Niveau: Supérieur, Doctorat, Bac+8
Stability and Interaction of Vortices in Two-Dimensional Viscous Flows Thierry Gallay Universite de Grenoble I Institut Fourier, UMR CNRS 5582 B.P. 74 F-38402 Saint-Martin-d'Heres, France January 5, 2012 Abstract The aim of these notes is to present in a comprehensive and relatively self-contained way some recent developments in the mathematical analysis of two-dimensional viscous flows. We consider the incompressible Navier-Stokes equations in the whole plane R2, and assume that the initial vorticity is a finite measure. This general setting includes vortex patches, vortex sheets, and point vortices. We first prove the existence of a unique global solution, for any value of the viscosity parameter, and we investigate its long-time behavior. We next consider the particular case where the initial flow is a finite collection of point vortices. In that situation, we show that the solution behaves, in the vanishing viscosity limit, as a superposition of Oseen vortices whose centers evolve according to the Helmholtz-Kirchhoff point vortex system. The proof requires a careful stability analysis of the Oseen vortices in the large Reynolds number regime, as well as a precise computation of the deformations of the vortex cores due to mutual interactions. Contents 1 Introduction 2 2 The Cauchy Problem for the 2D Vorticity Equation 4 2.1 General Properties of the Vorticity Equation .
- global solution
- dimensional biot-savart
- oseen vortices
- self-similar variables
- viscous fluid
- vortex cores
- navier stokes equations
- space
- kinematic viscosity
- vorticity equation