Energy conservation for inhomogeneous incompressible and compressible Euler equations

TL;DR

This paper establishes conditions under which weak solutions to inhomogeneous Euler equations conserve energy, requiring minimal regularity of the density, and extends known results to more general pressure laws and domains.

Contribution

It provides new sufficient conditions for energy conservation in inhomogeneous Euler equations with minimal density regularity and general pressure laws, extending classical results.

Findings

Energy conservation holds under specific regularity conditions.

Density regularity requirements are reduced to 2/3 for incompressible and 1/3 for compressible cases.

Results recover classical energy conservation when density is constant.

Abstract

Energy conservations are studied for inhomogeneous incompressible and compressible Euler equations with general pressure law in a torus or a bounded domain. We provide sufficient conditions for a weak solution to conserve the energy. By exploiting a suitable test function, the spatial regularity for the density is only required to be of order $2/3$ in the incompressible case, and of order $1/3$ in the compressible case. When the density is constant, we recover the existing results for classical incompressible Euler equation.