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Global solutions for the Navier-Stokes equations in the rotational framework

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Abstract

The existence of global unique solutions to the Navier-Stokes equations with the Coriolis force is established in the homogeneous Sobolev spaces \(\dot{H}^s (\mathbb R ^3)^3\) for \(1/2 < s < 3/4\) if the speed of rotation is sufficiently large. This phenomenon is so-called the global regularity. The relationship between the size of initial datum and the speed of rotation is also derived. The proof is based on the space time estimates of the Strichartz type for the semigroup associated with the linearized equations. In the scaling critical space \(\dot{H}^{\frac{1}{2}} (\mathbb R ^3)^3\), the global regularity is also shown.

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Acknowledgments

The authors would like to express his great thanks to Professor Hideo Kozono for his valuable advices and continuous encouragement. The authors would like to thank the referee for his constructive suggestions. The second author is partly supported by Research Fellow of the Japan Society for the Promotion of Science for Young Scientists.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Engineering, Chuo University, Tokyo, 112-8551, Japan

    Tsukasa Iwabuchi

  2. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Ryo Takada

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  1. Tsukasa Iwabuchi
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  2. Ryo Takada
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Correspondence to Tsukasa Iwabuchi.

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Iwabuchi, T., Takada, R. Global solutions for the Navier-Stokes equations in the rotational framework. Math. Ann. 357, 727–741 (2013). https://doi.org/10.1007/s00208-013-0923-4

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  • DOI: https://doi.org/10.1007/s00208-013-0923-4

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