In this colloquium talk, we study the Vlasov-Maxwell equations, a collisionless model in the field of kinetic theory. The model is a fundamental model for the dynamics of plasmas and was introduced in 1938 by Vlasov. Due to the hyperbolic nature of the system, there are still many open problems left for mathematical analysis including the global existence of classical solutions in 3D. The first part of the colloquium talk will include a brief introduction to the kinetic theory and a brief history of the mathematical analysis of the model. Then, in the rest of the talk, we introduce one kind of stability mechanism of the system, the magnetic confinement of plasmas. In the 2D annulus domain, we introduce proof for the global wellposedness of the Vlasov-Maxwell system if a huge (but finite-in-time) external magnetic potential is imposed near the boundary. The external magnetic potential well that we impose remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. This is a joint-work with Robert M. Strain and Tak Kwong Wong.
