Incompressible Inhomogeneous Viscous Fluid Flows: Existence, Uniqueness and Regularity

This thesis is devoted to the study of the solvability and regularity problems for the motion of incompressible inhomogeneous viscous fluid flows in the presence of variable viscosity coefficients.

Chapter 2 is devoted to 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.
The three-dimensional case under special symmetry assumptions is also considered.

Chapter 3 proves the existence, uniqueness, and regularity results of the two-dimensional evolutionary incompressible Boussinesq equations with temperature-dependent thermal and viscosity diffusion coefficients in general Sobolev spaces.

In addition to the above results in the domain of fluid mechanics, we study the turbulence cascades for a two-parameter family of damped Szeg\H{o} equations in Chapter 4.