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On the well‐posedness of the Cauchy problem for an MHD system in Besov spaces
Author(s) -
Miao Changxing,
Yuan Baoquan
Publication year - 2008
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1026
Subject(s) - besov space , mathematics , uniqueness , initial value problem , homogeneous , dimension (graph theory) , mathematical analysis , cauchy problem , space (punctuation) , compressibility , pure mathematics , interpolation space , combinatorics , physics , functional analysis , computer science , thermodynamics , biochemistry , chemistry , gene , operating system
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.