The role of density in the energy conservation for the isentropic compressible Euler equations
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 ConstantinETiti type and Lions type commutators on mollifying kernel.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.07267
 Bibcode:
 2021arXiv211007267W
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 17 pages