Analysis of a scalar conservation law with a flux function with discontinuous coefficients

Abstract

We study here a model of a conservation law with a flux function with discontinuous coefficients, namely the equation $u_t + (k(x)g(u)+f(u))_x=0$. We prove the existence and the uniqueness of an entropy solution in $L^{\infty}(\mathbb R_+ \times \mathbb R)$ for $u_0$, the initial condition, in $L^{\infty}(\mathbb R)$. We provide some physical background for the study of this equation. In particular, $g$ is assumed to be neither convex nor concave and $k$ is a discontinuous function.