Sharp interface limit for inhomogeneous incompressible Navier-Stokes/Allen-Cahn system in a bounded domain via a relative energy method

This paper concerns the sharp interface limit of solutions to the inhomogeneous incompressible Navier-Stokes/Allen-Cahn coupled system in a bounded domain $\Omega \subset \mathbb{R}^n,\ n =2,3$. Based on a relative energy method, we prove that the solutions to the Navier-Stokes/Allen-Cahn system converge to the corresponding solutions to a sharp interface model provided that the thickness of the diffuse interfacial zone goes to zero. It is noted that the relative energy method can avoid both the spectral estimates of the linearized Allen-Cahn operator and the construction of approximate solutions by the matched asymptotic expansion method in the study of the sharp interface limit process. And some suitable functionals are designed and estimated by elaborated energy methods accordingly.