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

Reinhard Farwig, Hideo Kozono, Hermann Sohr

    Research output: Chapter in Book/Report/Conference proceedingChapter

    Abstract

    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.

    Original languageEnglish
    Title of host publicationHandbook of Mathematical Analysis in Mechanics of Viscous Fluids
    PublisherSpringer International Publishing
    Pages419-459
    Number of pages41
    ISBN (Electronic)9783319133447
    ISBN (Print)9783319133430
    DOIs
    Publication statusPublished - 2018 Apr 19

    Fingerprint

    Very Weak Solutions
    Energy Inequality
    L-space
    Stokes
    Navier Stokes equations
    Weak Solution
    Navier-Stokes Equations
    Semigroup
    Stokes Operator
    Partial Regularity
    Sobolev spaces
    Navier-Stokes equation
    Navier-Stokes System
    Projection Operator
    Structure Theorem
    Mild Solution
    Hermann Von Helmholtz
    Strong Solution
    Sobolev space
    Sobolev Spaces

    ASJC Scopus subject areas

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

    Cite this

    Farwig, R., Kozono, H., & Sohr, H. (2018). Stokes semigroups, strong,weak, and very weak solutions for general domains. In 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.

    Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer International Publishing, 2018. p. 419-459.

    Research output: Chapter in Book/Report/Conference proceedingChapter

    Farwig, R, Kozono, H & Sohr, H 2018, Stokes semigroups, strong,weak, and very weak solutions for general domains. in 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
    Farwig R, Kozono H, Sohr H. Stokes semigroups, strong,weak, and very weak solutions for general domains. In Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer International Publishing. 2018. p. 419-459 https://doi.org/10.1007/978-3-319-13344-7_8
    Farwig, Reinhard ; Kozono, Hideo ; Sohr, Hermann. / Stokes semigroups, strong,weak, and very weak solutions for general domains. Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer International Publishing, 2018. pp. 419-459
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