Inviscid Limit for Vortex Patches in A Bounded Domain

2010/04/23 by Quansen Jiu, Yun Wang, Jiu, Quansen +1
Mathematics · #35Q31 #35Q35 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35Q31 #msc:35Q35

paper · pdf · doi:10.48550/arxiv.1004.4033

arxiv created 2010/04/23 · arxiv updated 2010/04/26

Abstract

In this paper, we consider the inviscid limit of the incompressible Navier-Stokes equations in a smooth, bounded and simply connected domain Ω⊂ ℝ^d, d=2,3. We prove that for a vortex patch initial data the weak Leray solutions of the incompressible Navier-Stokes equations with Navier boundary conditions will converge (locally in time for d=3 and globally in time for d=2) to a vortex patch solution of the incompressible Euler equation as the viscosity vanishes. In view of the results obtained in [1] and [19] which dealt with the case of the whole space, we derive an almost optimal convergence rate (νt)^\frac34-ε for any small ε>0 in L².

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