论文标题
关于两个半维的全球规律性问题的评论
Remarks on the global regularity issue of the two and a half dimensional Hall-magnetohydrodynamics system
论文作者
论文摘要
是否解决了$ 2 \ frac {1} {2} $ - 尺寸霍尔·马格尼特水力动力学系统从平滑初始数据开始保留其规律性一直是一个具有挑战性的开放问题。尽管有关Navier-Stokes方程和磁性水力动力学系统的规律性标准的组件减少的研究方向最近引起了很多关注，但Hall术语却带来了很大的困难。在本手稿中，我们在霍尔期限内发现了一定的取消，并获得了各种新的规律性标准：首先，就磁场的第三个成分的梯度而言；其次，仅在电流密度的第三个成分方面；第三，就速度场的第三个组成部分而言；第四，就速度字段的第一组和第二个组件而言。作为我们发现的取消的另一个结果，我们能够证明$ 2 \ frac {1} {1} {2} $ - 尺寸霍尔·麦格纳特摩毛水动力学系统具有超扩散仅用于水平方向的磁场；在三维情况下，我们还通过发现其他取消效果获得了类似的结果。这些结果扩展并改善了以前的各种工作。
Whether or not the solution to the $2\frac{1}{2}$-dimensional Hall-magnetohydrodynamics system starting from smooth initial data preserves its regularity for all time remains a challenging open problem. Although the research direction on component reduction of regularity criterion for Navier-Stokes equations and magnetohydrodynamics system have caught much attention recently, the Hall term has presented much difficulty. In this manuscript we discover a certain cancellation within the Hall term and obtain various new regularity criterion: first, in terms of a gradient of only the third component of the magnetic field; second, in terms of only the third component of the current density; third, in terms of only the third component of the velocity field; fourth, in terms of only the first and second components of the velocity field. As another consequence of the cancellation that we discovered, we are able to prove the global well-posedness of the $2\frac{1}{2}$-dimensional Hall-magnetohydrodynamics system with hyper-diffusion only for the magnetic field in the horizontal direction; we also obtained an analogous result in the 3-dimensional case via discovery of additional cancellations. These results extend and improve various previous works.
