Inviscid limit for vortex patches in a bounded domain

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 = 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) 3 4 −ε for any small ε > 0 in L.