Radon measures solving the Cauchy problem of the nonlinear transport equation

Abstract

The focus of interest is the Cauchy problem of the nonlinear transport equation d_t u + div (f(u, ·) u) = g(u, ·) u together with its distributional solutions u(·) whose values are positive Radon measures on the Euclidean space with compact support. The coefficients f(u, t), g(u, t) are assumed to be uniformly bounded and Lipschitz continuous vector fields on the Euclidean space. Sufficient conditions on the coefficients for existence, uniqueness and even for stability of these distributional solutions are presented. Starting from the well-known results about the corresponding linear problem, the step towards the nonlinear problem here relies on Aubin's mutational equations, i.e. dynamical systems in a metric space (with a new slight modification).