Energy balance for forced two-dimensional incompressible ideal fluid flow

TL;DR

This paper extends previous results on energy conservation in 2D Euler equations by including external forcing, demonstrating that physically realizable weak solutions still conserve energy under these conditions.

Contribution

It generalizes the energy conservation result to forced flows, broadening the understanding of weak solutions in 2D incompressible fluid dynamics.

Findings

Energy conservation holds for forced flows with bounded initial vorticity.

Extension of previous unforced flow results to include external forcing.

Conditions under which weak solutions preserve kinetic energy with forcing.

Abstract

In [Commun Math Phys 348(1), 129-143, 2016], Cheskidov et al. proved that physically realizable weak solutions of the incompressible 2D Euler equations on a torus conserve kinetic energy. Physically realizable weak solutions are those that can be obtained as limits of vanishing viscosity. The key hypothesis was boundedness of the initial vorticity in $L^p$, $p>1$. In this work we extend their result, by adding forcing to the flow.