标题： 无压 Euler--Poisson 方程中的临界阈值与背景状态 显示英文标题

标题： Critical thresholds in pressureless Euler--Poisson equations with background states

摘要： 我们研究了一类一维无压Euler-Poisson（EP）方程中的临界阈值现象，这些方程具有非零背景状态。 首先，我们在适合某种\textit{中立条件}成立的适当正则性空间中建立了局部时间适定性。 中性条件被证明是必要的：我们构造了表现出中性条件瞬时失效的光滑解，这进而导致在经典Sobolev空间$H^s({\mathbb R})$，$s \geq 2$中甚至局部时间内的解都不存在。 接下来，我们研究满足中性条件的无压EP系统中的临界阈值现象，在此我们区分两种情况。 我们证明，在吸引性强迫的情况下，中性条件可以进一步将亚临界区域限制在其边界上，即——亚临界区域在相平面上被缩减为一条直线。 然后，我们转向对具有变背景的排斥性EP系统中的临界阈值提供一个相当明确的答案。 作为应用，我们分析了冷等离子体离子动力学的阻尼EP系统的临界阈值，其中电子密度由\textit{麦克斯韦-玻尔兹曼关系}给出。 显示英文摘要

摘要： We investigate the critical threshold phenomena in a large class of one dimensional pressureless Euler--Poisson (EP) equations, with non-vanishing background states. First, we establish local-in-time well-posedness in proper regularity spaces, which are adapted for a certain \textit{neutrality condition} to hold. The neutrality condition is shown to be necessary: we construct smooth solutions that exhibit instantaneous failure of the neutrality condition, which in turn yields non-existence of solutions, even locally in time, in the classical Sobolev spaces $H^s({\mathbb R})$, $s \geq 2$. Next, we study the critical threshold phenomena in the neutrality-condition-satisfying pressureless EP systems, where we distinguish between two cases. We prove that in the case of attractive forcing, the neutrality condition can further restrict the sub-critical region into its borderline, namely -- the sub-critical region is reduced to a single line in the phase plane. We then turn to provide a rather definitive answer for the critical thresholds in the case of repulsive EP systems with variable backgrounds. As an application, we analyze the critical thresholds for the damped EP system for cold plasma ion dynamics, where the density of electrons is given by the \textit{Maxwell--Boltzmann relation}.