On the Well-posedness of 2-D Incompressible Navier-Stokes Equations with Variable Viscosity in Critical Spaces

In this paper, we first prove the local well-posedness of the 2-D incompressible Navier-Stokes equations with variable viscosity in critical Besov spaces with negative regularity indices, without smallness assumption on the variation of the density. The key is to prove for $p\in(1,4)$ and $a\in\dot{B}_{p,1}^{\frac2p}(\mathbb{R}^2)$ that the solution mapping $\mathcal{H}_a:F\mapsto\nablaΠ$ to the 2-D elliptic equation $\mathrm{div}\big((1+a)\nablaΠ\big)=\mathrm{div} F$ is bounded on $\dot{B}_{p,1}^{\frac2p-1}(\mathbb{R}^2)$. More precisely, we prove that $\|\nablaΠ\|_{\dot{B}_{p,1}^{\frac2p-1}}\leq C\big(1+\|a\|_{\dot{B}_{p,1}^{\frac2p}}\big)^2\|F\|_{\dot{B}_{p,1}^{\frac2p-1}}.$ The proof of the uniqueness of solution to (1.2) relies on a Lagrangian approach [15]-[17]. When the viscosity coefficient $μ(ρ)$ is a positive constant, we prove that (1.2) is globally well-posed.

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