Remarks on Sharp Interface Limit for an Incompressible Navier-Stokes and Allen-Cahn Coupled System

We are concerned with the sharp interface limit for an incompressible Navier-Stokes and Allen-Cahn coupled system in this paper. When the thickness of the diffuse interfacial zone, which is parameterized by $\varepsilon$, goes to zero, we prove that a solution of the incompressible Navier-Stokes and Allen-Cahn coupled system converges to a solution of a sharp interface model in the $L^\infty(L^2)\cap L^2(H^1)$ sense on a uniform time interval independent of the small parameter $\varepsilon$. The proof consists of two parts: one is the construction of a suitable approximate solution and another is the estimate of the error functions in Sobolev spaces. Besides the careful energy estimates, a spectral estimate of the linearized operator for the incompressible Navier-Stokes and Allen-Cahn coupled system around the approximate solution is essentially used to derive the uniform estimates of the error functions. The convergence of the velocity is well expected due to the fact that the layer of the velocity across the diffuse interfacial zone is relatively weak.