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A heat flow approach to Onsager's conjecture for the Euler equations on manifolds

Isett, Philip and Oh, Sung-Jin (2016) A heat flow approach to Onsager's conjecture for the Euler equations on manifolds. Transactions of the American Mathematical Society, 368 (9). pp. 6519-6537. ISSN 0002-9947. doi:10.1090/tran/6549.

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We give a simple proof of Onsager's conjecture concerning energy conservation for weak solutions to the Euler equations on any compact Riemannian manifold, extending the results of Constantin-E-Titi and Cheskidov-Constantin-Friedlander-Shvydkoy in the flat case. When restricted to T^d or R^d, our approach yields an alternative proof of the sharp result of the latter authors. Our method builds on a systematic use of a smoothing operator defined via a geometric heat flow, which was considered by Milgram-Rosenbloom as a means to establish the Hodge theorem. In particular, we present a simple and geometric way to prove the key nonlinear commutator estimate, whose proof previously relied on a delicate use of convolutions.

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Additional Information:© 2015 American Mathematical Society. Received by the editors May 4, 2014 and, in revised form, August 25, 2015. Article electronically published on November 17, 2015. The second author is a Miller research fellow, and would like to thank the Miller Institute for support.
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Miller Institute for Basic Research in ScienceUNSPECIFIED
Issue or Number:9
Classification Code:2010 Mathematics Subject Classification. Primary 58J35, 35Q31
Record Number:CaltechAUTHORS:20180627-084240647
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:87374
Deposited By: Tony Diaz
Deposited On:27 Jun 2018 16:15
Last Modified:15 Nov 2021 20:47

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