Global strong solution to the Cauchy problem of 1D compressible MHD equations with large initial data and vacuum

  • Yulin Ye
  • Zilai LiEmail author


In this paper, the Cauchy problem to the 1D compressible magnetohydrodynamics equations is considered. We establish the global existence and uniqueness of strong solution when the viscosity coefficient is assumed to be constant or density dependent. The analysis is based on some new mathematical techniques and the weighed Caffarelli–Kohn–Nirenberg inequality to get the \(L^p(1\le p<\infty )\) norm of the velocity u. Note that the initial data can be arbitrarily large and permit vacuum.


Compressible MHD equation Density-dependent viscosity Global strong solution Vacuum 

Mathematics Subject Classification

35Q35 (76W05, 35D35, 35A01) 



The authors would like to thank the anonymous referees for their helpful suggestions and kindly comments. Ye is supported by the NSFC (No. 11701145). Li is supported by the NSFC (Nos. 11601128, 11671319), Fund of HPU ( Nos. B2016-57, 2016XQG-12) and the Key Research Project of University in Henan Province (No.16A110015).


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHenan UniversityKaifengPeople’s Republic of China
  2. 2.School of Mathematics and Information ScienceHenan Polytechnic UniversityJiaozuoPeople’s Republic of China

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