Global regularity of solutions for the 3D non-resistive and non-diffusive MHD-Boussinesq system with axisymmetric data

Abstract

In this paper, we will show that the solution of the 3 dimensional non-diffusive MHD-Boussinesq system is globally regular if the initial data is axisymmetric and the swirl component of the velocity and the magnetic vorticity is trivial. Our method can also be applied to the magnetic Rayleigh-Bénard convection system.

One component regularity criteria for the axially symmetric MHD-Boussinesq system

“…Meanwhile, (3.2) follows from the standard L 2 estimate of the system (1.1), together with the result in (3.1). See also [27,Proposition 2.1]. We omit the details here.…”

“…Using the method in the proof of Proposition 2.2 of [27], the first 3 terms above can be estimated by…”

“…Recently there are more and more studies concerning the full 3D MHD-Boussinesq system. We refer readers to papers such as [18,22,3,27] for the regularity criteria, local and global well-posedness of weak and strong solutions of the MHD-Boussinesq system. The local well-posedness results were proved in Larios-Pei [18].…”

“…Recently, Bian-Pu [3] proved the global regularity of axially symmetric large solutions to the MHDB system (1.2) without the horizontal swirl component u θ of the velocity under the assumption that the support of the initial thermal fluctuation is away from the z-axis and its projection on to the z-axis is compact. Later, this result was improved in Pan [27] by removing the "support set" assumption on the initial data of the thermal fluctuation.…”

On the stability of hydrostatic equilibrium in a uniform magnetic field with partial dissipation

“…For more physics, numerical simulations, and experiments, the interested readers may refer to previous studies [1][2][3] and the references therein. Recently, important progress has been made on the global stability for Boussinesq-MHD system; see other works [4][5][6][7][8][9][10][11][12][13][14][15][16] and the references therein. And Bian et al 17 have proved the nonlinear stability of Couette flow in a uniform magnetic field for this system with only vertical dissipation, in which some ideas are shared from the paper of Deng et al 18 for 2D Boussinesq system with vertical dissipation.…”