标题： 关于整个空间上无电阻情况下Hall方程和电子磁流体动力学方程的不适定性 显示英文标题

标题： On illposedness of the Hall and electron magnetohydrodynamic equations without resistivity on the whole space

摘要： 在我们之前的工作中已经证明，不可压缩且不可抵抗的霍尔-电子磁流体动力学（MHD）方程在平坦域$M = \mathbb{R}^k \times \mathbb{T}^{3-k}$上是不适定的，对于$0 \le k \le 2$。 其中的数据和解被假设与一个坐标无关，这不仅显著简化了系统，还允许大量稳态的存在。 在本工作中，我们去除了独立性的假设，并得出紧支集数据在$\mathbb{R}^3$上的强不适定性。 这是通过在随时间变化的轴对称磁场附近构造退化波包来实现的。 一些主要的额外要素包括：对广义能量估计的更系统的应用、博戈夫斯基算子的使用，以及对霍尔-电子-MHD系统的轴对称解的先验估计。 显示英文摘要

摘要： It has been shown in our previous work that the incompressible and irresistive Hall- and electron-magnetohydrodynamic (MHD) equations are illposed on flat domains $M = \mathbb{R}^k \times \mathbb{T}^{3-k}$ for $0 \le k \le 2$. The data and solutions therein were assumed to be independent of one coordinate, which not only significantly simplifies the systems but also allows for a large class of steady states. In this work, we remove the assumption of independence and conclude strong illposedness for compactly supported data in $\mathbb{R}^3$. This is achieved by constructing degenerating wave packets for linearized systems around time-dependent axisymmetric magnetic fields. A few main additional ingredients are: a more systematic application of the generalized energy estimate, use of the Bogovski\v{i} operator, and a priori estimates for axisymmetric solutions to the Hall- and electron-MHD systems.