A Kato type criterion for the zero viscosity limit of the incompressible Navier-Stokes flows with vortex sheets data

Abstract

There are a few examples of solutions to the incompressible Euler equations which are piecewise smooth with a discontinuity of the tangential velocity across a hypersurface evolving in time: the so-called vortex sheets. An important open problem is to determine whether or not these solutions can be obtained as zero viscosity limits of the incompressible Navier-Stokes solutions in the energy space. In this paper we establish a couple of sufficient conditions similar to the one obtained by Kato in [T.~Kato. Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary. Seminar on nonlinear partial differential equations, 85-98, Math. Sci. Res. Inst. Publ., 2, 1984] for the convergence of Leray solutions to the Navier-Stokes equations in a bounded domain with no-slip condition towards smooth solutions to the Euler equation.