Fujita-Kato Theorem for the Inhomogeneous Incompressible Navier-Stokes Equations with Nonnegative Density (Communications in Mathematical Research, 2022)
In this paper, we prove the global existence and uniqueness of solutions for the inhomogeneous Navier-Stokes equations with the initial data $(ρ_0,u_0)\in L^∞× H^s_0$, $s>\frac{1}{2}$ and $||u_0||_{H^s_0}≤ ε_0$ in bounded domain $Ω \subset \mathbb{R}^3$, in which the density is assumed to be nonnegative. The regularity of initial data is weaker than the previous $(ρ_0,u_0)\in (W^{1,γ}∩L^∞)× H^1_0$ in [13] and $(ρ_0,u_0)\in L^∞× H^1_0$ 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.
Keywords: Inhomogeneous Navier-Stokes equations; nonnegative density; global existence and uniqueness.