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

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.

arXiv comment15 pages. Improved exposition, corrected typos. Added a criterion for energy conservation in terms of the Hölder norm in Theorem 1.2