Asymptotic Problems in Kinetic Theory

This project considers a set of problems in kinetic theory which deals with the motion of a vast number of particles, such as air or water flows, neutrons in the nuclear reactor, and even the collection of stars when modelling a galaxy. These immense systems can be analyzed on two scales: the microscopic scale, where first principles such as Newton's laws or the Schrödinger equation is employed to track the position and velocity of each individual particle, or the macroscopic scale, where statistical laws such as fluid mechanics and thermodynamics are utilized to predict the collective behavior of averaged properties like pressure and temperature. Kinetic theory serves as a bridge between these two approaches, utilizing probabilistic tools within the position-velocity space, also known as the phase space, to establish a mesoscopic description. The probability density of particles present in the phase space satisfies the Boltzmann-type or the Landau-type equations, which are evolutionary nonlinear partial differential equations. The broader impacts of the project consist of mentoring both graduate and undergraduate students, courses development, and running a summer school. This project focuses on the asymptotic problems in kinetic equations, emphasizing the rigorous analysis to connect the aforementioned three scales of descriptions, and to quantitatively determine the scope of their applicability. The goal is to develop novel mathematical tools to characterize the multi-scale behaviors of these particle systems in applications such as medical imaging, gas dynamics, and nuclear fusion. Specifically, this project concentrates on the theory of hydrodynamic limits, a key step to tackle the so-called "Hilbert's Sixth Problem" to treat physics in an axiomatic manner. Novel techniques will be developed to study the asymptotic behavior of kinetic equations when the Knudsen number or Strouhal number, which measures the spatial or temporal scaling between particles collisions, shrinks to zero. The investigator will develop new nonlinear energy method, regularization, and remainder splitting techniques to tackle the non-classical boundary layer effects and the “ghost” effects. 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.