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

Abstract

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 $a\in {\dot{B}}_{p,1}^{\frac{2}{p}}({\mathbb{R}}^2)$ that the solution mapping ${\mathcal{H}}_a:F\mapsto \nabla \Pi$ to the 2-D elliptic equation $\mathrm{div}((1+a)\nabla \Pi )=\mathrm{div}F$ is bounded on ${\dot{B}}_{p,1}^{\frac{2}{p}-1}({\mathbb{R}}^2)$. More precisely, we prove that $$\Vert \nabla \Pi {\Vert }_{{\dot{B}}_{p,1}^{\frac{2}{p}-1}}\le C(1+\Vert a{\Vert }_{{\dot{B}}_{p,1}^{\frac{2}{p}}}{)}^2\Vert F{\Vert }_{{\dot{B}}_{p,1}^{\frac{2}{p}-1}}.$$ The proof of the uniqueness of solution to (1.2) relies on a Lagrangian approach [15]-[17]. When the viscosity coefficient $\mu (\rho )$ is a positive constant, we prove that (1.2) is globally well-posed.