Mathematische Zeitschrift

, Volume 242, Issue 2, pp 251–278

The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations

  • Hideo Kozono
  • Takayoshi Ogawa
  • Yasushi Taniuchi
Original article

DOI: 10.1007/s002090100332

Cite this article as:
Kozono, H., Ogawa, T. & Taniuchi, Y. Math Z (2002) 242: 251. doi:10.1007/s002090100332

Abstract.

We show the critical Sobolev inequalities in the Besov spaces with the logarithmic form such as Brezis-Gallouet-Wainger and Beale-Kato-Majda. As an application of those inequalities, the regularity problem under the critical condition to the Navier-Stokes equations, the Euler equations in \({mathbb R}^n\) and the gradient flow to the harmonic map to the sphere are discussed. Namely the Serrin-Ohyama type regularity criteria are improved in the terms of the Besov spaces.

Mathematics Subject Classification (1991): 35Q05, 35L60, 75C05. 

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Hideo Kozono
    • 1
  • Takayoshi Ogawa
    • 2
  • Yasushi Taniuchi
    • 3
  1. 1.Mathematical Institute, Tohoku University, Sendai 980-8578, Japan JP
  2. 2.Faculty of Mathematics, Kyushu University, Fukuoka 812-8581, Japan JP
  3. 3.Department of Mathematical Sciences, Shinshu University, Matsumoto 390-8621, Japan JP

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