The Helmholtz-Hodge Decomposition in Lebesgue Spaces on Exterior Domains and Evolution Equations on the Whole Real Time Axis

The first topic of this thesis is the Helmholtz-Hodge decomposition of vector fields in Lebesgue spaces $L^p$ defined on three-dimensional exterior domains, i.e. a decomposition of vector fields into a gradient field, a harmonic vector field and a rotation field. Here, a full characterisation of the existence and uniqueness of the decomposition is given for two different kinds of boundary conditions and the full range of $p \in (1,\infty)$. As a part of the proof, a complete solution theory for systems of weak Poisson problems with partially vanishing boundary conditions is developed. The second part of the thesis is about bounded solutions to linear evolution equations on the whole real time axis which includes in particular periodic and almost periodic solutions. Building upon works of Yamazaki (2000) and Geissert, Hieber, Nguyen (2016), the existence of mild solutions and maximal continuous regularity of such equations is shown in an abstract setting of interpolation spaces under the assumption of suitable polynomial decay properties of the semigroup associated to the problem at hand