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2010, Volume 26, Issue 4: 1213-1240. Doi: 10.3934/dcds.2010.26.1213
This issue Previous Article Systems of Bellman equations to stochastic differential games with non-compact coupling Next Article Stability of solitary-wave solutions to the Hirota-Satsuma equation

Boundary layers in smooth curvilinear domains: Parabolic problems

  • 1.

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

  • 2.

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

Received:  February 28, 2009
Revised:  April 30, 2009
Published:  November 2010
Abstract Related Papers Cited by
    Abstract
  • 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.
    Mathematics Subject Classification: 35B25, 35C20, 35K05.

    Citation:
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