On the Onsager conjecture in two dimensions

TL;DR

This paper investigates energy conservation in 2D Euler equations, establishing conditions for conservation based on vorticity integrability, constructing a counterexample at the critical space, and showing energy conservation in vanishing viscosity limits.

Contribution

It provides a direct mollification-based proof of energy conservation for vorticity in L^{3/2}, constructs a sharp counterexample at the Onsager critical space, and proves energy conservation in supercritical regimes via vanishing viscosity.

Findings

Energy is conserved if vorticity is in L^{3/2}.

Constructed a counterexample with non-vanishing flux at the critical space.

Solutions from vanishing viscosity with vorticity in L^p (p>1) conserve energy.

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 $\omega = \nabla \times u \in L^{\frac32}$. An example of a 2D field in the class $\omega \in L^{\frac32 - \epsilon}$ for any $\epsilon>0$, and $u\in B^{1/3}_{3,\infty}$ (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.