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 , 1 2 p ( R 2 ) that the solution mapping H a : F ↦ ∇ Π to the 2-D elliptic equation div ( ( 1 + a ) ∇ Π ) = div F is bounded on B ˙ p , 1 2 p − 1 ( R 2 ) . More precisely, we prove that ‖ ∇ Π ‖ B ˙ p , 1 2 p − 1 ≤ C ( 1 + ‖ a ‖ B ˙ p , 1 2 p ) 2 ‖ F ‖ B ˙ p , 1 2 p − 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. [ABSTRACT FROM AUTHOR]