Dissipative Euler flows and Onsager's conjecture

Abstract

Building upon the techniques introduced in \cite{DS3}, for any $\theta<\frac{1}{10}$ we construct periodic weak solutions of the incompressible Euler equations which dissipate the total kinetic energy and are Hölder-continuous with exponent $\theta$. A famous conjecture of Onsager states the existence of such dissipative solutions with any Hölder exponent $\theta<\frac{1}{3}$. Our theorem is the first result in this direction.