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


Authors: Philip Isett and Sung-Jin Oh
Journal: Trans. Amer. Math. Soc. 368 (2016), 6519-6537
MSC (2010): Primary 58J35, 35Q31
Published electronically: November 17, 2015
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Abstract: 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 $ \mathbb{T}^{d}$ or $ \mathbb{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

Philip Isett
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: isett@math.mit.edu

Sung-Jin Oh
Affiliation: Department of Mathematics, University of California Berkeley, Berkeley, California 94720
Email: sjoh@math.berkeley.edu

DOI: https://doi.org/10.1090/tran/6549
Received by editor(s): May 4, 2014
Received by editor(s) in revised form: August 25, 2015
Published electronically: November 17, 2015
Additional Notes: The second author is a Miller research fellow, and would like to thank the Miller Institute for support
Article copyright: © Copyright 2015 American Mathematical Society