### Abstract

To solve the (Navier-)Stokes equations in general smooth domains Ως R^{n}, the spaces ~L ^{q}(Ω) defined as L^{q} nL^{2} when 2 ≤ q < ∞ and L^{q}+L^{2} when1 < q < 2 have shown to be a successful strategy. First, the main properties of the spaces ~L ^{q}(Ω) and related concepts for solenoidal subspaces, Sobolev spaces, Bochner spaces, and the corresponding Helmholtz projection and Stokes operator will be discussed. Then these concepts are used to construct and analyze very weak, weak, mild, and strong solutions to the instationary (Navier-)Stokes equations in general domains. In particular, the strategy allows to find weak solutions of the (Navier-)Stokes system satisfying the localized energy inequality and the strong energy inequality which are important in the context of Leray structure theorem and partial regularity results.

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

Publisher | Springer International Publishing |

Pages | 419-459 |

Number of pages | 41 |

ISBN (Electronic) | 9783319133447 |

ISBN (Print) | 9783319133430 |

DOIs | |

Publication status | Published - 2018 Apr 19 |

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### ASJC Scopus subject areas

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

### Cite this

*Handbook of Mathematical Analysis in Mechanics of Viscous Fluids*(pp. 419-459). Springer International Publishing. https://doi.org/10.1007/978-3-319-13344-7_8

**Stokes semigroups, strong,weak, and very weak solutions for general domains.** / Farwig, Reinhard; Kozono, Hideo; Sohr, Hermann.

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

*Handbook of Mathematical Analysis in Mechanics of Viscous Fluids.*Springer International Publishing, pp. 419-459. https://doi.org/10.1007/978-3-319-13344-7_8

}

TY - CHAP

T1 - Stokes semigroups, strong,weak, and very weak solutions for general domains

AU - Farwig, Reinhard

AU - Kozono, Hideo

AU - Sohr, Hermann

PY - 2018/4/19

Y1 - 2018/4/19

N2 - To solve the (Navier-)Stokes equations in general smooth domains Ως Rn, the spaces ~L q(Ω) defined as Lq nL2 when 2 ≤ q < ∞ and Lq+L2 when1 < q < 2 have shown to be a successful strategy. First, the main properties of the spaces ~L q(Ω) and related concepts for solenoidal subspaces, Sobolev spaces, Bochner spaces, and the corresponding Helmholtz projection and Stokes operator will be discussed. Then these concepts are used to construct and analyze very weak, weak, mild, and strong solutions to the instationary (Navier-)Stokes equations in general domains. In particular, the strategy allows to find weak solutions of the (Navier-)Stokes system satisfying the localized energy inequality and the strong energy inequality which are important in the context of Leray structure theorem and partial regularity results.

AB - To solve the (Navier-)Stokes equations in general smooth domains Ως Rn, the spaces ~L q(Ω) defined as Lq nL2 when 2 ≤ q < ∞ and Lq+L2 when1 < q < 2 have shown to be a successful strategy. First, the main properties of the spaces ~L q(Ω) and related concepts for solenoidal subspaces, Sobolev spaces, Bochner spaces, and the corresponding Helmholtz projection and Stokes operator will be discussed. Then these concepts are used to construct and analyze very weak, weak, mild, and strong solutions to the instationary (Navier-)Stokes equations in general domains. In particular, the strategy allows to find weak solutions of the (Navier-)Stokes system satisfying the localized energy inequality and the strong energy inequality which are important in the context of Leray structure theorem and partial regularity results.

UR - http://www.scopus.com/inward/record.url?scp=85054390927&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85054390927&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-13344-7_8

DO - 10.1007/978-3-319-13344-7_8

M3 - Chapter

SN - 9783319133430

SP - 419

EP - 459

BT - Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

PB - Springer International Publishing

ER -