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Geometric Hydrodynamics in Open Problems

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Abstract

Geometric Hydrodynamics has flourished ever since the celebrated 1966 paper of V. Arnold. In this paper we present a collection of open problems along with several new constructions in fluid dynamics and a concise survey of recent developments and achievements in this area. The topics discussed include variational settings for different types of fluids, models for invariant metrics, the Cauchy and boundary value problems, partial analyticity of solutions to the Euler equations, their steady and singular vorticity solutions, differential and Hamiltonian geometry of diffeomorphism groups, long-time behaviour of fluids, as well as mechanical models of direct and inverse cascades.

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Notes

  1. We thank the anonymous referee for this remark.

  2. V. Yudovich referred to this phenomenon as “regularity deterioration”.

  3. It is not clear if such an object has anything in common with the entropy of an invariant measure as typically defined in the theory of dynamical systems.

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Acknowledgements

We are indebted to the organizers of the IAS Symplectic Dynamics Program, where this paper was conceived, and in particular to Helmut Hofer for his constant encouragement. We are also grateful to Theodore Drivas and to the anonymous referee for many useful suggestions. B.K. was partially supported by a Simons Fellowship and an NSERC Discovery Grant. G.M. gratefully acknowledges support from the SCGP, Stony Brook University, where part of the work on this paper was done. Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, Canada

    Boris Khesin

  2. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Gerard Misiołek

  3. Department of Mathematics and Statistics, Concordia University, Montreal, Canada

    Alexander Shnirelman

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  1. Boris Khesin
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  2. Gerard Misiołek
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  3. Alexander Shnirelman
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Correspondence to Boris Khesin.

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Khesin, B., Misiołek, G. & Shnirelman, A. Geometric Hydrodynamics in Open Problems. Arch Rational Mech Anal 247, 15 (2023). https://doi.org/10.1007/s00205-023-01848-x

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