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Space-Time Regularity for the Three Dimensional Navier–Stokes and MHD Equations

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Abstract

In this paper, we investigate the space-time regularity of solutions to (1) the three dimensional incompressible Navier–Stokes equations for initial data \(u_{0}=(u_{0}^{h},u_{0}^{3}) \in \dot{B}_{p,r}^{ \frac{3}{p}-1} (\mathbb{R}^{3})\) with large initial vertical velocity component; and (2) the three dimensional incompressible magneto-hydrodynamic equations for initial datum \(u_{0}=(u_{0}^{h},u _{0}^{3})\in \dot{B}_{p,r}^{\frac{3}{p}-1} (\mathbb{R}^{3})\) with large initial vertical velocity component and \(b_{0}=(b_{0}^{h},b_{0}^{3}) \in \dot{B}_{p,r}^{\frac{3}{p}-1} (\mathbb{R}^{3})\) with large initial vertical magnetic field component.

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Acknowledgements

The authors sincerely acknowledge their gratefulness to the referee for the valuable comments and suggestions. W. Zhu is partially supported by the National Natural Science Foundation of China (11571381). J. Zhao is partially supported by the National Natural Science Foundation of China (11501453) and the Natural Science Foundation of Shaanxi Province (2018JM1004).

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Authors and Affiliations

  1. Department of Mathematics, Sun Yat-Sen University, Guangzhou, 510275, China

    Weipeng Zhu

  2. School of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji, Shaanxi, 721013, China

    Jihong Zhao

  3. College of Science, Northwest A&F University, Yangling, Shaanxi, 712100, China

    Jihong Zhao

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  1. Weipeng Zhu
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Correspondence to Jihong Zhao.

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Zhu, W., Zhao, J. Space-Time Regularity for the Three Dimensional Navier–Stokes and MHD Equations. Acta Appl Math 163, 157–184 (2019). https://doi.org/10.1007/s10440-018-0218-6

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