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

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