电子-离子碰撞冷等离子体动力学的精确阈值
Exact thresholds in the dynamics of cold plasma with electron-ion collisions
摘要：
本文考虑一个准线性双曲型方程组，它描述了冷等离子体中电子的一维非相对论振荡，并考虑了电子-离子碰撞。考虑到碰撞导致出现类似于机械系统中的干摩擦的术语，导致总能量的减少。我们得到了Cauchy问题在时间上整体光滑解存在的一个判据。它允许准确地将初始数据分成两类：一类对应于时间上的全局光滑解，另一类导致有限时间爆破。研究了电子碰撞频率$ ν$对解的影响。它示出，有一个阈值，超过该阈值后，阻尼振荡的制度是由单调阻尼的制度所取代。对应于柯西问题的全局时间光滑解的初始数据集随着$ ν$的增加而扩展，然而，在任意大的值处，存在光滑初始数据，对于该光滑初始数据，解在有限时间内形成奇点，并且当$ ν$趋于无穷大时，该时间趋于零。数值例子说明了新出现的奇异性的性质。
英文摘要：
We consider a quasilinear system of hyperbolic equations that describes plane one-dimensional non-relativistic oscillations of electrons in a cold plasma with allowance for electron-ion collisions. Accounting for collisions leads to the appearance of a term analogous to dry friction in a mechanical system, leading to a decrease in the total energy. We obtain a criterion for the existence of a global in time smooth solution to the Cauchy problem. It allows to accurately separate the initial data into two classes: one corresponds to a globally in time smooth solutions, and the other leads to a finite-time blowup. The influence of electron collision frequency $ ν$ on the solution is investigated. It is shown that there is a threshold value, after exceeding which the regime of damped oscillations is replaced by the regime of monotonic damping. The set of initial data corresponding to a globally in time smooth solution of the Cauchy problem expands with increasing $ ν$, however, at an arbitrarily large value there are smooth initial data for which the solution forms a singularity in a finite time, and this time tends to zero as $ ν$ tends to infinity. The character of the emerging singularities is illustrated by numerical examples.
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