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∈(1,4) and a∈B˙p,12p(R2) that the solution mapping Ha:F↦∇Π to the 2-D elliptic equation div((1+a)∇Π)=divF is bounded on B˙p,12p−1(R2). More precisely, we prove that‖∇Π‖B˙p,12p−1≤C(1+‖a‖B˙p,12p)2‖F‖B˙p,12p−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.