Kinetic Equations in Bounded Domains

Kinetic theory concerns the dynamics of a large number of particles, such as flows of air or water, plasmas, neutron transport, and radiation transfer. In classical mechanics, such systems can be described under different scales. In the microscopic scale, Newton's laws track the position and velocity of each particle.

In the macroscopic scale, fluid mechanics and thermodynamics provide effective tools to predict the behaviors of averaged statistical properties like pressure and temperature. Kinetic theory forms a bridge between these two approaches and utilizes probabilistic tools in the position-velocity space, the so-called phase space, to obtain a mesoscopic description.

The probability density of particles present in the phase space satisfies the Boltzmann equation or the Landau equation, which are evolutionary partial differential equations. This project focuses on the kinetic equations in bounded domains, where the particles may be reflected or absorbed by the physical boundary. The purpose is to develop novel mathematical tools to characterize the multi-scale behaviors of these particle systems in applications such as medical imaging, fluid mechanics, and nuclear fusion. The project provides opportunities for research training of graduate students.

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. The aim is to study the asymptotic behavior of kinetic equations when the Knudsen number, which measures the relative distance a particle can travel between two collisions, shrinks to zero.

The focus is in justifying the validity of asymptotic approximations of kinetic equations in the presence of boundary layer effects, where the geometric correction of the boundary is non-negligible. The investigator will develop new boundary layer decomposition, regularization, and reflection extension techniques.

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.