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Erschienen in:
Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids Kapitel lesen Erstes Kapitel lesen

2018 | OriginalPaper | Buchkapitel

3. The Stokes Equation in the L p -Setting: Well-Posedness and Regularity Properties

verfasst von : Matthias Hieber, Jürgen Saal

Erschienen in: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Verlag: Springer International Publishing

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Abstract

This article discusses the Stokes equation in various classes of domains \(\Omega \subset \mathbb{R}^{n}\) within the L p -setting for 1 ≤ p ≤ ∞ from the point of view of evolution equations. Classical as well as modern approaches to well-posedness results for strong solutions to the Stokes equation, to the Helmholtz decomposition, to the Stokes semigroup, and to mixed maximal L q − L p -regularity results for 1 ¡ p, q < ∞ are presented via the theory of \(\mathcal{R}\)-sectorial operators. Of concern are domains having compact or noncompact, smooth or nonsmooth boundaries, as well as various classes of boundary conditions including energy preserving boundary conditions. In addition, the endpoints of the L p -scale, i.e., p = 1 and p = ∞ are considered and recent well-posedness results for the case p = ∞ are described. Results on L p − L q -smoothing properties of the associated Stokes semigroups and on variants of the Stokes equation (e.g., nonconstant viscosity, Lorentz spaces, Stokes-Oseen system, flow past rotating obstacles, hydrostatic Stokes equation) complete this survey article.

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Vorheriges Kapitel Variational Modeling and Complex Fluids
Nächstes Kapitel Stokes Problems in Irregular Domains with Various Boundary Conditions
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Metadaten
Titel
The Stokes Equation in the L p -Setting: Well-Posedness and Regularity Properties
verfasst von
Matthias Hieber
Jürgen Saal
Copyright-Jahr
2018
Buch
Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Print ISBN: 978-3-319-13343-0

Electronic ISBN: 978-3-319-13344-7

Copyright-Jahr: 2018

DOI
https://doi.org/10.1007/978-3-319-13344-7_3