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

Abstract

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.