The Initial-boundary value problem for the Lifshitz-Slyozov equation with non-smooth rates at the boundary

Abstract

We prove existence and uniqueness of solutions to the initialboundary value problem for the Lifshitz-Slyozov equation (a non-linear transport equation on the half-line) focusing on the case of kinetic rates that are not Lipschitz continuous at the origin. Our theory covers in particular those cases with rates behaving as power laws at the origin, for which an inflow behaviour is expected and a boundary condition describing nucleation phenomena needs to be imposed. The method we introduce here to prove existence is based on a mild formulation with a careful analysis on the behavior of characteristics near the singular boundary. Uniqueness exploits monotonicity properties of the associated transport equation. Contents