Global well-posedness of 3-D density-dependent Navier-Stokes system with variable viscosity

Abstract

Given initial data (ρ ₀, u ₀) satisfying 0 < m ⩽ ρ ₀ ⩽ M, $\rho _0 - 1 \in L^2 \cap \dot W^{1,r} (R^3 )$ and $u_0 \in \dot H^{ - 2\delta } \cap H^1 (\mathbb{R}^3 )$ for δ ∈]1/4, 1/2[ and r ∈]6, 3/1 − 2δ[, we prove that: there exists a small positive constant ɛ ₁, which depends on the norm of the initial data, so that the 3-D incompressible inhomogeneous Navier-Stokes system with variable viscosity has a unique global strong solution (ρ, u) whenever $\left\| {u_0 } \right\|_{L^2 } \left\| {\nabla u_0 } \right\|_{L^2 }$ and $\left\| {\mu (\rho _0 ) - 1} \right\|_{L^\infty } \leqslant \varepsilon _0$ for some uniform small constant ɛ ₀. Furthermore, with smoother initial data and viscosity coefficient, we can prove the propagation of the regularities for such strong solution.