Global Strong Solution with Vacuum to the Two Dimensional Density-dependent Navier--Stokes System

Abstract

In this paper, we consider the two-dimensional density-dependent Navier--Stokes equations over bounded domains. First, we derive a new blow-up criterion for strong solutions with vacuum, which involves only the $L^p$-norm of $\nabla \mu(\rho)$. As a corollary, the global existence of strong solutions with vacuum is derived for the constant viscosity. The key idea is the utilization of a lemma proved by Desjardins [Arch. Rational Mech. Anal., 137 (1997), pp. 135--158]. Second, in addition to the blow-up criterion, strong solutions are proved to exist globally when the $L^p$-norm of $\nabla \mu(\rho_0)$ is suitably small, even in the presence of vacuum. This improves all of the previous global existence result where the density needs to be strictly positive.