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

Abstract

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. For some stationary symmetric flows, we reformulate the Navier–Stokes equations as ordinary differential equations and give explicit examples of weak solutions. We present some further (ir-)regularity results in the case of piecewise-constant viscosity coefficients.