Well‐posedness and large‐time behavior for the two‐phase system with magnetic field in critical Besov spaces

Abstract This paper is dedicated to the study of the two‐phase system with magnetic field in multi‐dimensional spaces . We investigate the local and global well‐posedness of strong solutions to the Cauchy problem in critical Besov spaces. Moreover, a Lyapunov‐type energy argument can be developed, which leads to the time‐decay estimates of solutions without additional smallness assumptions. We improve the previous effects obtained by Wen and Zhu (JDE‐2018), Wang and Cheng (AMS‐2022), where they established the global well‐posedness of strong solutions in the ‐norm Sobolev spaces.