This 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.
