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

Helmut Abels

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arXiv · 2102.10057

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

Helmut Abels

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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.

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Helmut Abels. 2021-02-19. (Non-)Convergence of Solutions of the Convective Allen-Cahn Equation. https://arxiv.org/abs/2102.10057

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