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.