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

## Abstract

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.

Item Type: Ph.D. Thesis
Erschienen: 2018
Creators: Seyfert, Anton
Title: The Helmholtz-Hodge Decomposition in Lebesgue Spaces on Exterior Domains and Evolution Equations on the Whole Real Time Axis
Language: English
Abstract:

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.

Divisions: 04 Department of Mathematics
04 Department of Mathematics > Analysis
04 Department of Mathematics > Analysis > Angewandte Analysis
Date Deposited: 02 Sep 2018 19:55
Das erste Thema der vorliegenden Arbeit ist die Helmholtz-Hodge-Zerlegung von Vektorfeldern in Lebesgue-Räumen $L^p$ definiert auf dreidimensionalen Außenraumgebieten. Das heißt, es geht um Zerlegungen von Vektorfeldern in ein Gradientenfeld, ein harmonisches Vektorfeld und ein Rotationsfeld. Es wird eine komplette Existenz- und Eindeutigkeitstheorie in Abhängigkeit von den Randbedingungen der einzelnen Komponenten und der Integrationsordnung $p \in (1,\infty)$ hergeleitet. Einen großen Teil des Beweises macht dabei die vollständige Lösungstheorie zu einem System schwacher Poisson-Probleme mit partiellen Dirichlet-Randbedingungen aus. Das zweite Thema der Arbeit sind lineare Evolutionsgleichungen auf der gesamten reellen Zeitachse. Unter Annahme geeigneter polynomieller Abklingbedingungen an die zugehörige Halbgruppe werden die Existenz milder Lösungen oder maximale stetige Regularität in Interpolationsräumen nachgewiesen.German