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Regularity condition by mean oscillation to a weak solution of the 2-dimensional Harmonic heat flow into sphere

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Abstract

We show a regularity criterion to the harmonic heat flow from 2-dimensional Riemannian manifold M into a sphere. It is shown that a weak solution of the harmonic heat flow from 2-dimensional manifold into a sphere is regular under the criterion

$$\int\limits_0^T\|\nabla u(\tau)\|_{BMO_r}^{2}d \tau$$

where BMO r is the space of bounded mean oscillations on M. A sharp version of the Sobolev inequality of the Brezis–Gallouet type is introduced on M. A monotonicity formula by the mean oscillation is established and applied for proving such a regularity criterion for weak solutions as above.

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Authors and Affiliations

  1. Department of Mathematics, School of Science, Kumamoto University, Kumamoto, 860-8555, Japan

    Masashi Misawa

  2. Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

    Takayoshi Ogawa

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  1. Masashi Misawa
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  2. Takayoshi Ogawa
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Correspondence to Masashi Misawa.

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Misawa, M., Ogawa, T. Regularity condition by mean oscillation to a weak solution of the 2-dimensional Harmonic heat flow into sphere. Calc. Var. 33, 391–415 (2008). https://doi.org/10.1007/s00526-008-0166-5

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  • DOI: https://doi.org/10.1007/s00526-008-0166-5

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