On the well‐posedness of the Cauchy problem for an MHD system in Besov spaces

This paper is devoted to the study of the Cauchy problem of incompressible magneto‐hydrodynamics system in the framework of Besov spaces. In the case of spatial dimension n ⩾3, we establish the global well‐posedness of the Cauchy problem of an incompressible magneto‐hydrodynamics system for small data and the local one for large data in the Besov space Ḃ   p,r n / p −1(ℝ n ), 1⩽ p <∞ and 1⩽ r ⩽∞. Meanwhile, we also prove the weak–strong uniqueness of solutions with data in Ḃ   p,r n / p −1(ℝ n )∩ L 2 (ℝ n ) for n /2 p +2/ r >1. In the case of n =2, we establish the global well‐posedness of solutions for large initial data in homogeneous Besov space Ḃ   p,r 2/ p −1(ℝ 2 ) for 2< p <∞ and 1⩽ r <∞. Copyright © 2008 John Wiley & Sons, Ltd.