Global small solutions to 2-D incompressible MHD system
Global small solutions to 2-D incompressible MHD system
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DOI：
10.1016/j.jde.2015.06.034
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发表时间：
2013-02
期刊：
影响因子：
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通讯作者：
F. Lin;Li Xu;Ping Zhang
中科院分区：
文献类型：
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作者：
F. Lin;Li Xu;Ping Zhang
本文研究了初始数据光滑且接近非平凡定态的二维不可压缩磁流体力学系统的全局适定性。它是一个介于Navier-Stokes方程和具有普遍非线性耦合结构的自由输运方程之间的耦合系统。证明的主要困难在于探索系统的耗散机制。为了实现这一点并避免传播自由输运方程的各向异性正则性的困难，我们首先在拉格朗日坐标(2.19)中重新表示我们的系统(1.1)。然后利用各向异性Littlewood-Paley分析建立了拉格朗日速度场Yt的关键先验L 1(R+；Lip(R2))估计.利用这个估计，我们可以用能量方法证明(2.19)在光滑和小的初始数据下的整体适定性.我们强调，(2.19)的代数结构是证明工作的关键。然后，原始系统(1.1)的全局完好性之后是变量的适当变化。
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.