Partial Differential Equation Methods in Kinetic Theory and Their Applications

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A plasma is a collection of dilute, fast moving charged particles. It is believed that more than 95% of the matter in the universe is in the form of a plasma (the fourth state). The main motivation of this research is to promote design of nuclear fusion devices (such as a tokamak), in which a confined plasma is accelerated at high speed to produce high energy. This project focuses on basic mathematical research related to the study of a plasma in a tokamak. Research topics include study of plasma-wall interaction, dynamic stability of plasma configurations, as well as shock waves in plasma. All of such topics are fundamental theoretically in the plasma control for a nuclear fusion device with potential applications.More specifically, the project is aimed at establishing regularity, and the validity of boundary layer expansion with geometric corrections for the Boltzmann theory in a convex domain. The key methodology is based on the principal investigator's previous work on regularity of similar boundary layer expansions for the neutron transport equations. Another direction to be pursued is well-posedness for the Vlasov-Landau equations in a bounded domain. This system of equations is fundamental in modeling a collisional plasma, and the boundary value problem models the plasma-wall interaction. Among other problems to be investigated are: The Landau damping and long-time stability for the BGK waves in a Vlasov-Poisson system; the long-time behavior of Euler-Maxwell system in the presence of a non-zero vorticity that describes the fundamental two-fluid models in a plasma; the dynamics of moving contact lines in fluids.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.