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Zygmund Spaces, Inviscid Limit and Uniqueness of Euler Flows

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Abstract

The paper improves the classical uniqueness result for the incompressible Euler system in the n dimensional case assuming that \(\nabla u^E \in L_1(0,T;{\it BMO}(\Omega))\) , only. Moreover the rate of the convergence for the inviscid limit of solutions to the Navier-Stokes equations is obtained, under the same regularity of the limit Eulerian flow. A key element of the proof is a logarithmic inequality between the Hardy and L 1 spaces which is a consequence of the basic properties of the Zygmund space L ln L.

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Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, Warsaw University, ul. Banacha 2, 02-097, Warszawa, Poland

    Piotr Bogusław Mucha

  2. School of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis, 55455, MN, USA

    Walter M. Rusin

Authors
  1. Piotr Bogusław Mucha
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  2. Walter M. Rusin
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Correspondence to Walter M. Rusin.

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Communicated by P. Constantin

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Mucha, P.B., Rusin, W.M. Zygmund Spaces, Inviscid Limit and Uniqueness of Euler Flows. Commun. Math. Phys. 280, 831–841 (2008). https://doi.org/10.1007/s00220-008-0452-2

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  • DOI: https://doi.org/10.1007/s00220-008-0452-2

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