On the Onsager conjecture in two dimensions

Abstract

This note addresses the question of energy conservation for the 2D Euler system with an $L^p$-control on vorticity. We provide a direct argument, based on a mollification in physical space, to show that the energy of a weak solution is conserved if $\omega =\nabla \times u\in L^{\frac{3}{2}}$. An example of a 2D field in the class $\omega \in L^{\frac{3}{2}-\epsilon }$ for any $\epsilon >0$, and $u\in B_{3,\infty }^{1/3}$ (Onsager critical space) is constructed with non-vanishing energy flux. This demonstrates sharpness of the kinematic argument. Finally we prove that any solution to the Euler equation produced via a vanishing viscosity limit from Navier-Stokes, with $\omega \in L^p$, for $p>1$, conserves energy. This is an Onsager-supercritical condition under which the energy is still conserved, pointing to a new mechanism of energy balance restoration.

14 pages