Partial differential equation methods in kinetic theory and their applications

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GuoDMS-1611695 Many important scientific applications require understanding dynamics of dilute gases of particles, e.g., re-entry of a space shuttle in the air, flows passing an airplane, or a plasma inside a tokamak device for nuclear fusion. Kinetic theory describes the dynamics of a dilute gas or plasma. Motivated by problems with various scientific applications in the real world, the investigator aims to understand the following basic and important phenomena and questions from a mathematical standpoint: the interaction of a dilute gas with its surrounding walls, interaction of a dilute plasma with its surrounding walls, interactions of neutron beams with surrounding walls, characterization of the long-time behavior of a collisionless plasma, dynamics of contact lines in fluids (e.g., the coffee edge on the side of a coffee cup), and the validity of boundary layer theory for flows with high Reynolds number. Graduate students are involved in the project. Kinetic theory is at the center of multi-scale modeling, which connects the microscopic particle models with macroscopic fluid models. Even though boundary effects play an important role in kinetic theory, they have not yet been studied deeply due to their challenging characteristic nature. As one of his main goals, the investigator and his collaborators continue to study such boundary value problems. More specifically, the investigator further studies regularity of Boltzmann solutions in convex domains in the presence of external forces, well-posedness for the Landau equation in a bounded domain, a general theory for geometric correction to the boundary layer theory in kinetic theory, flows passing an obstacle as a diffusive limit of Boltzmann theory, time decay around stable BGK waves in a collisionless Vlasov theory for a plasma, and the well-posedness of the recent Ren-E model for contact line dynamics. He also seeks to verify the validity of steady Prandtl boundary layer expansions for flows with high Reynolds number. Graduate students are involved in the project.