Global refined Fujita-Kato solution of 3-D inhomogeneous incompressible Navier-Stokes equations with large density

.We investigate the global unique Fujita-Kato solution to the 3-D inhomogeneous incompressible Navier-Stokes equations, (Equation1.1(1.1) {∂tρ+div(ρu)=0(∀(t,x)∈R+×R3),ρ(∂tu+u⋅∇u)−Δu+∇Π=0,divu=0,(ρ,u)|t=0=(ρ0,u0),(1.1) ), with initial velocity u0 being sufficiently small in B˙2,∞12 and with initial density being bounded from above and below. We first prove the global existence of the Fujita-Kato solution to the system (Equation1.1(1.1) {∂tρ+div(ρu)=0(∀(t,x)∈R+×R3),ρ(∂tu+u⋅∇u)−Δu+∇Π=0,divu=0,(ρ,u)|t=0=(ρ0,u0),(1.1) ) if we assume in addition that the initial velocity u0∈H˙12. While under the additional assumptions that the initial velocity u0∈B˙2,112 and initial density ρ0 satisfying ρ0−1−1∈B˙6,112, we prove that ∥ρ−1−1∥L~∞(R+;B˙6,112) and ∥u∥L~∞(R+;B˙2,112)∩L1(R+;B˙2,152) are controlled by the norm of the initial data. Our results not only improve the smallness condition in the previous references for the initial velocity concerning the global Fujita-Kato solution of the system (Equation1.1(1.1) {∂tρ+div(ρu)=0(∀(t,x)∈R+×R3),ρ(∂tu+u⋅∇u)−Δu+∇Π=0,divu=0,(ρ,u)|t=0=(ρ0,u0),(1.1) ) but also improve the exponential-in-time growth estimate for the solution in [Citation1] to be the uniform-in-time estimate (Equation1.11(1.11) ∥ρ−1−1∥L~∞(R+;B˙6,112)+∥u∥L~∞(R+;B˙2,112)+∥u∥L1(R+;B˙2,152)≲∥ρ0−1−1∥B˙6,112+∥u0∥B˙2,112.(1.11) ).