带点电荷的Vlasov-Poisson系统的适定性及解的渐近性态研究

The Vlasov-Poisson system with point charges is an important kinetic equation, it is mainly used to model the time evolution of a collisionless plasma with point charges. Due to the long range interactions among the particles and the repulsive or attractive interactions between the point charges and the particles, it is difficult to describe the motion of the plasma. This project is devoted to the investigation on the well-posedness and the asymptotic behavior of the solutions to the Vlasov-Poisson system with point charges. In the situation of repulsive interactions between the point charges and the plasma, by virtue of detailed region segmentation and velocity-spatial moments, we obtain the support estimate of the classical solution and establish the velocity moment propagation; For the infinite mass system, we firstly consider the existence of global classical solution for the screened Vlasov-Poisson system with point charges. Then we study the limit as the screening is removed, to show the existence of Vlasov-Poisson system with point charges. In the more complex attractive situation, by constructing local energy functional, we prove the global existence of classical solution for the screening Vlasov-Poisson system with point charges.. The achievement of this project may enrich the research of the Vlasov-Poisson system with point charges. Furthermore, it will broaden out theory research on kinetic equation, promoting the development of controlled thermonuclear fusion，high-temperature plasma physics and related disciplines.

带点电荷的Vlasov-Poisson系统是一类重要的动理学方程，可以描述加入点电荷的无碰撞等离子体的运动分布规律。由于粒子之间相互作用的长程库伦位势，点电荷与粒子之间的斥力或引力相互作用，使得等离子体的运动变得很复杂。本项目主要研究带点电荷的Vlasov-Poisson系统的适定性及解的相关性态。在斥力场情形，借助细致的区域分割和速度空间联合矩，建立解的速度支柱估计及矩传播；对于质量无限系统，首先建立带屏蔽效应的系统解的存在性，然后研究屏蔽强度趋于0时解的极限，间接证明非屏蔽系统解的存在性。在复杂的引力场情形，构造局部能量函数证明带点电荷的屏蔽Vlasov-Poisson系统存在整体经典解。. 本项目的预期成果将进一步丰富带点电荷的Vlasov-Poisson系统的研究结果，同时拓宽动理学方程的研究，促进受控热核聚变、高温等离子体理论等相关学科的发展。

本项目主要研究带点电荷的Vlasov-Poisson系统的适定性及解的相关性态。在斥力场情形，假设初始质量密度函数具有衰减性，通过构造局部能量函数，证明了带点电荷的Vlasov-Poisson系统无限质量整体经典解的存在性；假设初始微观密度函数支柱非紧但具有衰减性，借助局部能量函数和相空间分割，证明了无限速度的带点电荷的Vlasov-Poisson系统无限质量整体经典解的存在性；对于环上的三维Vlasov-Poisson系统，充分利用能量守恒率和渐近方法，证明了弱解三阶以上速度矩的传播。同时，我们还利用调和分析技巧研究了分数阶Navier-Stokes方程，在Fourier-Herz空间中建立了整体解的稳定性；利用伽乐金方法证明了广义的Novikov方程弱解的整体存在性和唯一性。该研究项目得到的相关结论，完善并丰富了Vlasov-Poisson系统的理论研究，而且为带点电荷的Vlasov-Poisson系统的柱面对称解、数值解等问题提供了新的研究思路。