(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_ε =m₀ε ^θ m ε = m 0 ε θ as ε → 0 ε → 0 . In the case θ >2 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 θ = 0 . Moreover, we show that an associated mean curvature functional does not converge to the corresponding functional for the sharp interface. Finally, we discuss the convergence in the case θ =0,1 θ = 0 , 1 by the method of formally matched asymptotics.