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A Note on Necessary Conditions for Blow-up of Energy Solutions to the Navier-Stokes Equations

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Parabolic Problems

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 80))

Abstract

In the present note, we address the question about behavior of L 3-norm of the velocity field as time t approaches blow-up time T. It is known that the upper limit of the above norm must be equal to infinity. We show that, for blow-ups of type I, the lower limit of L 3-norm equals to infinity as well.

Mathematics Subject Classification (2000). 35K, 76D.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, 24–29 St. Giles, Oxford, OX1 3LB, UK

    Gregory Seregin

Authors
  1. Gregory Seregin
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Correspondence to Gregory Seregin .

Editor information

Editors and Affiliations

  1. Inst. Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Joachim Escher

  2. School of Physical Sciences, Dept. Mathematics, University of California, Irvine, Irvine, 92697-3875, California, USA

    Patrick Guidotti

  3. , Fachbereich Mathematik, TU Darmstadt, Schlossgartenstr 7, Darmstadt, 64289, Germany

    Matthias Hieber

  4. Inst. Applied Mathematics & Mechanics, Warsaw University, ul. Banacha 2, Warszawa, 02-097, Poland

    Piotr Mucha

  5. , FB Mathematik und Informatik, Universität Halle-Wittenberg, Theodor-Lieser-Strasse 5, Halle, 06099, Germany

    Jan W. Prüss

  6. 3-4-1, Ohkubo Shinjuku-ku, Tokyo, 169-8555, Japan

    Yoshihiro Shibata

  7. , Deparment of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, USA

    Gieri Simonett

  8. Inst. Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Christoph Walker

  9. Fac. Mathematics & Information Sciences, Warsaw University of Technology, pl. Politechniki 1, Warszawa, 00-661, Poland

    Wojciech Zajaczkowski

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Dedicated to Herbert Amann

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Seregin, G. (2011). A Note on Necessary Conditions for Blow-up of Energy Solutions to the Navier-Stokes Equations. In: Escher, J., et al. Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 80. Springer, Basel. https://doi.org/10.1007/978-3-0348-0075-4_31

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