Fujita-Kato Theorem for the Inhomogeneous Incompressible Navier-Stokes Equations with Nonnegative Density

Abstract

In this paper, we prove the global existence and uniqueness of solutions for the inhomogeneous Navier-Stokes equations with the initial data $(\rho_0,u_0)\in L^∞\times H^s_0$, $s>\frac{1}{2}$ and $||u_0||_{H^s_0}\leq \varepsilon_0$ in bounded domain $\Omega \subset \mathbb{R}^3$, in which the density is assumed to be nonnegative. The regularity of initial data is weaker than the previous $(\rho_0,u_0)\in (W^{1,\gamma}∩L^∞)\times H^1_0$ in [13] and $(\rho_0,u_0)\in L^∞\times H^1_0$ in [7], which constitutes a positive answer to the question raised by Danchin and Mucha in [7]. The methods used in this paper are mainly the classical time weighted energy estimate and Lagrangian approach, and the continuity argument and shift of integrability method are applied to complete our proof.