Navier-Stokes equations with Navier boundary conditions for a bounded domain in the plane

Abstract

We consider solutions to the Navier-Stokes equations with Navier boundary conditions in a bounded domain in the plane with a C^2-boundary. Navier boundary conditions can be expressed in the form w = (2 K - A) v . T and v . n = 0 on the boundary, where v is the velocity, w the vorticity, n a unit normal vector, T a unit tangent vector, and A is a bounded measurable function on the boundary. Such solutions have been considered for simply connected domains by Clopeau, Mikeli\'{c}, and Robert under the assumption that the initial vorticity is bounded, and by Lopes Filho, Nussenzveig Lopes, and Planas under the assumption that the initial vorticity lies in L^p for some p > 2. We extend the results of these authors to non-simply connected domains. Assuming a particular bound on the growth of the L^p-norms of the initial vorticity with p, and also assuming that the boundary and the function A have fractionally greater smoothness, we obtain a bound on the rate of convergence in L^2 uniform over any finite time interval to the solution of the Euler equations in the vanishing viscosity limit. We also show that if the initial velocity is in H^3 and the boundary is C^3, then solutions to the Navier-Stokes equations with Navier boundary conditions converge in L^2 uniformly over any finite time interval to the solution to the Navier-Stokes equations with the usual no-slip boundary conditions as we let the function A grow large uniformly on the boundary.