Energy conservation for 2D Euler with vorticity in $L(\log L)^\alpha$

TL;DR

This paper investigates energy conservation in weak solutions of 2D Euler equations with vorticity in a specific integrability class, establishing conditions under which energy is conserved based on approximation methods and initial vorticity regularity.

Contribution

It proves that canonical approximations yield energy-conserving solutions for initial vorticity in the class $L( ext{log} L)^eta$ with $eta > 1/2$, extending previous results.

Findings

Energy conservation holds for vorticity in $L^ ext{infty}_t L^p_x$, $p \\geq 3/2$.

Canonical approximations preserve energy for initial vorticity in $L( ext{log} L)^eta$, $eta > 1/2$.

Conservation may depend on approximation methods for less integrable vorticity.

Abstract

In these notes we discuss the conservation of the energy for weak solutions of the two-dimensional incompressible Euler equations. Weak solutions with vorticity in $L^\infty_t L^p_x$ with $p\geq 3/2$ are always conservative, while for less integrable vorticity the conservation of the energy may depend on the approximation method used to construct the solution. Here we prove that the canonical approximations introduced by DiPerna and Majda provide conservative solutions when the initial vorticity is in the class $L(\log L)^\alpha$ with $\alpha>1/2$.