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

Reinhard Farwig; Hideo Kozono; Hermann Sohr

doi:10.1007/978-3-319-13344-7_8

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

Reinhard Farwig^*, Hideo Kozono, Hermann Sohr

^*Corresponding author for this work

School of Fundamental Science and Engineering

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

4 Citations (Scopus)

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
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	https://doi.org/10.1007/978-3-319-13344-7_8
Publication status	Published - 2018 Apr 19

ASJC Scopus subject areas

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

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

Cite this

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title = "Stokes semigroups, strong,weak, and very weak solutions for general domains",

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. ",

author = "Reinhard Farwig and Hideo Kozono and Hermann Sohr",

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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 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.

AB - 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.

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BT - Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

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

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