Mathematical Analysis of Regularity Propagation in Variable-Viscosity Fluid Models

In this thesis we investigate several variable-viscosity fluid models, including incompressible viscous fluids with usual or odd viscosity and compressible viscous fluids. We mostly focus on the regularity propagation of sharp interfaces between two immiscible, viscous fluids. This thesis is divided into three parts.
The first part (Chapter 2) investigates the existence of weak solutions to the two-dimensional inhomogeneous incompressible Navier-Stokes equations with variable, odd viscosity. Odd or Hall viscosity is the anti-symmetric part of the viscosity tensor, and it is present in fluids with broken microscopic time-reversal symmetry and broken parity. We prove the existence of weak solutions in both the evolutionary and stationary cases. Furthermore, we study the limit of the weak solutions as the odd viscosity coefficient converges to a constant. Lastly, examples of stationary solutions for parallel, concentric and radial flows, are considered.
The second part (Chapter 3) addresses the two-dimensional incompressible Navier-Stokes equations with freely transported viscosity coefficient. Under a suitable smallness assumption on the initial velocity, a global-in-time well-posedness result is established which allows for discontinuous density and viscosity coefficients without size restriction on the jumps. ... mehrAs an application, the global-in-time well-posedness of the two-dimensional inhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity, and the local-in-time well-posedness of the two-dimensional incompressible Boussinesq equations with temperature-dependent viscosity is proven. Both results allow for discontinuous, largely varying viscosity. The regularity of an interface of discontinuity and the tangential regularity of the density/temperature along this interface is also shown to persist over time.
The third part (Chapter 4) concerns the two-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefficients. Under a suitable smallness assumption on the initial velocity and initial energy of the system, a global-in-time well-posedness result is proven in a framework allowing for discontinuous density and viscosity coefficients without size restriction on the jumps. We also show the persistence of regularity of the discontinuity curve as well as the tangential regularity of the density over time.