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October  2010, 26(4): 1213-1240. doi: 10.3934/dcds.2010.26.1213

Boundary layers in smooth curvilinear domains: Parabolic problems

Gung-Min Gie 1, , Makram Hamouda 1, and  Roger Témam 2,

1. 

The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 East Third Street, Bloomington, Indiana 47405, United States, United States
2. 

Institute for Scientific Computing and Applied Mathematics, Indiana University, Rawles Hall, 831 E. Third Street, Bloomington, IN 47405

Received  March 2009 Revised  May 2009 Published  December 2009

The goal of this article is to study the boundary layer of the heat equation with thermal diffusivity in a general (curved), bounded and smooth domain in $\mathbb{R}^{d}$, $d \geq 2$, when the diffusivity parameter ε is small. Using a curvilinear coordinate system fitting the boundary, an asymptotic expansion, with respect to ε, of the heat solution is obtained at all orders. It appears that unlike the case of a straight boundary, because of the curvature of the boundary, two correctors in powers of ε and ε1/2 must be introduced at each order. The convergence results, between the exact and approximate solutions, seem optimal. Beside the intrinsic interest of the results presented in the article, we believe that some of the methods introduced here should be useful to study boundary layers for other problems involving curved boundaries.
Keywords: heat equation, singular perturbations, boundary layers, curvilinear coordinates..
Mathematics Subject Classification: 35B25, 35C20, 35K0.
Citation: Gung-Min Gie, Makram Hamouda, Roger Témam. Boundary layers in smooth curvilinear domains: Parabolic problems. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1213-1240. doi: 10.3934/dcds.2010.26.1213
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