A new boundary condition for the three-dimensional Navier-Stokes equation and the vanishing viscosity limit

Abstract

In this paper, we propose a new vorticity boundary condition for the three-dimensional incompressible Navier-Stokes equation for a general smooth domain in R3. This boundary condition is motivated by the generalized Navier-slip boundary condition involving the vorticity. It is shown first that such an initial boundary value problem is well-posed at least local in time. Furthermore, more importantly, we obtain some estimates on rate of convergence in C([0, T], H1(Ω)) and C([0, T], H2(Ω)) of the solutions to the corresponding solutions of the ideal Euler equations with the standard slip boundary condition.