Vlasov-type models deal with continua of particles where the electric charges dominate the collisions, so that the collisions are ignored. They occur in physical plasmas, including astrophysical plasmas and fusion reactors. There are many examples of astrophysical plasmas of this type, such as the solar wind. When a fusion reactor is very hot, the relevant times scales are so short that collisions can be ignored. Vlasov theory also models systems where the dominant force is gravity, such as clusters of stars or galaxies.
Problem 1: Stability of Equilibria. There are a lot of steady states in the Vlasov theory and their stability has been a major focus of interest. Stability requires both very long-time computations as well as theoretical analysis in order to make progress. A fundamental open problem in Vlasov theory is the question of whether there are any periodic BGK modes that are stable under arbitrary perturbations of the same period. Another open problem is to find unstable galaxy configurations in stellar dynamics. Collaborations in these problems between numerical and theoretical researchers are particularly crucial.
Problem 2: Boundary Effects. Boundary effects play an major role physical plasmas. Many effects can occur at boundaries, for instance, chemical reactions with the surface material. An important focus of our program will be to design numerical schemes to capture possible singularity formation at boundaries and the propagation of singularities in a general 3D domain.
Problem 3: Landau Damping. The concept of Landau damping has been a major source of controversy in the physics community for decades. In principle it could be checked numerically except it requires very long-time computations. The need for great accuracy is a challenge to the numerical community. Recent work has established Landau damping for the case of analytic data for the Vlasov-Poisson system. A central question is to investigate a possible Landau damping effect in the presence of a magnetic field.
Problem 4: Well-posedness. A major open problem is whether the three- dimensional Vlasov-Maxwell system is globally well-posed as a Cauchy problem. All that is known is, on the one hand, global existence but not uniqueness of weak solutions and, on the other, well-posedness and regularity of solutions assuming either some symmetry or almost neutrality. Computation of asymmetric solutions should shed light on the general problem.