-045

Compensated Compactness for 2D Conservation Laws

Abstract: 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 {\em 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) $\ueps$ 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 {\em 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.

Paper:

Available as PDF (304 Kbytes).

Author(s):

Eitan Tadmor, email: tadmor at cscamm dot umd dot edu

Michel Rascle, email: rascle at math dot unice dot fr

Patrizia Bagnerini, email: pbagneri at dimat dot unipv dot it

Publishing information:

J. Hyperbolic DEs 2(3) (2005) 697-712

Comments:

Submitted by:

tadmor at cscamm dot umd dot edu November 3 2005.