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Bulletin of the American Mathematical Society

Published by the American Mathematical Society, the Bulletin of the American Mathematical Society (BULL) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.47.

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Geometric hydrodynamics and infinite-dimensional Newton’s equations
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by Boris Khesin, Gerard Misiołek and Klas Modin HTML | PDF
Bull. Amer. Math. Soc. 58 (2021), 377-442 Request permission

Abstract:

We revisit the geodesic approach to ideal hydrodynamics and present a related geometric framework for Newton’s equations on groups of diffeomorphisms and spaces of probability densities. The latter setting is sufficiently general to include equations of compressible and incompressible fluid dynamics, magnetohydrodynamics, shallow water systems and equations of relativistic fluids. We illustrate this with a survey of selected examples, as well as with new results, using the tools of infinite-dimensional information geometry, optimal transport, the Madelung transform, and the formalism of symplectic and Poisson reduction.
References
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Additional Information
  • Boris Khesin
  • Affiliation: Department of Mathematics, University of Toronto, Toronto, Canada
  • MR Author ID: 238631
  • Gerard Misiołek
  • Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana
  • Klas Modin
  • Affiliation: Department of Mathematics, Chalmers University of Technology, Gothenburg, Sweden; and University of Gothenburg, Gothenburg, Sweden
  • MR Author ID: 798387
  • ORCID: 0000-0001-6900-1122
  • Received by editor(s): February 3, 2020
  • Published electronically: June 2, 2021
  • Additional Notes: Part of this work was done while the first author held the Pierre Bonelli Chair at the IHES. He was also partially supported by an NSERC research grant and a Simons Fellowship.
    Part of this work was done while the second author held the Ulam Chair Visiting Professorship in University of Colorado at Boulder.
    The third author was supported by the Swedish Foundation for International Cooperation in Research and Higher Eduction (STINT) grant No. PT2014-5823, by the Swedish Research Council (VR) grant No. 2017-05040, and by the Knut and Alice Wallenberg Foundation grant No. WAF2019.0201.

  • Dedicated: To the memory of Vladimir Arnold and Jerry Marsden, pioneers of geometric hydrodynamics, who left in 2010, ten years ago.
  • © Copyright 2021 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 58 (2021), 377-442
  • MSC (2010): Primary 35Q35, 58B20, 76-02
  • DOI: https://doi.org/10.1090/bull/1728
  • MathSciNet review: 4273106