Abstract
n this paper, we focus on the energy conservation for the weak solutions
of inhomogeneous Navier-Stokes equations. It is proved that if the
function of density belongs to
$L^{\infty}(0,T;L^{\infty}(\mathbb{T}^N))\cap
L^p(0,T;W^{1,p}(\mathbb{T}^N))$, and the
function of velocity belongs to
$L^s(0,T;L^r(\mathbb{T}^N))$ with
$\f2s+\f2r=1$, then the energy equality
holds. This result can be seen as a inhomogeneous version for Shinbrot’s
criterion.