Boundary layers for the 2D linearized primitive equations

Makram Hamouda; Chang-Yeol Jung; Roger Temam

doi:10.3934/cpaa.2009.8.335

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2009, Volume 8, Issue 1: 335-359. Doi: 10.3934/cpaa.2009.8.335

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Boundary layers for the 2D linearized primitive equations

1.
Faculty of Sciences of Bizerte, Department of Mathematics,7021 Zarzouna, Bizerte, Tunisia, The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 E. 3rd St., Rawles Hall, Bloomington, IN 47405
2.
Department of Mathematics, Arizona State University,Tempe, AZ 85287-1804, The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 E. 3rd St., Rawles Hall, Bloomington, IN 47405
3.
The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 E. 3rd St., Rawles Hall, Bloomington, IN 47405

Received: June 30, 2008

Revised: August 31, 2008

Published: December 2008

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Abstract

In this article, we establish the asymptotic behavior, when the viscosity goes to zero, of the solutions of the Linearized Primitive Equations (LPEs) in space dimension $2$. More precisely, we prove that the LPEs solution behaves like the corresponding inviscid problem solution inside the domain plus an explicit corrector function in the neighborhood of some parts of the boundary. Two cases are considered, the subcritical and supercritical modes depending on the fact that the frequency mode is less or greater than the ratio between the reference stratified flow (around which we linearized) and the buoyancy frequency. The problem of boundary layers for the LPEs is of a new type since the corresponding limit problem displays a set of (unusual) nonlocal boundary conditions.

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Mathematics Subject Classification: 35L50, 47D06, 76D10, 86A05.

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