The stokes equations in spaces of bounded functions and spaces of bounded mean oscillation

The Navier-Stokes equations are the fundamental equations in mathematical fluid dynamics, which provide a wide field of research within mathematics. An important step in understanding the Navier-Stokes equations is to investigate its linearized version, the Stokes equations. The author Martin Bolkart considers these equations in function spaces which in this setting are non-classical. In the first part he contributes to the theory of the Stokes equations in spaces of bounded functions. The existence of solutions of the stationary Stokes problem in layer domains and the boundedness in exterior domains are considered. In the second part he introduces a new kind of spaces of bounded mean oscillation and considers the Stokes equations within these spaces. He proves several equivalences and relations between different norms and based on these results he shows the analyticity of the Stokes semigroup in these spaces. An application of this theory is to prove that the Helmholtz projection is not necessary for obtaining analyticity results on the Stokes equations.