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

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 $m_\varepsilon=m_0\varepsilon^θ$ as $\varepsilon\to 0$. 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.