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Étude de l'approximation hydrostatique de Stokes & d'une équation dégénérée

Abstract : In this work, we study some elliptic partial differential equations problems modelling fluid motion, such as global oceanographic circulation. The thesis is divided into three parts. Part 1 is dedicated to the so called « hydrostatic » Stokes problem in dimension three, set in a bounded domain non necessarily cylindrical. The originality of this work relies in the fact that we consider non homogeneous data, not only in the mass conservation equation but also in the boundary condition carried by the vertical velocity. To handle this new situation, we prove that the difficulty is reduced to solve a non homogeneous linearized primitive equations system, that we solve with an entirely functional and optimal approach given the framework we consider. Therefore, we give two cases of existence and uniqueness of a weak solution to the hydrostatic Stokes problem with non homogeneous conditions. Part 2 and 3 are dedicated to the study of an elliptic model with a diffusion coefficient having a possible degenerated behavior. We can also find these equations in geophysical problems, such as in global circulation modelling questions or seepage and porous media problems. We study the half-space case for which we obtain an optimal regularity theory of weak solutions. Finally, we deal with the general case for which we establish an existence and uniqueness proof of a weak solution, jointly with a regularity result.
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Submitted on : Thursday, January 7, 2010 - 2:15:56 PM
Last modification on : Friday, January 15, 2021 - 9:24:37 AM
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Fabien Dahoumane. Étude de l'approximation hydrostatique de Stokes & d'une équation dégénérée. Mathématiques [math]. Université de Pau et des Pays de l'Adour, 2009. Français. ⟨tel-00444885⟩

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