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Global Small Solutions to an MHD‐Type System: The Three‐Dimensional Case
Author(s) -
Lin Fanghua,
Zhang Ping
Publication year - 2014
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21506
Subject(s) - dissipative system , mathematics , magnetohydrodynamics , vector field , nonlinear system , compressibility , mathematical analysis , divergence (linguistics) , coupling (piping) , type (biology) , anisotropy , magnetic field , physics , mechanics , geometry , mechanical engineering , ecology , linguistics , philosophy , engineering , biology , quantum mechanics
In this paper, we consider the global well‐posedness of a three‐dimensional incompressible MHD type system with smooth initial data that is close to some nontrivial steady state. It is a coupled system between the Navier‐Stokes equations and a free transport equation with a universal nonlinear coupling structure. The main difficulty of the proof lies in exploring the dissipative mechanism of the system due to the fact that there is a free transport equation of ϕ in the coupled equations and only the horizontal derivatives of ϕ is dissipative with respect to time. To achieve this, we first employ anisotropic Littlewood‐Paley analysis to establish the key L 1 (ℝ + ; Lip(ℝ 3 )) estimate to the third component of the velocity field. Then we prove the global well‐posedness to this system by the energy method, which depends crucially on the divergence‐free condition of the velocity field. © 2014 Wiley Periodicals, Inc.