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Time-Periodic Solutions to the Navier-Stokes Equations

Kyed, Mads (2012):
Time-Periodic Solutions to the Navier-Stokes Equations.
[Online-Edition: http://tuprints.ulb.tu-darmstadt.de/3309/],
[Habilitation]

Abstract

The three-dimensional Navier-Stokes system in the whole space with time-periodic data is investigated. Both the case of a vanishing and that of a non-vanishing velocity field at spatial infinity are treated. In the first part of the thesis, a maximal regularity framework for the linearized system is developed in a general L^q-setting. A function space with the property that the corresponding linear operator maps this space homeomorphically onto L^q is identified. Existence of a strong solution herein is then shown for sufficiently "small" data. Moreover, regularity and uniqueness properties are established. In the following part, an asymptotic expansion at spatial infinity of the strong solution is carried out. In particular, the asymptotic profile is completely identified. In the final part, existence of a weak solution is proved without any restriction on the "size" of the data. Furthermore, a decomposition of the weak solution into a time-independent part and a time-periodic part with finite kinetic energy is obtained. On the basis of this decomposition, regularity properties of the weak solution are derived.

Item Type: Habilitation
Erschienen: 2012
Creators: Kyed, Mads
Title: Time-Periodic Solutions to the Navier-Stokes Equations
Language: English
Abstract:

The three-dimensional Navier-Stokes system in the whole space with time-periodic data is investigated. Both the case of a vanishing and that of a non-vanishing velocity field at spatial infinity are treated. In the first part of the thesis, a maximal regularity framework for the linearized system is developed in a general L^q-setting. A function space with the property that the corresponding linear operator maps this space homeomorphically onto L^q is identified. Existence of a strong solution herein is then shown for sufficiently "small" data. Moreover, regularity and uniqueness properties are established. In the following part, an asymptotic expansion at spatial infinity of the strong solution is carried out. In particular, the asymptotic profile is completely identified. In the final part, existence of a weak solution is proved without any restriction on the "size" of the data. Furthermore, a decomposition of the weak solution into a time-independent part and a time-periodic part with finite kinetic energy is obtained. On the basis of this decomposition, regularity properties of the weak solution are derived.

Uncontrolled Keywords: Navier-Stokes, time periodic, maximal regularity, Asymptotic expansion
Divisions: 04 Department of Mathematics
04 Department of Mathematics > Analysis
Date Deposited: 18 Mar 2013 16:27
Official URL: http://tuprints.ulb.tu-darmstadt.de/3309/
URN: urn:nbn:de:tuda-tuprints-33095
License: Creative Commons: Attribution-No Derivative Works 3.0
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