Dynamics of Partial Differential Equations

Volume 15 (2018)

Number 2

Regularity criteria to the axisymmetric incompressible Magneto-hydrodynamics equations

Pages: 109 – 126

DOI: https://dx.doi.org/10.4310/DPDE.2018.v15.n2.a2


Quansen Jiu (School of Mathematical Sciences, Capital Normal University, Beijing, China)

Jitao Liu (College of Applied Sciences, Beijing University of Technology, Beijing, China)


In this paper, we investigate the regularity criteria of axisymmetric solutions to the incompressible MHD equations, which have the form $u = u_r e_r + u_\theta e_\theta + u_z e_z$ and $b = b_\theta e_\theta$. Through establishing some innovative estimates, we obtain some new regularity criteria that are scaling invariant and independent of $b_\theta$. To some extent, our work can be seen as a generation of the result by D. Chae and J. Lee on the axisymmetric incompressible Navier–Stokes equations.


regularity criteria, incompressible MHD equations, axisymmetric

2010 Mathematics Subject Classification

Primary 35B65. Secondary 76D03, 76W05.

The authors would like to thank the anonymous referees for their valuable suggestions in improving original manuscript. The authors would also like to thank Professor Zhouping Xin for his valuable discussions and constant encouragements. Q. Jiu is supported by National Natural Sciences Foundation of China (No. 11671273). J. Liu is supported by youth backbone individual program of the organization department of Beijing (No. 2017000020124G052), and the Connotation Development Funds of Beijing University of Technology.

Received 3 March 2016

Published 29 May 2018