Noninear PDE models in kinetic theory

Kinetic theory provides a powerful mathematical framework for modeling systems involving large number of interacting particles or agents. Rather than tracking individual particles, kinetic models describe the evolution of particle densities over time using nonlinear partial differential equations (PDEs). These models are central to understanding the behavior of gases and plasmas, which play key roles in both natural phenomena and modern technologies. This project focuses on two classical kinetic models: the Boltzmann and Landau equations, and their variants. The research will advance mathematical understanding in three main directions: First, it will investigate the regularity and potential breakdown of solutions for the inhomogeneous non-cutoff Boltzmann equation via improved continuation criteria and ruling out specific breakdown mechanisms. Second, it will extend existing results on the existence and regularity of solutions to the classical Landau model to its relativistic counterpart, motivated by physical applications and new mathematical challenges posed by relativistic collisions. Third, it will examine delicate regularity issues that arise when the Boltzmann or Landau equations are coupled with the Maxwell system via a mean-field Vlasov term. 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.