Dissipative Euler flows with Onsager-critical spatial regularity

TL;DR

This paper constructs continuous, energy-dissipating weak solutions to the Euler equations with spatial regularity just below the Onsager critical exponent, advancing understanding of turbulence and energy conservation.

Contribution

It demonstrates the existence of dissipative Euler solutions with spatial regularity at the Onsager-critical threshold, extending previous results to the borderline case.

Findings

Existence of continuous dissipative solutions with $C^{1/3-\epsilon}$ regularity.

Solutions do not conserve kinetic energy.

Solutions are in $L^1_t(C_x^{1/3-\epsilon})$ space.

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 $L^1_t (C_x^{\frac{1}{3}-\epsilon})$, namely $x\mapsto v (x,t)$ is $(\frac{1}{3}-\epsilon)$-H\"older 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 $L^\infty_t (C_x^{\frac{1}{3}-\epsilon})$.