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

TL;DR

This paper establishes conditions under which solutions to the Navier-Stokes equations with vortex sheet data converge to Euler solutions as viscosity approaches zero, extending Kato's criterion to vortex sheets.

Contribution

It introduces a Kato-type criterion for the zero viscosity limit of Navier-Stokes flows with vortex sheet initial data, addressing a key open problem.

Findings

Provides sufficient conditions for convergence of Navier-Stokes to Euler solutions with vortex sheets.

Extends Kato's criterion to flows with discontinuous tangential velocities.

Clarifies the role of vortex sheets in the zero viscosity limit.

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.