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Onsager’s Conjecture with Physical Boundaries and an Application to the Vanishing Viscosity Limit

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Abstract

We consider the incompressible Euler equations in a bounded domain in three space dimensions. Recently, the first two authors proved Onsager’s conjecture for bounded domains, i.e., that the energy of a solution to these equations is conserved provided the solution is Hölder continuous with exponent greater than 1/3, uniformly up to the boundary. In this contribution we relax this assumption, requiring only interior Hölder regularity and continuity of the normal component of the energy flux near the boundary. The significance of this improvement is given by the fact that our new condition is consistent with the possible formation of a Prandtl-type boundary layer in the vanishing viscosity limit.

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Acknowledgements

The authors would like to thank the anonymous referee for the constructive remarks that have contributed to the improvement of this revised version of this work. C.B. and E.S.T. would like to thank the “South China Research Center for Applied Mathematics and Interdisciplinary Studies (CAMIS)", South China Normal University, for its warm hospitality where this work was completed. The work of E.S.T. was supported in part by the ONR Grant N00014-15-1-2333, and by the Einstein Stiftung/Foundation - Berlin, through the Einstein Visiting Fellow Program, and by the John Simon Guggenheim Memorial Foundation.

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Authors and Affiliations

  1. Laboratoire J.-L. Lions, BP187, 75252, Paris Cedex 05, France

    Claude Bardos

  2. Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, TX, 77843-3368, USA

    Edriss S. Titi

  3. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK

    Edriss S. Titi

  4. Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, 76100, Rehovot, Israel

    Edriss S. Titi

  5. Institut für Angewandte Analysis, Universität Ulm, Helmholtzstraße 18, 89081, Ulm, Germany

    Emil Wiedemann

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  1. Claude Bardos
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  2. Edriss S. Titi
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  3. Emil Wiedemann
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Correspondence to Claude Bardos.

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Communicated by C. Mouhot

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Bardos, C., Titi, E.S. & Wiedemann, E. Onsager’s Conjecture with Physical Boundaries and an Application to the Vanishing Viscosity Limit. Commun. Math. Phys. 370, 291–310 (2019). https://doi.org/10.1007/s00220-019-03493-6

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  • DOI: https://doi.org/10.1007/s00220-019-03493-6

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