Communications in Mathematical Sciences

Volume 20 (2022)

Number 6

Global well-posedness and decay of the low-regularity solution to the 3D density-dependent magnetohydrodynamic equations with vacuum

Pages: 1517 – 1540



Shengquan Liu (School of Mathematics, Liaoning University, Shenyang, Liaoning, China)

Qiao Liu (School of Mathematics and Statistics, Hunan Key Laboratory of Analytical Mathematics and Applications (HNP-LAMA), Central South University, Changsha, Hunan, China)


In this paper, we consider the initial-boundary value problem of the 3D density-dependent magnetohydrodynamic equations with a low-regularity initial data. Assume that the initial density $\rho_0 \geq 0$ is bounded, and the scaling invariant quantity\[{\left({\bigl\lVert \rho^{1/2}_0 u_0 \bigr\rVert}^2_{L^2} + {\bigl\lVert H_0 \bigr\rVert}^2_{L^2} \right)}{\left({\bigl\lVert \nabla u_0 \bigr\rVert}^2_{L^2} + {\bigl\lVert \nabla H_0 \bigr\rVert}^2_{L^2} \right)}\]is sufficiently small, then we prove that this system admits a unique global low-regularity solution. Here, no compatibility conditions are imposed on the initial data, and the initial density is allowed to vanish. In particular, we also obtain the exponential decay of the solution by introducing a delicate time-weighted estimate.


magnetohydrodynamic equations, global existence and uniqueness, exponential decay, vacuum

2010 Mathematics Subject Classification

35B40, 76N10, 76W05

S. Liu is partially supported by the Foundation of Education Department of Liaoning Province (No. LQN201901), and the Natural Science Foundation of Liaoning Province (No. 2019-MS-152).

Q. Liu is supported by the National Natural Science Foundation of China (No. 12071122), and the Research Foundation of Education Bureau of Hunan Province, China (No. 19A313).

Received 21 February 2021

Received revised 5 December 2021

Accepted 7 January 2022

Published 14 September 2022