Unique solvability for the density-dependent Navier–Stokes equations

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Global well-posedness for the incompressible Navier–Stokes equations with density-dependent viscosity coefficient (2015, background): “4), Cho and Kim [5] assumed that −div(μ(ρ0)∇u0)+∇ 0 = ρ 0 g for some (∇ 0, g) ∈ L2, (1.”
Existence results for viscous polytropic fluids with vacuum (2006, methodology): “For instance, in case of bounded domains, we may apply a semi-discrete Galerkin method (as in [2]) or the method of continuity (as in [31]).”
Strong Solutions of the Navier–Stokes Equations for Nonhomogeneous Incompressible Fluids (2003, methodology): “A local existence result with a blow-up criterion is proved by Cho and Kim ( [0000] ).”