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Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains

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Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Abstract

To solve the (Navier-)Stokes equations in general smooth domains \(\Omega \subset \mathbb{R}^{n}\), the spaces \(\tilde{L}^{q}(\Omega )\) defined as Lq ∩ L2 when 2 ≤ q < ∞ and Lq + L2 when 1 < q < 2 have shown to be a successful strategy. First, the main properties of the spaces \(\tilde{L}^{q}(\Omega )\) 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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Author information

Authors and Affiliations

  1. Department of Mathematics, Darmstadt University of Technology, Castle Garden Road 7, 64289, Darmstadt, Germany

    Reinhard Farwig

  2. International Research Training Group Darmstadt-Tokyo (IRTG 1529), 64289, Darmstadt, Germany

    Reinhard Farwig

  3. Department of Mathematics, Waseda University, 3-4-1 Okubo, Shinjuku-ku, 169-8555, Tokyo, Japan

    Hideo Kozono

  4. Japanese-German Graduate Externship Program, Japan Society of Promotion of Science, Tokyo, Japan

    Hideo Kozono

  5. Faculty of Electrical Engineering, Informatics and Mathematics, Department of Mathematics, University of Paderborn, Warburger Str. 100, 33098, Paderborn, Germany

    Hermann Sohr

Authors
  1. Reinhard Farwig
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  2. Hideo Kozono
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  3. Hermann Sohr
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Corresponding author

Correspondence to Reinhard Farwig .

Editor information

Editors and Affiliations

  1. Graduate School of Mathematical Sciences, University of Tokyo, Meguro-ku, Tokyo, Japan

    Yoshikazu Giga

  2. Université de Toulon IMATH, Toulon, France

    Antonín Novotný

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Farwig, R., Kozono, H., Sohr, H. (2018). Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains. In: Giga, Y., Novotný, A. (eds) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Cham. https://doi.org/10.1007/978-3-319-13344-7_8

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