Existence, uniqueness and Large time behavior of a viscous conservation law with discontinuous flux

Abstract

In this paper, we study a viscous conservation law with discontinuous flux. We define a weak solution concept, show its existence via an explicit formula, and prove that the weak solution is unique. In addition, we obtain the large-time behavior of the solution. The asymptotic limit of the solution is a steady-state solution. Here, the large-time asymptotic governed by the convection terms either converges to a constant steady state or to a non-constant steady state, depending on the behavior of the flux functions at a threshold value, which is the meeting point of two fluxes. This phenomenon does not occur in the single-flux case.