Maximum principles for the primitive equations of the atmosphere
Brian D. Ewald Roger Témam
Discrete & Continuous Dynamical Systems - A 2001, 7(2): 343-362 doi: 10.3934/dcds.2001.7.343
In this article, maximum principles are derived for a suitably modified form of the equation of temperature for the primitive equations of the atmosphere; we consider both the limited domain case in Cartesian coordinates and the ow of the whole atmosphere in spherical coordinates.
keywords: primitive equations of the atmosphere maximum principles
Boundary layers in smooth curvilinear domains: Parabolic problems
Gung-Min Gie Makram Hamouda Roger Témam
Discrete & Continuous Dynamical Systems - A 2010, 26(4): 1213-1240 doi: 10.3934/dcds.2010.26.1213
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.

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