Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density

Abstract

In this paper, we consider the initial-boundary value problem to the nonhomogeneous incompressible Navier-Stokes equations. Local strong solutions are established, for any initial data $(\rho_0, u_0)\in (W^{1,\gamma} \cap L^\infty)\times H_{0,\sigma}^1$, with $\gamma>1$, and if $\gamma\geq2$, then the strong solution is unique. The initial density is allowed to be nonnegative, and in particular, the initial vacuum is allowed. The assumption on the initial data is weaker than the previous widely used one that $(\rho_0, u_0)\in (H^1 \cap L^\infty )\times(H_{0,\sigma}^1 \cap H^2)$, and no compatibility condition is required.