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

2013/10/29 by Philip Isett, Isett, Philip, Sung‐Jin Oh +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1310.7947

openalex publication_date 2013/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a simple proof of Onsager's conjecture concerning energy conservation\nfor weak solutions to the Euler equations on any compact Riemannian manifold,\nextending the results of Constantin-E-Titi and\nCheskidov-Constantin-Friedlander-Shvydkoy in the flat case. When restricted to\n mathbbT^d or \ℝ^d, our approach yields an alternative proof\nof the sharp result of the latter authors.\n Our method builds on a systematic use of a smoothing operator defined via a\ngeometric heat flow, which was considered by Milgram-Rosenbloom as a means to\nestablish the Hodge theorem. In particular, we present a simple and geometric\nway to prove the key nonlinear commutator estimate, whose proof previously\nrelied on a delicate use of convolutions.\n

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