On the zero-viscosity limit of the Navier–Stokes equations in [formula omitted] without analyticity

Abstract

We consider the zero viscosity limit of the incompressible Navier–Stokes equations with non-slip boundary condition in R+3 for the initial vorticity located away from the boundary. Unlike 2-D case, this kind of data is not analytic in (x,y) for y close to the boundary in 3-D. Maekawa proved the local in time convergence of the Navier–Stokes equations in R+2 to the Euler equations outside a boundary layer and to the Prandtl equations in the boundary layer. His proof used Cauchy–Kowaleskaya theorem, where the vorticity formulation and Euler–Prandtl decomposition play an important role. In this paper, we generalize Maekawa's result to R+3 by using a direct energy method, which may be applicable for general physical domain.