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

  • Weipeng Zhu
  • Jihong Zhao
Article
  • 17 Downloads

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.

Keywords

Navier–Stokes equations MHD equations Space-time regularity Large initial components 

Mathematics Subject Classification

35B65 35Q30 35Q35 76D03 

Notes

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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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsSun Yat-Sen UniversityGuangzhouChina
  2. 2.School of Mathematics and Information ScienceBaoji University of Arts and SciencesBaojiChina
  3. 3.College of ScienceNorthwest A&F UniversityYanglingChina

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