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
Td or Rd, 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.Comment: 15 pages. Improved exposition, corrected typos. Added a criterion for
energy conservation in terms of the H\"older norm in Theorem 1.