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


Author: Franck Sueur
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 35B25, 35B30, 76D05, 76D10
DOI: https://doi.org/10.1090/proc/14160
Published electronically: July 23, 2018
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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 [Math. Sci. Res. Inst. Publ., 2, (1984), pp. 85-98] for the convergence of Leray solutions to the Navier-Stokes equations in a bounded domain with a no-slip condition toward smooth solutions to the Euler equation.


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Franck Sueur
Affiliation: Institut de Mathématiques de Bordeaux, UMR CNRS 5251, Université de Bordeaux
Email: Franck.Sueur@math.u-bordeaux.fr

DOI: https://doi.org/10.1090/proc/14160
Keywords: Vanishing viscosity, vortex sheets, boundary layer theory
Received by editor(s): August 29, 2017
Received by editor(s) in revised form: March 14, 2018
Published electronically: July 23, 2018
Additional Notes: The author thanks the Agence Nationale de la Recherche, Project IFSMACS, grant ANR-15-CE40-0010 and Project BORDS, grant ANR-16-CE40-0027-01.
Communicated by: Catherine Sulem
Article copyright: © Copyright 2018 American Mathematical Society

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