On the Endpoint Regularity in Onsager's Conjecture

TL;DR

This paper advances the understanding of Onsager's conjecture by constructing energy non-conserving solutions with regularity approaching the critical exponent, using innovative convex integration techniques and localization methods.

Contribution

It introduces a new convex integration approach with optimized regularity, improving previous results towards the critical endpoint case of Onsager's conjecture.

Findings

Constructed solutions with regularity approaching the critical exponent [0,1/3)

Developed a new method to optimize regularity in convex integration schemes

Proved that energy dissipating solutions cannot satisfy Kolmogorov-Obukhov scaling under certain conditions

Abstract

Onsager's conjecture states that the conservation of energy may fail for $3D$ incompressible Euler flows with H\"{o}lder regularity below $1/3$. This conjecture was recently solved by the author, yet the endpoint case remains an interesting open question with further connections to turbulence theory. In this work, we construct energy non-conserving solutions to the $3D$ incompressible Euler equations with space-time H\"{o}lder regularity converging to the critical exponent at small spatial scales and containing the entire range of exponents $[0,1/3)$. Our construction 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 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 \cite{IOnonpd} to modify the convex integration scheme. We also prove results on general solutions at the critical regularity that may not conserve energy. These include a theorem on intermittency stating roughly that energy dissipating solutions cannot have absolute structure functions satisfying the Kolmogorov-Obukhov scaling for any $p > 3$ if their singular supports have space-time Lebesgue measure zero.