On the global well-posedness of 2-D inhomogeneous incompressible Navier–Stokes system with variable viscous coefficient

Given solenoidal vector u0∈H˙−2δ∩H1(R2), ρ0−1∈L2(R2)ρ0−1∈L2(R2), and ρ0∈L∞∩W˙1,r(R2) with a positive lower bound for δ∈(0,12) and 2