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

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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 \({mathbb R}^n\) 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.

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Received: 21 September 2000; in final form: 16 Feburary 2001 / Published online: 18 January 2002

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Kozono, H., Ogawa, T. & Taniuchi, Y. The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations. Math Z 242, 251–278 (2002). https://doi.org/10.1007/s002090100332

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  • DOI: https://doi.org/10.1007/s002090100332

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