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

2021/02/19 by Abels, Helmut
#35Q30 #35Q35 #35R35 #76D05 #76D45 #76T99 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2102.10057

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_ε=m₀ε^θ as ε→ 0. In the case θ>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 θ=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 θ=0,1 by the method of formally matched asymptotics.

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