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Volume 25, Issue 1
Boundary Layers Associated with a Coupled Navier-Stokes/Allem-Cahn System: the Non-characteristic Boundary Case

Xiaoqiang Xie

DOI: 10.4208/jpde.v25.n1.5

J. Part. Diff. Eq., 25 (2012), pp. 66-78.

Published online: 2012-03

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  • Abstract

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.

  • Keywords

Boundary layers Navier-Stokes Euler equations Allen-Cahn vanishing viscosity limit

  • AMS Subject Headings

35Q35, 35K55, 76D05

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

xiaoqiangxie021@gmail.com (Xiaoqiang Xie)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-25-66, author = {Xie , Xiaoqiang}, title = {Boundary Layers Associated with a Coupled Navier-Stokes/Allem-Cahn System: the Non-characteristic Boundary Case}, journal = {Journal of Partial Differential Equations}, year = {2012}, volume = {25}, number = {1}, pages = {66--78}, abstract = {

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.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v25.n1.5}, url = {http://global-sci.org/intro/article_detail/jpde/5175.html} }
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 -

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.

Xiaoqiang Xie. (2019). Boundary Layers Associated with a Coupled Navier-Stokes/Allem-Cahn System: the Non-characteristic Boundary Case. Journal of Partial Differential Equations. 25 (1). 66-78. doi:10.4208/jpde.v25.n1.5
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