### Abstract

This article discusses the Stokes equation in various classes of domains Ω C 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 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^{q} -L^{p}-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 language | English |
---|---|

Title of host publication | Handbook of Mathematical Analysis in Mechanics of Viscous Fluids |

Publisher | Springer International Publishing |

Pages | 117-206 |

Number of pages | 90 |

ISBN (Electronic) | 9783319133447 |

ISBN (Print) | 9783319133430 |

DOIs | |

Publication status | Published - 2018 Apr 19 |

Externally published | Yes |

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### 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

^{p}-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 L ^{p}-setting : Well-posedness and regularity properties.** / Hieber, Matthias Georg; Saal, Jürgen.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

^{p}-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

^{p}-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

}

TY - CHAP

T1 - The stokes equation in the Lp-setting

T2 - Well-posedness and regularity properties

AU - Hieber, Matthias Georg

AU - Saal, Jürgen

PY - 2018/4/19

Y1 - 2018/4/19

N2 - 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.

AB - 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.

KW - Helmholtz decomposition

KW - L-L-smoothing

KW - L∞-estimates

KW - Maximal L-regularity

KW - Stokes equations

KW - Stokes semigroup

UR - http://www.scopus.com/inward/record.url?scp=85054361538&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85054361538&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-13344-7_3

DO - 10.1007/978-3-319-13344-7_3

M3 - Chapter

SN - 9783319133430

SP - 117

EP - 206

BT - Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

PB - Springer International Publishing

ER -