Solvability of the two-dimensional stationary incompressible inhomogeneous Navier–Stokes equations with variable viscosity coefficient

We show the existence and the regularity properties of (a class of) weak solutions to the two-dimensional stationary incompressible inhomogeneous Navier–Stokes equations with density-dependent viscosity coefficients, by analyzing a fourth-order nonlinear elliptic equation for the stream function.
We formulate the stationary solutions for the parallel, concentric and radial flows respectively, and we give some (ir-)regularity results as well as some explicit examples in the case of piecewise-constant viscosity coefficients.