Analysis of Nonlinear Partial Differential Equations including Boundary Value problems in Kinetic Theory, Free Boundary Fluid Dynamics, and the Einstein-Boltzmann system

This project aims to advance the mathematical analysis of non-linear partial differential equations used in a wide range of applications. The first part of this project involves the study of fluid dynamics problems with free boundaries, aimed at enhancing the understanding of water waves, tsunamis, and hurricanes. The second part of this project investigates the dynamics of gasses and plasmas under physical kinetic boundary conditions, which is expected to provide insight into important physical phenomena such as the solar wind, galactic nebulae, and the Van Allen radiation belt.

The third part of this project explores the physical interactions between relativistic kinetic theory and gravitational models bringing potential to increase knowledge in astrophysics, such as in systems of galaxies, supernova explosions, models of the early universe, and the study of hot gases and plasmas. This project will support the education and training of postdoctoral researchers, graduate students, and undergraduate students through research mentoring and seminars.

It aims to further the goal of developing a diverse and globally competitive STEM workforce and to improve STEM education at the collegiate level.

This research will focus on improving the local-in-time well-posedness for large initial data and the global-in-time well-posedness near equilibrium for various fundamental non-linear partial differential equations. It involves developing new methods for analyzing several different physical models. One part of this work is to study fluid dynamics problems with free boundaries, such as the study of the Muskat bubble problem in 2D and 3D.

Another part of this work examines problems related to the non-cutoff Boltzmann equation and the Landau equation from kinetic theory with the physical kinetic boundary conditions. The third part studies the relativistic Boltzmann equation and the Einstein-Boltzmann system. These developments are expected to benefit both mathematical and physical research in the future.

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.