On the existence and compactness of a two-dimensional resonant system of conservation laws

Abstract

We prove the existence of a weak solution to a two-dimensional resonant 3 × 3 system of conservation laws with BV initial data. Due to possible resonance (coinciding eigenvalues), spatial BV estimates are in general not available. Instead, we use an entropy dissipation bound combined with the time translation invariance property of the system to prove existence based on a two-dimensional compensated compactness argument adapted from [E. Tadmor, M. Rascle, and P. Bagnerini, J. Hyperbolic Differ. Equ., 2(3), 697-712, 2005]. Existence is proved under the assumption that the flux functions in the two directions are linearly independent. © 2007 International Press.