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arXiv · 1012.3529

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

Liyun Zhao·Boling Guo·Haiyang Huang
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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}.

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Liyun Zhao, Boling Guo, Haiyang Huang. 2010-12-16. Vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system. https://arxiv.org/abs/1012.3529

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