Mathematical Problems in Collisionless Kinetic Theory

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).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 approaches 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. For 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. 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 possess solutions (under appropriate conditions), determine their qualitative behavior, and approximate them numerically (so that one can predict behavior in future situations with certainty). A proof that the Vlasov-Maxwell system has a "nice" solution would also confirm that the system of equations is the "right" one to describe plasma-related phenomena.