New prospectives and challenges in collisional kinetic theory

The mathematical description of interacting particles is a fundamental and challenging problem with broad implications across physics and engineering. While microscopic models involve large systems of ordinary differential equations, such formulations become impractical due to the prohibitively large size of the system. At larger scales, effective models take the form of kinetic equations –such as the Boltzmann and Landau equations- and at the macroscopic level, continuum equations like the Euler and Navier-Stokes equations. Importantly, these equations are connected: in certain asymptotic settings fluid equations can be rigorously derived from kinetic equations. Due to the complexity of the kinetic Boltzmann and Landau equations, recent research has been devoted to simplified kinetic models. These simplifications, while advantageous for the understanding of collision from a mathematical perspective, have limited direct applicability. With a solid mathematical foundation for these simplified models now in place, this project advances the field by studying more complex kinetic models that incorporate different physical phenomena. The objective of this research project is to gain a comprehensive mathematical understanding of several fundamental kinetic models arising in physics. The first project centers on the quantum Landau equation, with particular focus on the global well-posedness, singularity formation, and regularity of its solutions. Progress on these questions is expected to open new research directions within the field. The second project addresses the existence of regular solutions to the inhomogeneous Landau equation. This work involves novel approximation methods that could provide new insights into this well-known and challenging problem in non-linear kinetic theory. The mathematical methods and techniques developed through these studies are expected to serve as a foundation for tackling more complex models in the physics literature and to help generate specifically designed problems suitable for training junior researchers, including graduate students and postdoctoral fellows. 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.