Quelques problèmes relatifs à la dynamique des points vortex dans les équations d'Euler et de Ginzburg-Landau complexe

Abstract : In this dissertation, we study some problems related to vortex dynamics in two equations for two-dimensional fluids or superfluids. The first part is devoted to the incompressible Euler equations. We analyze the so-called Vortex-Wave system, introduced by Marchioro and Pulvirenti, in which the vorticity is given by the superposition of point vortices and of a smoother part. We first examine the link between the lagrangian and eulerian points of view. We then tackle the question of uniqueness. We also study the large time behavior of the support of the vorticity. Finally, we address the problem of existence for more singular vorticities. In the second part of the thesis, we focus on a complex Ginzburg-Landau equation that has the form of a Gross-Pitaevskii equation with some dissipation added. We first study the Cauchy problem in the corresponding energy space. We then turn to the study of vortex dynamics for very well prepared data. At last, we consider another asymptotic regime without vortices for perturbations of the vacuum. We show that the perturbation essentially behaves according to a damped wave-like equation.
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Evelyne Miot. Quelques problèmes relatifs à la dynamique des points vortex dans les équations d'Euler et de Ginzburg-Landau complexe. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2009. Français. ⟨tel-00444820⟩

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