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Viscosité évanescente dans les équations de la mécanique des Fluides bidimensionnels.

Abstract : My thesis is devoted to the study of some problems related to the stability of the vortex patches structures in the two-dimensional incompressible Navier-Stokes equations. In the first chapter we prove, in particular, that if the initial vorticity is the characteristic function of a bounded domain (vortex patch) whose boundary is $C^{1+EE}$ (Hölderian spaces), then its image through the viscous flow preserves for all time this regularity. We show in the second chapter that the viscous vorticity goes strongly in $L^p$ to the Eulerian one in the case of initial vortex patches with a boundary having a null measure. The last chapter generalizes for the Navier-Stokes system a result due to J.-Y. Chemin, concerning the singular vortex patches. We prove that if the boundary of the initial vortex patch is regular outside a closed set, then its image through the viscous flow is regular outside the image of the singular set. We also prove that the Navier-Stokes solution is Lipschitzian far from this set, with uniform estimates in terms of the viscosity.
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Submitted on : Wednesday, July 21, 2010 - 1:05:42 PM
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  • HAL Id : pastel-00000827, version 1



Taoufik Hmidi. Viscosité évanescente dans les équations de la mécanique des Fluides bidimensionnels.. Mathématiques [math]. Ecole Polytechnique X, 2003. Français. ⟨pastel-00000827⟩



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