Convergence of a Convective Allen-Cahn Equation to a Transport Equation

Abstract

We show convergence of solutions of a convective Allen-Cahn equation for a given smooth and divergence free velocity field to a transport equation for an evolving interface in the case when the thickness of the diffuse interface tends to zero and the mobility coefficient is proportional to the interfacial thickness. This is done for well-prepared initial data by estimating the difference of the exact and an approximate solution of the convective Allen-Cahn equation with the aid of a uniform lower bound for the linearized Allen-Cahn operator. The approximate solution is constructed with the aid of three terms from formally matched asymptotics calculations close to the interface.