On the endpoint regularity in Onsager’sconjecture

Abstract

Onsager's conjecture states that the conservation of energy may fail for $3D$\nincompressible Euler flows with H\\"{o}lder regularity below $1/3$. This\nconjecture was recently solved by the author, yet the endpoint case remains an\ninteresting open question with further connections to turbulence theory. In\nthis work, we construct energy non-conserving solutions to the $3D$\nincompressible Euler equations with space-time H\\"{o}lder regularity converging\nto the critical exponent at small spatial scales and containing the entire\nrange of exponents $[0,1/3)$.\n Our construction improves the author's previous result towards the endpoint\ncase. To obtain this improvement, we introduce a new method for optimizing the\nregularity that can be achieved by a convex integration scheme. A crucial point\nis to avoid power-losses in frequency in the estimates of the iteration. This\ngoal is achieved using localization techniques of \\cite{IOnonpd} to modify the\nconvex integration scheme.\n We also prove results on general solutions at the critical regularity that\nmay not conserve energy. These include a theorem on intermittency stating\nroughly that energy dissipating solutions cannot have absolute structure\nfunctions satisfying the Kolmogorov-Obukhov scaling for any $p > 3$ if their\nsingular supports have space-time Lebesgue measure zero.\n