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On the L^p-theory of the Navier-Stokes equation on Lipschitz domains

Tolksdorf, Patrick :
On the L^p-theory of the Navier-Stokes equation on Lipschitz domains.
[Online-Edition: http://tuprints.ulb.tu-darmstadt.de/5960]
Technische Universität , Darmstadt
[Ph.D. Thesis], (2016)

Official URL: http://tuprints.ulb.tu-darmstadt.de/5960

Abstract

In this thesis, we investigate the Stokes operator on bounded Lipschitz domains in L^p. We proof important L^p-L^q-estimates for the Stokes semigroup as well as maximal regularity of the Stokes operator. These facts are used to derive the existence of solutions to the Navier-Stokes equations in L^p. Furthermore, we show that the techniques to derive the maximal regularity can be used in a simplified way in order to prove maximal regularity of higher-order elliptic systems subject to mixed boundary conditions.

Item Type: Ph.D. Thesis
Erschienen: 2017
Creators: Tolksdorf, Patrick
Title: On the L^p-theory of the Navier-Stokes equation on Lipschitz domains
Language: English
Abstract:

In this thesis, we investigate the Stokes operator on bounded Lipschitz domains in L^p. We proof important L^p-L^q-estimates for the Stokes semigroup as well as maximal regularity of the Stokes operator. These facts are used to derive the existence of solutions to the Navier-Stokes equations in L^p. Furthermore, we show that the techniques to derive the maximal regularity can be used in a simplified way in order to prove maximal regularity of higher-order elliptic systems subject to mixed boundary conditions.

Place of Publication: Darmstadt
Divisions: 04 Department of Mathematics > Analysis > Angewandte Analysis
04 Department of Mathematics > Analysis
04 Department of Mathematics
Date Deposited: 05 Feb 2017 20:55
Official URL: http://tuprints.ulb.tu-darmstadt.de/5960
URN: urn:nbn:de:tuda-tuprints-59609
Referees: Haller-Dintelmann, Prof. Dr. Robert and Farwig, Prof. Dr. Reinhard and Saal, Prof. Dr. Jürgen
Refereed / Verteidigung / mdl. Prüfung: 31 August 2016
Alternative Abstract:
Alternative abstract Language
In dieser Doktorarbeit wird der Stokes-Operator auf beschränkten Lipschitz-Geibeten in L^p untersucht. Es werden unter anderem wichtige L^p-L^q-Abschätzungen der Stokes-Halbgruppe sowie die Eigenschaft der maximalen Regularität des Stokes-Operators bewiesen. Dies wird benutzt, um die Existenz von Lösungen der Navier-Stokes-Gleichungen in L^p zu zeigen. Des Weiteren wird anhand von elliptischen Systemen höherer Ordnung mit gemischen Randbedingungen gezeigt, dass die gleiche Methode wie im Falle des Stokes-Operators in einer vereinfachten Weise benutzt werden kann, um maximale Regularität für diese Operatorenklasse nachzuweisen.German
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