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

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 $L (\operatorname{log} L)^\alpha$ with $\alpha \gt 1/2$.

Keywords

2D Euler equations, vanishing viscosity, vortex methods, conservation of energy

2010 Mathematics Subject Classification

35Q31, 35Q35

Full Text (PDF format)

Received 3 March 2021

Received revised 28 September 2021

Accepted 6 October 2021

Published 21 March 2022