Mathematical methods in the kinetic theory of plasmas and gravitating systems

提案ID：0604946 PI：Lancellotti，CarloInstitution：纽约市立史泰登岛标题：等离子体和引力系统动力学理论中的数学方法摘要本项目研究等离子体和引力系统动力学理论中的一些开放性数学问题。 描述粒子系统通过引力（牛顿）或静电（库仑）相互作用的动力学方程在天体物理学和等离子体物理学中至关重要，但它们对数学分析相当不敏感。该研究建立在PI最近的一些突破的基础上，并遵循两条攻击线：1）通过主方程方法研究（线性或非线性）Fokker-Planck类型的空间齐次方程的数学性质，其中动力学方程近似为精心选择的$N$体随机过程的线性Kolmogorov方程的无限粒子极限。 后者取代了“物理”$N$-体系统的（复杂得多的）基本哈密顿动力学，并使得有可能提取关于动力学方程解的存在性、它们衰减到平衡态的速率等的新信息。2）该项目还探索了轨道平均方法在天体物理学和等离子体物理学中出现的空间非均匀弗拉索夫-朗道-泊松方程中的应用。 这些方法利用了某些物理系统中存在的不同时间尺度，以大大减少自变量的数量，从而便于方程的数学分析。动力学方程描述了气体、等离子体和星团等多体系统的演化。这种类型的方程在许多应用中起着至关重要的作用，从气体动力学到聚变等离子体，从天体物理学到物理化学，从交通流量到半导体。 该研究项目侧重于系统中的粒子通过长程力相互作用的情况，如电离气体（等离子体）和恒星的情况。 在这些情况下，数学分析变得非常复杂，因为长程力允许许多粒子同时相互作用，而普通气体只是由粒子对之间的“二元”（一对一）碰撞驱动。该项目的主要目标是获得这些方程的解的行为的精确数学估计。 例如，当研究实验室等离子体时，非常希望知道所使用的数学模型具有明确定义的解，并且还希望估计外部干扰将多快消失等。（并支持有效性）的数值模拟，在研究聚变反应堆，地球的磁层，球状星团和许多其他应用。

Proposal ID: 0604946PI: Lancellotti, CarloInstitution: CUNY Staten IslandTitle: Mathematical methods in the kinetic theory of plasmas and gravitating systemsAbstractThis project investigates some open mathematical problems in the kinetic theory of plasmas and gravitating systems. The kinetic equations that describe systems of particles interacting via gravitational (Newton) or electrostatic (Coulomb) forces are crucially important in astrophysics and plasma physics, but they have been quite impervious to mathematical analysis. The investigation builds on some recent breakthroughs by the PI and follows two lines of attack:1) The mathematical properties of spatially homogeneous equations of the (linear or nonlinear) Fokker-Planck type are being studied via a master equation approach, in which a kinetic equation is approximated as the infinite-particle limit of a linear Kolmogorov equation for a carefully selected $N$-body stochastic process. This latter replaces the (much more complex) underlying Hamiltonian dynamics of the ``physical'' $N$-body system and makes it possible to extract new information about existence of solutions to a kinetic equation, their rate of decay to equilibrium etc. 2) This project also explores the application of orbit-averaging methods to the spatially inhomogeneous Vlasov-Landau-Poisson equations that arise in astrophysics and plasma physics. These methods take advantage of different time scales present in certain physical systems in order to greatly reduce the number of independent variables and thus facilitate the mathematical analysis of the equations.Kinetic equations describe the evolution of many-body systems such as gases, plasmas and clusters of stars. Equations of this type play an essential role in many applications, ranging from gas dynamics to fusion plasma, from astrophysics to physical chemistry, from traffic flow to semiconductors. This research project focuses on those situations in which the particles in the system interact via long-range forces, as is the case for ionized gases (plasmas) and for stars. In these cases, the mathematical analysisis greatly complicated by the fact that long-range forces allow many particles to interact with each other at the same time, whereas ordinary gases are driven just by "binary" (one-on-one) collisions between pairs of particles. The main goal of the project is to obtain precise mathematical estimates of the behavior of solutions to these equations. For instance, when studying a laboratory plasma it is very desirable to know that the mathematical model being used has well-defined solutions, and also to estimate how quickly external disturbances will fade away etc. Beside their intrinsic mathematical value, results of this type also aid the development (and support the validity) of the numerical simulations that play an important role in the study of fusion reactors, of the earth's magnetosphere, of globular clusters of stars and many other applications.