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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Homogeneous solutions to the 3D Euler system

Author: Roman Shvydkoy
Journal: Trans. Amer. Math. Soc. 370 (2018), 2517-2535
MSC (2010): Primary 76B99, 37J45
Published electronically: October 5, 2017
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Abstract: We study stationary homogeneous solutions to the 3D Euler equation. The problem is motivated by recent exclusions of self-similar blowup for Euler and its relation to the Onsager conjecture and intermittency. We reveal several new classes of solutions and prove rigidity properties in specific categories of genuinely 3D solutions. In particular, irrotational solutions are characterized by vanishing of the Bernoulli function, and tangential flows are necessarily 2D axisymmetric pure rotations. In several cases solutions are excluded altogether. The arguments reveal geodesic features of the Euler flow on the sphere. We further show that in the case when homogeneity corresponds to the Onsager-critical state, the anomalous energy flux at the singularity vanishes, which is suggestive of absence of extreme 0-dimensional intermittencies in dissipative flows.

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Roman Shvydkoy
Affiliation: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 322 Science and Engineering Offices (M/C 249), 851 S. Morgan Street, Chicago, Illinois 60607-7045

Keywords: Euler equation, homogeneous solution, Onsager conjecture, Landau solution
Received by editor(s): January 7, 2016
Received by editor(s) in revised form: July 15, 2016
Published electronically: October 5, 2017
Additional Notes: The work of the author was partially supported by NSF grants DMS-1210896 and DMS-1515705.
Article copyright: © Copyright 2017 American Mathematical Society