On the Cauchy Problem for the Hall and Electron Magnetohydrodynamic Equations Without Resistivity I: Illposedness Near Degenerate Stationary Solutions
On the Cauchy Problem for the Hall and Electron Magnetohydrodynamic Equations Without Resistivity I: Illposedness Near Degenerate Stationary Solutions
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DOI：
10.1007/s40818-022-00134-5
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发表时间：
2019-02
期刊：
影响因子：
2.8
通讯作者：
In-Jee Jeong;Sung-Jin Oh
中科院分区：
文献类型：
--
作者：
In-Jee Jeong;Sung-Jin Oh
在本文中，我们证明了无电阻率的不可压缩霍尔方程和电子磁流体动力学方程（MHD）的柯西问题的各种病态结果。这些偏微分方程是等离子体的流体描述，其中忽略了碰撞的影响（无电阻率），而考虑了电子相对于离子的运动（霍尔电流项）。霍尔电流项使磁场方程具有拟线性色散特性，这是病态机制的关键。也许本文最引人注目的结论是，在一个平移对称下，Hall-MHD（粘性或非粘性）和电子- mhd方程的Cauchy问题在任何足够高正则性Sobolev空间甚至任何Gevrey空间中都是在平凡解附近的不适定问题。尽管在平凡解附近的线性化方程具有明显的适位性，并且非线性能量守恒，即解的norm（能量）随时间保持恒定，但这个结果仍然成立。核心病态（或不稳定性）机制是某些高频波包解的退化，从而导致这些方程的一类线性退化平稳解的线性化，这些方程本质上是具有退化主符号的色散方程。这项工作中开发的方法是尖锐和鲁棒的，因为我们还证明了在存在小于1阶的分数耗散的情况下非线性不适定性（任意高），与先前已知的适定性结果相匹配。本文的结果得到了一项配套工作的补充，其中我们提供了初始磁场的几何条件，以确保不可压缩霍尔方程和电子- mhd方程的柯西问题的适位性（!）。特别地，与这里的结果形成鲜明对比的是，在同伴工作中表明，非线性柯西问题在任何非零常数磁场附近都是适定的。
In this article, we prove various illposedness results for the Cauchy problem for the incompressible Hall- and electron-magnetohydrodynamic (MHD) equations without resistivity. These PDEs are fluid descriptions of plasmas, where the effect of collisions is neglected (no resistivity), while the motion of the electrons relative to the ions (Hall current term) is taken into account. The Hall current term endows the magnetic field equation with a quasilinear dispersive character, which is key to our mechanism for illposedness. Perhaps the most striking conclusion of this article is that the Cauchy problems for the Hall-MHD (either viscous or inviscid) and the electron-MHD equations, under one translational symmetry, are ill-posed near the trivial solution in any sufficiently high regularity Sobolev spaceand even in any Gevrey spaces. This result holds despite obvious wellposedness of the linearized equations near the trivial solution, as well as conservation of the nonlinear energy, by which thenorm (energy) of the solution stays constant in time. The core illposedness (or instability) mechanism is degeneration of certain high frequency wave packet solutions to the linearization around a class of linearly degenerate stationary solutions of these equations, which are essentially dispersive equations with degenerate principal symbols. The method developed in this work is sharp and robust, in that we also prove nonlinear-illposedness (forsarbitrarily high) in the presence of fractional dissipation of any order less than 1, matching the previously known wellposedness results. The results in this article are complemented by a companion work, where we provide geometric conditions on the initial magnetic field that ensure wellposedness(!) of the Cauchy problems for the incompressible Hall and electron-MHD equations. In particular, in stark contrast to the results here, it is shown in the companion work that the nonlinear Cauchy problems are well-posed near any nonzero constant magnetic field.