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Local Regularity Results for the Instationary Navier-Stokes Equations Based on Besov Space Type Criteria

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Recent Developments of Mathematical Fluid Mechanics

Part of the book series: Advances in Mathematical Fluid Mechanics ((AMFM))

Abstract

Consider a weak instationary solution u of the Navier-Stokes equations in a domain \(\Omega \subset \mathbb{R}^{3}\), i.e.,

$$u \in L^{\infty }\big(0,T;L^{2}(\Omega )\big) \cap L^{2}\big(0,T;W_{0}^{1,2}(\Omega )\big)$$

and u solves the Navier-Stokes system in the sense of distributions. It is a famous open problem whether weak solutions are unique and smooth. A main step in the analysis of this problem is to show that the given weak solution is a strong one in the sense of J. Serrin, i.e., \(u \in L^{s}\big(0,T;L^{q}(\Omega )\big)\) where s > 2, q > 3 and \(\frac{2} {s} + \frac{3} {q} = 1\). In this review we report on recent results on this problem, considering first of all optimal initial values u(0) to yield a local in time strong solution, then criteria to prove regularity locally on subintervals of [0, T). Special emphasis is put on results for smooth bounded and also general unbounded domains. Most criteria are based on conditions of Besov space type.

To our colleague Prof. Yoshihiro Shibata on the occasion of his 60th birthday

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Acknowledgements

This work was completed with the support of the International Research Training Group on Mathematical Fluid Mechanics Darmstadt-Tokyo (IRTG 1529).

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Authors and Affiliations

  1. Fachbereich Mathematik and International Research Training Group (IRTG 1529), Darmstadt-Tokyo, Technische Universität Darmstadt, Schlossgartenstr. 7, 64283, Darmstadt, Germany

    Reinhard Farwig

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  1. Reinhard Farwig
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Correspondence to Reinhard Farwig .

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Editors and Affiliations

  1. Universität of Zürich, Institut für Mathematik, Zürich, Switzerland

    Herbert Amann

  2. University of Tokyo, Graduate School of Mathematical Sciences, Tokyo, Japan

    Yoshikazu Giga

  3. Department of Mathematics, Waseda University, Tokyo, Japan

    Hideo Kozono

  4. Research Institute for Mathematical Sciences, Kyoto University Kitashirakawa, Kyoto, Japan

    Hisashi Okamoto

  5. Department of Mathematics, Waseda University, Tokyo, Japan

    Masao Yamazaki

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Farwig, R. (2016). Local Regularity Results for the Instationary Navier-Stokes Equations Based on Besov Space Type Criteria. In: Amann, H., Giga, Y., Kozono, H., Okamoto, H., Yamazaki, M. (eds) Recent Developments of Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0939-9_11

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