Inviscid limit for vortex patches in a bounded domain

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 Abidi and Danchin (2004) [5] and Masmoudi (2007) [3] which dealt with the case of the whole space, we derive an almost optimal convergence rate ( ν t ) 3 4 − ε for any small ε > 0 in L 2 .