The thesis explores a significant mathematical breakthrough by Lev D. Landau in 1944, focusing on a non-trivial solution to stationary Navier-Stokes flow in three-dimensional space. This solution is characterized by its axial symmetry and a unique condition where the velocity decays linearly. The work delves into the implications and applications of this finding in fluid dynamics, showcasing its relevance in the field of mathematical analysis.
The Navier-Stokes equations are fundamental in fluid dynamics, modeling the flow of incompressible Newtonian fluids. Despite advancements in research, these equations remain poorly understood. Recent literature has addressed specific issues in unbounded domains, where compactness arguments fail, often requiring decay conditions at spatial infinity. However, there is limited research on partially spatially periodic setups, such as periodic flows in pipes or models for fluids with intrinsic periodic structures. In his thesis, Jonas Sauer employs a range of mathematical tools to develop a comprehensive theory in weighted Lebesgue spaces for these partially periodic scenarios. He introduces Muckenhoupt weights on locally compact abelian groups and explores their relevance to partially periodic evolution equations. The author establishes weighted resolvent estimates for solutions to linear Stokes equations across various domains, including the whole space, half space, and cylindrical domains, while also addressing weak Neumann problems to introduce the Helmholtz-Leray projection. In the final section, Sauer investigates a nonlinear, partially periodic model for nematic liquid crystal flows, leveraging the linear theory to demonstrate maximal regularity of the partially periodic Stokes operator through the connection of Muckenhoupt weights to evolution equations.