Mathematical Analysis: Long-time dynamics in kinetic theory

The Vlasov-Maxwell and Vlasov-Poisson systems of equations describe the behaviour of plasmas such as those one can find in fusion reactors. The long-time behaviour of these systems is an important problem in physics and engineering, and has therefore attracted significant interest in the mathematics community (pure, applied and numerical). In this project you will first familiarize yourself with the state-of-the-art in the field and then explore the following questions:
• A gas of electrons disperses due to the repulsive nature of the electromagnetic force. Can we obtain an optimal rate for this dispersion?
• Existence and uniqueness of solutions is known for plasmas where the magnetic force is neglected (the Vlasov-Poisson system) but when the full set of Maxwell’s equations is introduced this is still an open problem. Can we push the limits of what’s known and achieve a better understanding of the effect of the magnetic field?
• There are some explicit steady-state solutions of these systems. However, are these solutions stable, or would a slight fluctuation result in a totally different solution? This is an extremely important question for engineers who are attempting to build fusion reactors.