On the Endpoint Regularity in Onsager's Conjecture

Philip Isett

Published 2017-06-05Version 1

Onsager's conjecture states that the conservation of energy may fail for $3D$ incompressible Euler flows with H\"older regularity below $1/3$. This conjecture was recently solved by the author, yet the endpoint case remains an interesting open question. In this work, we construct energy non-conserving solutions to the $3D$ incompressible Euler equations with space-time H\"older regularity converging to the critical exponent at small spatial scales and containing the entire range of exponents $[0,1/3)$. This result improves the author's previous result towards the endpoint case. To obtain this improvement, we introduce a new method for optimizing the regularity that can be achieved by a general convex integration scheme. A crucial point is to avoid power-losses in frequency in the estimates of the iteration. This goal is achieved using localization techniques of [Isett-Oh, 2014] to modify the convex integration scheme.