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A Note on the Vanishing Viscosity Limit in the Yudovich Class

Part of: Equations of mathematical physics and other areas of application Incompressible viscous fluids

Published online by Cambridge University Press:  24 April 2020

Christian Seis
Christian Seis*
Affiliation:
Institut für Analysis und Numerik, Westfälische Wilhelms-Universität Münster, Orléans-Ring 10, 48149Münster, Germany
*
e-mail: seis@wwu.de
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Abstract

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We consider the inviscid limit for the two-dimensional Navier–Stokes equations in the class of integrable and bounded vorticity fields. It is expected that the difference between the Navier–Stokes and Euler velocity fields vanishes in $L^2$ with an order proportional to the square root of the viscosity constant $\nu $. Here, we provide an order $ (\nu /|\log \nu | )^{\frac 12\exp (-Ct)}$ bound, which slightly improves upon earlier results by Chemin.

Keywords

Vanishing viscosityYudovichEulerNavier-StokesstabilityWasserstein

MSC classification

Primary: 35Q30: Navier-Stokes equations
Secondary: 76D09: Viscous-inviscid interaction
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 1 , March 2021 , pp. 112 - 122
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Canadian Mathematical Society 2020

Footnotes

This work is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure.

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