Vanishing viscosity solutions for conservation laws with regulated flux

摘要

In this paper we introduce a concept of "regulated function" v(t, x) of two variables, which reduces to the classical definition when v is independent of t. We then consider a scalar conservation law of the form u(t) + F(v(t, x), u)(x) = 0, where F is smooth and vis a regulated function, possibly discontinuous w.r.t. both t and x. By adding a small viscosity, one obtains a well posed parabolic equation. As the viscous term goes to zero, the uniqueness of the vanishing viscosity limit is proved, relying on comparison estimates for solutions to the corresponding Hamilton-Jacobi equation.