Navier-Stokes Flow in Partially Periodic Domains

Beschreibung

One of the fundamental models in fluid dynamics are the Navier-Stokes equations, modelling the fluid flow of an incompressible Newtonian fluid. Despite recent progress in scientific research, the Navier-Stokes equations continue to be far from well-understood. Nevertheless, during the last two decades many results have appeared in the literature dealing with special problems that arise in the case of unbounded domains, where compactness arguments break down. Typically, these results impose in one way or the other a decay condition towards spatial infinity. However, there are not many results concerning (partially) spatially periodic setups, e.g., periodic flows in pipes with periodically repetitive profile or models for certain fluids with intrinsical periodic structures. In his thesis, Jonas Sauer uses a wide spectrum of mathematical tools in order to develop a full theory in weighted Lebesgue spaces for such partially periodic set-ups. The concept of Muckenhoupt weights on locally compact abelian groups is introduced and its connection to partially periodic evolution equations is highlighted. The author then continues to establish weighted resolvent estimates for solutions to the linear Stokes equations in different domains such as the whole space, the half space and cylindrical domains. Also weak Neumann problems are treated in order to introduce the Helmholtz-Leray projection. In the last part, the author investigates a non-linear, partially periodic model describing the dynamics of nematic liquid crystal flows. Here, the full power of the linear theory is exploited by proving maximal regularity of the partially periodic Stokes operator via the connection of Muckenhoupt weights to evolution equations.

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Jonas Sauer

Themen in »Navier-Stokes Flow in Partially Periodic Domains«

Extrapolation Theorem Locally Compact Abelian Groups Maximal Regularity Muckenhoupt weights Navier-Stokes Equations Nematic Liquid Crystal Flow Periodic solutions

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