Energy equality in the isentropic compressible NavierStokes equations allowing vacuum
Abstract
It is wellknown that a LerayHopf weak solution in $L^4 (0,T; L^4(\Omega))$ for the incompressible NavierStokes system is persistence of energy due to Lions [19]. In this paper, it is shown that Lions's condition for energy balance is also valid for the weak solutions of the isentropic compressible NavierStokes equations allowing vacuum under suitable integrability conditions on the density and its derivative. This allows us to establish various sufficient conditions implying energy equality for the compressible flow as well as the nonhomogenous incompressible NavierStokes equations. This is an improvement of corresponding results obtained by Yu in [32, Arch. Ration. Mech. Anal., 225 (2017)], and our criterion via the gradient of the velocity partially answers a question posed by Liang in [18, Proc. Roy. Soc. Edinburgh Sect. A (2020)].
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.09425
 Bibcode:
 2021arXiv210809425Y
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 18 pages