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

Hideo Kozono; Takayoshi Ogawa; Yasushi Taniuchi

doi:10.1007/s002090100332

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

Hideo Kozono^*, Takayoshi Ogawa, Yasushi Taniuchi

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

249 Citations (Scopus)

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 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.

Original language	English
Pages (from-to)	251-278
Number of pages	28
Journal	Mathematische Zeitschrift
Volume	242
Issue number	2
DOIs	https://doi.org/10.1007/s002090100332
Publication status	Published - 2002 Dec 1
Externally published	Yes

ASJC Scopus subject areas

Mathematics(all)

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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 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.",

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AU - Ogawa, Takayoshi

AU - Taniuchi, Yasushi

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Y1 - 2002/12/1

N2 - 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 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.

AB - 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 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.

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

Abstract

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