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A Heuristic Approach to Convex Integration for the Euler Equations

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Progress in Mathematical Fluid Dynamics

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2272))

Abstract

The purpose of these lecture notes is to employ a heuristic approach in designing a convex integration scheme that produces non-conservative weak solutions to the Euler equations.

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Notes

  1. 1.

    In practice is often simpler to assume that the frequencies grow super-exponentially (cf. [2, 5, 12, 17]). However, for the purpose of heuristics, geometric growth simplifies some of the calculations.

  2. 2.

    It should be noted however that in order for the scheme described here to close, one should replace the geometric growth of frequencies λ q described in (1.7) with superexponential growth. A scheme involving geometric growth of frequencies is slightly more delicate to describe.

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Acknowledgements

T.B. was supported by the NSF grant DMS-1900149 and a Simons Foundation Mathematical and Physical Sciences Collaborative Grant. V.V. was supported by the NSF grant DMS-1911413.

Author information

Authors and Affiliations

  1. Department of Mathematics, Princeton University, Princeton, NJ, USA

    Tristan Buckmaster

  2. Courant Institute for Mathematical Sciences, New York University, New York, NY, USA

    Vlad Vicol

Authors
  1. Tristan Buckmaster
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  2. Vlad Vicol
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Correspondence to Tristan Buckmaster .

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Editors and Affiliations

  1. Department of Mathematics, University of Pisa, Pisa, Italy

    Luigi C. Berselli

  2. Abteilung für Angewandte Mathematik, Albert-Ludwigs-Universität Freiburg, Freiburg, Germany

    Michael Růžička

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Buckmaster, T., Vicol, V. (2020). A Heuristic Approach to Convex Integration for the Euler Equations. In: Berselli, L.C., Růžička, M. (eds) Progress in Mathematical Fluid Dynamics. Lecture Notes in Mathematics(), vol 2272. Springer, Cham. https://doi.org/10.1007/978-3-030-54899-5_1

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