(Non-)Convergence of Solutions of the Convective Allen-Cahn Equation

TL;DR

This paper investigates the convergence behavior of solutions to the convective Allen-Cahn equation in the sharp interface limit, revealing different regimes depending on the mobility scaling and highlighting non-convergence phenomena.

Contribution

It provides new insights into the convergence properties of the convective Allen-Cahn equation under various mobility scalings, including non-convergence results for certain regimes.

Findings

Concentrations converge to a transport equation solution for >2.

Rescaled optimal profile behavior fails for >2.

Associated mean curvature functional does not converge for >2.

Abstract

We consider the sharp interface limit of a convective Allen-Cahn equation, which can be part of a Navier-Stokes/Allen-Cahn system, for different scalings of the mobility $m_\varepsilon=m_0\varepsilon^\theta$ as $\varepsilon\to 0$. In the case $\theta>2$ we show a (non-)convergence result in the sense that the concentrations converge to the solution of a transport equation, but they do not behave like a rescaled optimal profile in normal direction to the interface as in the case $\theta=0$. Moreover, we show that an associated mean curvature functional does not converge the corresponding functional for the sharp interface. Finally, we discuss the convergence in the case $\theta=0,1$ by the method of formally matched asymptotics.