Vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system

TL;DR

This paper investigates the vanishing viscosity limit of a coupled Navier-Stokes/Allen-Cahn system, proving convergence to Euler/Allen-Cahn solutions under certain conditions and establishing a convergence rate in 2D.

Contribution

It introduces a modified Galerkin method and boundary layer analysis to prove convergence of Navier-Stokes/Allen-Cahn solutions to Euler/Allen-Cahn solutions as viscosity vanishes.

Findings

Solutions converge in small time intervals as viscosity tends to zero.

Boundary layer functions effectively handle boundary condition mismatches.

Convergence rate in 2D is proportional to the square root of viscosity.

Abstract

In this paper, we study the vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system in a bounded domain. We first show the local existence of smooth solutions of the Euler/Allen-Cahn equations by modified Galerkin method. Then using the boundary layer function to deal with the mismatch of the boundary conditions between Navier-Stokes and Euler equations, and assuming that the energy dissipation for Navier-Stokes equation in the boundary layer goes to zero as the viscosity tends to zero, we prove that the solutions of the Navier-Stokes/Allen-Cahn system converge to that of the Euler/Allen-Cahn system in a proper small time interval. In addition, for strong solutions of the Navier-Stokes/Allen-Cahn system in 2D, the convergence rate is c\nu^{1/2}.