Strong time regularity and decay of $L^{\infty }$ solutions to $2\times 2$ systems of conservation laws

Abstract

We consider ${\mathbf{L}}^{\infty }$ solutions to $2\times 2$ systems of conservation laws. For genuinely nonlinear systems we prove that finite entropy solutions (in particular entropy solutions, if a uniformly convex entropy exists) belong to $C^0({\mathbb{R}}^{+};{\mathbf{L}}_{loc}^1(\mathbb{R}))$. Our second result establishes a dispersive-type decay estimate for vanishing viscosity solutions. Both results are unified by the use of a kinetic formulation.