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

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

2010-12-16

arxiv.org/abs/1012.3529v2

J. Math. Anal. Appl. 384(2011) 232-245

Mathematics

Analysis of PDEs

19 pages

Scientific paper

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}.

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