Dissipative Euler flows with Onsager-critical spatial regularity

Abstract

For any $\epsilon >0$ 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 $L_t^1(C_x^{\frac{1}{3}-\epsilon })$, namely $x\mapsto v(x,t)$ is $(\frac{1}{3}-\epsilon )$-Hölder continuous in space at a.e. time $t$ and the integral $\int [v(\cdot ,t){]}_{\frac{1}{3}-\epsilon }dt$ is finite. A well-known open conjecture of L. Onsager claims that such solutions exist even in the class $L_t^{\infty }(C_x^{\frac{1}{3}-\epsilon })$.

65 pages

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