Communications in Mathematical Sciences

Volume 18 (2020)

Number 3

Global strong solutions to the Cauchy problem of 1D compressible MHD equations with no resistivity

Pages: 851 – 873



Zilai Li (School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, China)

Huaqiao Wang (College of Mathematics and Statistics, Chongqing University, Chongqing, China)

Yulin Ye (School of Mathematics and Statistics, Henan University, Kaifeng, China)


We consider the Cauchy problem to the 1D non-resistive compressible magnetohydrodynamics (MHD) equations. We establish the global existence and uniqueness of strong solutions for large initial data and vacuum when the viscosity coefficient is assumed to be constant or density-dependent. The analysis is based on the full use of effective viscous flux and the Caffarelli–Kohn–Nirenberg weighted inequality to get the higher-order estimates of the solutions. This result could be viewed as the first one on the global well-posedness of strong solutions to the Cauchy problem of 1D non-resistive compressible MHD equations while the initial data may be arbitrarily large and permit vacuum.


1D compressible MHD equations, zero-resistivity, Cauchy problem, global strong solutions, vacuum

2010 Mathematics Subject Classification

35D35, 35Q35, 76N10, 76W05

Z. Li is supported by the NSFC (No.11601128, No.11671319, No.11931013), Fund of HPU (No.B2016-57, No.2016XQG-12) and the Key Research project of university in Henan Province (No.16A110015).

H. Wang’s research is supported by the NSFC (Grant No. 11901066) and the Natural Science Foundation of Chongqing (Grant No. cstc2019jcyj-msxmX0167) and Project No. 2019CDXYST0015 supported by the Fundamental Research Funds for the Central Universities.

Y. Ye is partially supported by NSFC (No.11701145, No.11971147).

Received 5 December 2018

Accepted 6 December 2019

Published 30 June 2020