The stokes equation in the Lp-setting: Well-posedness and regularity properties

Matthias Georg Hieber, Jürgen Saal

Research output: Chapter in Book/Report/Conference proceedingChapter

1 Citation (Scopus)

Abstract

This article discusses the Stokes equation in various classes of domains Ω C Rn within the Lp-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 Lq -Lp-regularity results for 1 < p; q < ∞ are presented via the theory of 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 Lp-scale, i.e., p=1 and p=∞ are considered and recent well-posedness results for the case p =∞ are described. Results on Lq -Lp-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.

Original languageEnglish
Title of host publicationHandbook of Mathematical Analysis in Mechanics of Viscous Fluids
PublisherSpringer International Publishing
Pages117-206
Number of pages90
ISBN (Electronic)9783319133447
ISBN (Print)9783319133430
DOIs
Publication statusPublished - 2018 Apr 19
Externally publishedYes

Fingerprint

Regularity Properties
Stokes Equations
regularity
Well-posedness
Stokes
Boundary conditions
Semigroup
Helmholtz Decomposition
Viscosity
Sectorial Operator
Decomposition
Hydrostatics
Lorentz Spaces
boundary conditions
Strong Solution
Evolution Equation
Smoothing
Rotating
hydrostatics
smoothing

Keywords

  • Helmholtz decomposition
  • L-L-smoothing
  • L∞-estimates
  • Maximal L-regularity
  • Stokes equations
  • Stokes semigroup

ASJC Scopus subject areas

  • Mathematics(all)
  • Physics and Astronomy(all)
  • Engineering(all)

Cite this

Hieber, M. G., & Saal, J. (2018). The stokes equation in the Lp-setting: Well-posedness and regularity properties. In Handbook of Mathematical Analysis in Mechanics of Viscous Fluids (pp. 117-206). Springer International Publishing. https://doi.org/10.1007/978-3-319-13344-7_3

The stokes equation in the Lp-setting : Well-posedness and regularity properties. / Hieber, Matthias Georg; Saal, Jürgen.

Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer International Publishing, 2018. p. 117-206.

Research output: Chapter in Book/Report/Conference proceedingChapter

Hieber, MG & Saal, J 2018, The stokes equation in the Lp-setting: Well-posedness and regularity properties. in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer International Publishing, pp. 117-206. https://doi.org/10.1007/978-3-319-13344-7_3
Hieber MG, Saal J. The stokes equation in the Lp-setting: Well-posedness and regularity properties. In Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer International Publishing. 2018. p. 117-206 https://doi.org/10.1007/978-3-319-13344-7_3
Hieber, Matthias Georg ; Saal, Jürgen. / The stokes equation in the Lp-setting : Well-posedness and regularity properties. Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer International Publishing, 2018. pp. 117-206
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