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Global regularity for the supercritical active scalars

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Abstract

This paper focuses on the global regularity problem for a family of active scalar equations with fractional dissipation. We obtain an explicit lower bound on the local existence of solutions, the eventual regularity of solutions and the global regularity of solutions for large initial data for the supercritical active scalar equations. In addition, a new regularity criterion is presented for the supercritical case which is used to study the eventual regularity.

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Acknowledgements

This work was partially supported by NSFC (No. 11201124, No. 11471103), Foundation for University Key Teacher by the Henan Province (No. 2015GGJS-070) and Outstanding Youth Foundation of Henan Polytechnic University (No. J2014-03).

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  1. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454000, People’s Republic of China

    Haifeng Shang, Yana Guo & Mengmeng Song

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  1. Haifeng Shang
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  2. Yana Guo
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  3. Mengmeng Song
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Correspondence to Haifeng Shang.

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Shang, H., Guo, Y. & Song, M. Global regularity for the supercritical active scalars. Z. Angew. Math. Phys. 68, 64 (2017). https://doi.org/10.1007/s00033-017-0810-z

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