The role of density in the energy conservation for the isentropic compressible Euler equations

TL;DR

This paper investigates how the integrability of density affects energy conservation in the isentropic compressible Euler equations, extending known results beyond the bounded density case.

Contribution

It establishes a new energy conservation criterion involving density integrability, revealing the interplay between density and velocity regularity.

Findings

Lower density integrability requires higher velocity integrability for energy conservation.

Results extend energy conservation criteria to densities in L^k(0,T;L^l).

Proof uses commutator estimates of Constantin-E-Titi and Lions types.

Abstract

In this paper, we study Onsager's conjecture on the energy conservation for the isentropic compressible Euler equations via establishing the energy conservation criterion involving the density $\varrho\in L^{k}(0,T;L^{l}(\mathbb{T}^{d}))$. The motivation is to analysis the role of the integrability of density of the weak solutions keeping energy in this system, since almost all known corresponding results require $\varrho\in L^{\infty}(0,T;L^{\infty}(\mathbb{T}^{d}))$. Our results imply that the lower integrability of the density $\varrho$ means that more integrability of the velocity $v$ are necessary in energy conservation and the inverse is also true. The proof relies on the Constantin-E-Titi type and Lions type commutators on mollifying kernel.