Calculus of Variations and Geometric Measure Theory

T. Buckmaster - C. De Lellis - L. J. Székelyhidi

Dissipative Euler flows with Onsager-critical spatial regularity

created by delellis on 05 May 2014
modified on 04 Sep 2017


Published Paper

Inserted: 5 may 2014
Last Updated: 4 sep 2017

Journal: Comm. Pure Appl. Math.
Volume: 69
Pages: 1613-1670
Year: 2016


For any $\varepsilon >0$ we show the existence of continuous periodic weak solutions $v$ of the Euler equations which do not conserve the kinetic energy and belong to the space $L^1_t (C_x^{1/3-\varepsilon})$, namely $x\mapsto v (x,t)$ is $(1/3-\varepsilon)$-H\"older continuous in space at a.e. time $t$ and the integral $\int [v(\cdot, t)]_{1/3-\varepsilon}\, dt$ is finite. A well-known open conjecture of L. Onsager claims that such solutions exist even in the class $L^\infty_t (C_x^{1/3-\varepsilon})$