Nonlinear Partial Differential Equations in Kinetic Theory

Indiana University, Bloomington IN

Investigators

Abstract

A collisionless plasma is a fully ionized gas in which electromagnetic forces dominate collisional effects. The motion of a high temperature, low density collisionless plasma is described by the Vlasov-Maxwell equations, a nonlinear system of hyperbolic partial differential equations. In this setting collisions are ignored while the charge and current densities (which drive the Maxwell system) are determined in a self-consistent manner from velocity moments of solutions to the Vlasov equation. The major question to be studied is this: are there shocks in a three dimensional collisionless plasma? That is, could a singularity develop from smoothly prescribed initial values as time progresses? Smooth global solutions are known to exist in lower dimensional situations (e.g., two space and velocity variables) and when the data are "small." Kinetic Theory includes the study of the motion and properties of plasmas. Plasmas are often called the fourth state of matter (after solids, liquids and gases); they account for practically all of the material in the universe. Plasmas are charged gases and hence are excellent conductors of electricity. "Plasma engines" have been used recently to power some NASA spacecraft. Notable examples of collisionless plasmas include the solar wind, the ionosphere, galactic nebulae and comet tails. The motion of a plasma is described by a number of complicated equations dictated by physics. Among the mathematician's goals are to show that these equations have solutions (under appropriate conditions) and to approximate them numerically (so that one can predict certain behavior in future situations). To show that the Vlasov-Maxwell system has a "nice" solution would at least partially confirm this system as the "right" one to describe these phenomena.