In this talk, we will present some existence, uniqueness and regularity results for the motion of two-dimensional incompressible inhomogeneous viscous fluid flows in presence of a density-/temperature-dependent viscosity coefficient.

Firstly, we will discuss the boundary value problem for the stationary Navier-Stokes equation, where the viscosity coefficient is density-dependent. We will give some explicit solutions with piecewise constant viscosity coefficients, where some regularity and irregularity results will be considered.

We will also discuss the initial value problem for the evolutionary Boussinesq equation, which is a nonlinear coupling between a heat equation and a Navier-Stokes type of equation. In this case, the viscosity coefficient is temperature-dependent.

This talk is based on joint work with Xian Liao (KIT).

Firstly, we will discuss the boundary value problem for the stationary Navier-Stokes equation, where the viscosity coefficient is density-dependent. We will give some explicit solutions with piecewise constant viscosity coefficients, where some regularity and irregularity results will be considered.

We will also discuss the initial value problem for the evolutionary Boussinesq equation, which is a nonlinear coupling between a heat equation and a Navier-Stokes type of equation. In this case, the viscosity coefficient is temperature-dependent.

This talk is based on joint work with Xian Liao (KIT).