Effet dispersif pour les fluides anisotropes avec viscosité évanescente en rotation rapide

Abstract : In this work, we study the global existence of strong solutions of anisotropic rotating fluid systems in the whole space $\mathbb{R}^3$, in the case where the data are large and where there are no viscosity in the vertical direction and a small viscosity in the horizontal direction (of size $\varepsilon^{\alpha}$ with $0 < \alpha \leq \alpha_0$, for some $\alpha_0 > 0$). Using Strichartz-type estimates, we prove global existence of strong solutions when the Rossby number $\varepsilon$ is small enough. In the last chapter of this thesis, we prove the analyticity of the global solution of the system of fluids of second grade, for small analytic data. In the first part (third chapter), we consider the Navier-Stokes equations with rotating term $\frac{u\wedge e_3}\varepsilon$, with no vertical viscosity and with horizontal viscosity of size $\varepsilon^\alpha$, $\alpha > 0$. We prove the global existence of a unique, strong solution for large data, provided that $\varepsilon$ is small enough. Using a method of J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, we decompose the system into two parts: a linear system with smooth data and a nonlinear system with small data. An important part of the work is to adapt Strichartz-type estimates and to find new estimates in order to deal with the small viscosities. For the nonlinear system, we use a ``bootstrap'' argument, more delicate than in the classic case because of small viscosities. Always in the third chapter, we consider the rotating fluids system in the ill-prepared case. By adding a ``friction'' term to the system, we prove that we could obtain good dissipative estimates and good properties for the limiting system, which imply global existence of strong solutions for large data. In the last section of this chapter, we study an application of the previously mentioned method for the case of rotating fluids between two infinite parallel plates. More precisely, we prove that the result of E. Grenier and N. Masmoudi also applies in the case where the viscosity is of size $\varepsilon^\alpha$, $\alpha > 0$. The fourth chapter is devoted to study the primitive equations in $\mathbb{R}^3$, in the case where there is no vertical viscosity and where the horizontal viscosity is $\varepsilon^\alpha$, $\alpha > 0$. In this chapter, we apply the method using previously for the primitive equations and we adapt in the anisotropic case the calculation developed by F. Charve for the isotropic case. This enables us to prove the global existence of strong solutions for large data with no quasi-geostrophic part. In the fifth chapter, we study the anisotropic rotating magneto-hydrodynamic system in the whole space $\mathbb{R}^3$. We first prove the local existence (global for small data) and uniqueness results of strong solutions. Then, with precise choices of parameters, we prove certain symmetries of the system, which allow us to use the previous method to prove global existence of strong solutions for large data. Finally, in the last chapter, we consider the problem of propagation of regularity for the system of fluids of second grade on the torus $\TT^3$. Using a technique recently developed by J.-Y. Chemin, we prove that, if the initial data is small in an appropriate Gevrey class, the solution of the system of fluids of second grade exists globally in time, stays in a certain Gevrey class for any positive time, and thus is analytic.
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Van-Sang Ngo. Effet dispersif pour les fluides anisotropes avec viscosité évanescente en rotation rapide. Mathématiques [math]. Université Paris Sud - Paris XI, 2009. Français. ⟨tel-00466698⟩

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