标题： 关于三维磁流体动力学的全局动力学：阿尔芬波的非线性稳定性 显示英文标题

标题： On global dynamics of three dimensional magnetohydrodynamics: nonlinear stability of Alfvén waves

摘要： 我们构建并研究了三维不可压缩MHD系统的全局解，其粘性任意小。 特别是，我们为在许多情况下观察到的动力学现象提供了严格的验证：解最初表现得像非色散波，解的形状在非常长的时间内（与雷诺数成正比）保持不变，之后，由于扩散效应的长时间积累，解将被阻尼，最终，系统的总能量相对于粘性变得极小，以至于扩散占主导地位，解随后迅速随时间衰减。 我们不对对称性或涡度做任何假设。 数据的大小和先验估计不依赖于粘性。 证明建立在基本能量恒等式的创新使用以及特征超曲面的几何研究之上。 该方法部分受到Christodoulou-Klainerman在广义相对论中证明闵可夫斯基空间非线性稳定性的启发。 显示英文摘要

摘要： We construct and study global solutions for the 3-dimensional incompressible MHD systems with arbitrary small viscosity. In particular, we provide a rigorous justification for the following dynamical phenomenon observed in many contexts: the solution initially behaves like non-dispersive waves and the shape of the solution persists for a very long time (proportional to the Reynolds number), thereafter, the solution will be damped due to the long-time accumulation of the diffusive effects, eventually, the total energy of the system becomes extremely small compared to the viscosity so that the diffusion takes over and the solution afterwards decays fast in time. We do not assume any condition on the symmetry or on the vorticity. The size of data and the a priori estimates do not depend on viscosity. The proof is builded upon a novel use of the basic energy identity and a geometric study of the characteristic hypersurfaces. The approach is partly inspired by Christodoulou-Klainerman's proof of the nonlinear stability of Minkowski space in general relativity.