Description
- Abstract:
- This dissertation is devoted to the study of some topics in the theory of kinetic equations and nonlinear dispersive systems. It is divided into two parts. In the first part, we consider the Vlasov-Poisson-Landau system, a classical model for a dilute collisional plasma interacting through Coulombic collisions and with its self-consistent electrostatic field. We establish global stability and well-posedness near the Maxwellian equilibrium state with decay in time and some regularity results for small initial perturbations, in any general bounded domain (including a torus as in a tokamak device), in the presence of specular-reflection boundary condition. We provide a new improved $L^{2}\rightarrow L^{\infty}$ framework: $L^{2}$ energy estimate combines only with $S^{p}$ estimate for ultra-parabolic equations. The second part concerns the long-time behavior of a Wave-Klein-Gordon coupled system in 3D, which is a simplified model for the global nonlinear stability of the Minkowski space-time for self-gravitating massive fields. We study the global behavior of solutions to such systems, and prove modified wave operators for small and smooth data with mild decay at infinity. The key novelty comes from a crucial observation that the asymptotic dynamics is dictated by the resonant interactions. As a consequence, our main results include the derivation of a resonant system with good error bounds, and a detailed description of the asymptotic dynamics of such quasilinear evolution system of hyperbolic and dispersive type.
- Notes:
- Thesis (Ph. D.)--Brown University, 2021
Citation
Ouyang, Zhimeng,
"Some Topics in Kinetic and Dispersive PDEs"
(2021).
Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://repository.library.brown.edu/studio/item/bdr:j8rxm7dc/
