Communications in Mathematical Sciences

Volume 20 (2022)

Number 5

Global strong solution to the cauchy problem of 2D density-dependent Boussinesq equations for magnetohydrodynamics convection with thermal diffusion

Pages: 1437 – 1458



Min Liu (Faculty of Science, Beijing University of Technology, Beijing, China)


In this paper, we study the Cauchy problem of density-dependent Boussinesq equations for magnetohydrodynamics convection on the whole 2D space. We first establish global and unique strong solution for the 2D Cauchy problem when the initial density includes vacuum state. Furthermore, we consider that the initial data can be arbitrarily large. We derive a consistent priori estimate by the energy method, and extend the local strong solutions to the global strong solutions. Finally, we obtain the large-time decay rates of the gradients of velocity, temperature field, magnetic field and pressure.


KMHD-Boussinesq equation, global strong solution, density-dependent, large-time behavior, vacuum

2010 Mathematics Subject Classification

35Q35, 76D03

Received 25 November 2020

Received revised 1 December 2021

Accepted 24 December 2021

Published 26 May 2022