A collisionless plasma is a fully ionized gas in which electromagnetic forces are strong enough to 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 neglected 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 collisionless plasma? That is, could a singularity develop from smoothly prescribed initial values as time progresses? In some cases, such as in lower dimensional, relativistic formulations (e.g., one space and two velocity variables), smooth global solutions are known to exist. Another problem to be investigated concerns the long-time behavior of the charge and current densities and electromagnetic fields in the system. More specifically, do dispersive effects in the equations cause these quantities to decay over time, or is there sufficient interaction so as to sustain their strength even as time tends to infinity?
Kinetic Theory includes the study of the motion and properties of plasma. Plasmas are often referred to as the fourth state of matter (after solids, liquids and gases) and account for 99.99% of all material in the universe. They are of great practical interest because they are charged gases, and thus serve as excellent conductors of electricity. As an example, plasma engines have been developed by a number of space agencies and recently used to power some NASA spacecraft. Additionally, the use of plasmas through nuclear fusion as a source of clean energy is currently of immense scientific interest. Notable examples of collisionless plasmas include the solar wind, the Earth's ionosphere, galactic nebulae, low-density fusion reactors, and comet tails. A complete understanding of the solar wind would also be extremely useful, as this natural phenomenon dictates the intensity of "space weather", which is often responsible for expensive damage to satellites orbiting the Earth. The motion of a plasma is described by a number of complicated differential equations dictated by physics. Among the goals of the current project are to show that these equations possess solutions (under appropriate conditions), determine their qualitative behavior, and approximate them computationally so that one can predict behavior in future situations with certainty.
