Global small solutions of 2-D incompressible MHD system

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:259

Article

Lin, Fanghua¹  Xu, Li²  Zhang, Ping^3,4 

[1] NYU, Courant Inst, New York, NY 10012 USA

[2] Chinese Acad Sci, Acad Math & Syst Sci, Inst Computat Math, LSEC, Beijing 100190, Peoples R China

[3] Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China

[4] Chinese Acad Sci, Hua Loo Keng Key Lab Math, Beijing 100190, Peoples R China

关键词: Inviscid MHD system;    Anisotropic Littlewood-Paley theory;    Dissipative estimates;    Lagrangian coordinates;   

DOI  :  10.1016/j.jde.2015.06.034

来源: Elsevier

【 摘 要 】

In this paper, we consider the global wellposedness of 2-D incompressible magneto-hydrodynamical system with smooth initial data which is close to some non-trivial 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. To achieve this and to avoid the difficulty of propagating anisotropic regularity for the free transport equation, we first reformulate our system (1.1) in the Lagrangian coordinates (2.19). Then we employ anisotropic Littlewood-Paley analysis to establish the key a priori L-1(R+; Lip(R-2)) estimate for the Lagrangian velocity field Y-t. With this estimate, we can prove the global wellposedness of (2.19) with smooth and small initial data by using the energy method. We emphasize that the algebraic structure of (2.19) is crucial for the proofs to work. The global wellposedness of the original system (1.1) then follows by a suitable change of variables. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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