Survival Threshold for Collective Plasma Oscillations

The main objective of this project is to investigate questions about the final states of matter consisting of a sufficiently large number of interacting particles such as plasmas in plasma physics and condensates in quantum mechanics. The research will advance the understanding of turbulence in plasma physics and quantum mechanics, provide foundational mathematics to tackle unsolved problems in physics, and push the boundaries of current mathematical techniques. The research will contribute new techniques to the theory of partial differential equations, mathematical physics, dynamical systems, and applied mathematics. The project includes activities aimed at training graduate students and young researchers. The project will prove longstanding conjectures concerning the large time behavior of solutions to the mathematical mean field models that are used in plasma physics and quantum mechanics. The primary mathematical models under investigation include the relativistic Vlasov-Maxwell system and the Hartree equations used to model the nonlinear collective effects of infinitely many interacting particles. The research will rigorously validate nonlinear physical phenomena including plasma oscillations, phase transition, phase mixing, Landau damping, and the formation of coherent structures. The scattering theory as well as the formation of periodic structures for the Vlasov and Hartree equations near nontrivial translation-invariant equilibria will be established. The work of the project involves mathematical techniques from spectral theory, resolvent analysis, Fourier analysis, dispersive PDEs, probability, and statistical physics. 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.