Asymptotic Problems on Kinetic and Dispersive Equations

About This Grant

This project seeks to advance the mathematical analysis of plasma physics and rotating fluids. Plasmas are ubiquitous in astronomy and in geophysical fluids. They are also central to many challenges and opportunities facing the modern world, from neon lamps to fusion reactors. The many scales involved, and the very unstable nature of plasmas represent formidable challenges and require many different levels of description whose precise relationship (when to use one instead of the other, and which to believe if the results disagree) is still poorly understood. Since the Earth spins, all questions involving geofluids, such as weather predictions, necessitate delicate analysis of rotating fluids. One of the throughlines of this project is that, in some special cases, the more complicated settings may actually be better behaved or more stable than one would a priori expect; for example, a fluid in rotation may be more predictable than a fluid (almost) quiescent. Isolating such favorable situations in rotating fluids and plasma physics, and understanding how and why predictions are easier then, is one of the main goals of the project. Some of the problems addressed relate to the interaction between a charged, hot gas and a solid wall bounding it; how, in the absence of boundary, a hot plasma may be stabilized under a reversible phenomenon called ``Landau damping'' or whether a faster spinning Earth would have a simpler weather system. The project provides research training opportunities for graduate students and postdoctoral scholars. This project studies partial differential equations inspired by physics. The first part of the project concerns the description of the asymptotic behavior of solutions to non-collisional kinetic equations. The Principal Investigator (PI) studies the stability of homogeneous equilibria for the Vlasov-Poisson system, starting with the case of fat-tail equilibria, as well as the stability of vacuum for the same system in the presence of boundaries. Extensions to more involved models (e.g. Vlasov-Maxwell), will also be considered. The second part of the project addresses problems related to quasilinear dispersive equations. The PI studies the derivation of the Euler-Poisson system for ions from the two-fluid model in the case of confined domain and also investigates the stability of a constant background at rest for the compressible Euler equations in a rotating environment. These problems are addressed using tools from finite dimensional Hamiltonian dynamical systems, harmonic analysis, atomic spaces, functional analysis and more standard partial differential equations tools like the maximum principle, energy estimates, bilinear estimates and precise dispersion analysis. 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.