A heat flow approach to Onsager’s conjecture for the Euler equations on manifolds
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- by Philip Isett and Sung-Jin Oh PDF
- Trans. Amer. Math. Soc. 368 (2016), 6519-6537 Request permission
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
- MR Author ID: 1060366
- ORCID: setImmediate$0.8624889592195648$2
- Email: sjoh@math.berkeley.edu
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 6519-6537
- MSC (2010): Primary 58J35, 35Q31
- DOI: https://doi.org/10.1090/tran/6549
- MathSciNet review: 3461041