TY - JOUR
T1 - Boundary Layers Associated with a Coupled Navier-Stokes/Allem-Cahn System: the Non-characteristic Boundary Case
AU - Xie , Xiaoqiang
JO - Journal of Partial Differential Equations
VL - 1
SP - 66
EP - 78
PY - 2012
DA - 2012/03
SN - 25
DO - http://doi.org/10.4208/jpde.v25.n1.5
UR - https://global-sci.org/intro/article_detail/jpde/5175.html
KW - Boundary layers
KW - Navier-Stokes
KW - Euler equations
KW - Allen-Cahn
KW - vanishing viscosity limit
AB - <p>The goal of this article is to study the boundary layer of Navier-Stokes/Allen- Cahn system in a channel at small viscosity. We prove that there exists a boundary layer at the outlet (down-wind) of thickness n, where n is the kinematic viscosity. The convergence in L^2 of the solutions of the Navier-Stokes/Allen-Cahn equations to that of the Euler/Allen-Cahn equations at the vanishing viscosity was established. In two dimensional case we are able to derive the physically relevant uniform in space and time estimates, which is derived by the idea of better control on the tangential derivative and the use of an anisotropic Sobolve imbedding.</p>