带固定背景的三维Vlasov-Poisson系统质量无限的经典解的存在唯一性

The Vlasov-Poisson system is one of the most basic kinetic models in plasma physics and astrophysics. It simulates the evolution of an ensemble of plasmas or astronomical bodies which interact by a repulsive potential or gravitational potential determined by themselves. After proving that a large initial data lead to the existence of global classical solutions for the standard three-dimensional Vlasov-Poisson system, the Vlasov-Poisson system with a positive charge background has gained wide attention. When the microscopic density of the background depends only upon velocity, this system models the evolution of mobile negative ions which balanced the positive ions as the space variable going to infinity. It follows from the infinite mass of positive ions that the mass of negative ions is also infinite, which is very different from the classic system with mass consevation. When the spatial asymptotics between the initial negative ions and the positive ions are so weak that the difference between their initial macroscopic densities is non-integratable, an unique smooth solution with infinite mass will be shown to exist locally in time by obtaining a strong estimate of characteristic flows, and a criterion for the continuation of the solution will be also established in this project. At last, by controlling the velocity support of solutions we will also establish the global existence and uniqueness of solutions for the system with a background, the microscopic density of which is compactly supported in velocity.

Vlasov-Poisson系统是等离子体物理和天体物理中最基本的动力学模型之一。它模拟了大量无碰撞的等离子体(或天体)由于相互之间斥力(或引力)的作用而随时间的演化。在标准的三维系统大初值问题经典解的整体存在性被证明之后，一类带正离子背景的Vlasov-Poisson系统得到广泛关注。该系统是在背景介质的微观密度仅和速度有关的情形下，模拟在空间无穷远处逼近正离子密度分布的负离子的演化。由于此时正离子的总质量无限，所以该负离子的总质量也无限，这与质量守恒的标准系统有很大的区别。在初始负离子与正离子的空间渐进性非常弱以至于它们初始宏观密度之差不可积时，本项目将通过建立特征流的强衰减估计来得到该三维系统质量无限的经典解的局部存在唯一性，并建立解的延拓准则。最后，本项目将在正离子背景的微观密度有紧支柱的情形下，通过控制解的速度支柱来建立该三维系统质量无限的经典解的整体存在唯一性。

在本项目中，我们主要研究了具有空间渐近行为的三维Vlasov-Poisson系统。假设固定的正离子的微观密度仅与速度有关，且运动的负离子在空间变量趋于无穷大时逼近正离子，所以正离子和负离子的总电量都是正无穷大。在非常弱的条件下（该假设蕴含总电量无穷大的情形），我们建立了具有空间弱渐近行为的光滑解的存在唯一性，并且建立了一个几乎最优的延拓准则。这个结果发表在Journal of Mathematical Physics上。另外，我们还研究了稀薄气体中的另一类重要动理学模型：BGK方程。在得到一类新的矩引理以及加权L^{\infty}解的存在唯一性之后，我们建立了具有无穷能量的BGK方程的几类全局解。该结果发表于Mathematical Methods in the Applied Sciences。