Analysis and PDE Seminar——The Cauchy problem for Manton's Chern-Simons-Schrodinger equation
报告人：Jason Zhao (UC Berkeley)
时间：2026-05-18 16:00-17:00
地点：智华楼313
Abstract: The Chern-Simons-Schrodinger equation arises (in temporal gauge) as the Hamiltonian flow of the abelian Higgs energy on R^2, which is known to admit a rich class of critical points known as magnetic vortices. Manton (1997) conjectured that, in a low energy regime, the dynamics of these vortices under the Chern-Simons-Schrodinger flow could be modeled by a flow on the moduli space of a particular class of vortices, namely the self-dual vortices of Jaffe-Taubes (1980). As a first step towards understanding Manton's conjecture, we formulate the Cauchy problem in DeTurck gauge within the natural energy space and prove global well-posedness. The main difficulty of the problem lies in the non-linear structure of the phase space, which stems from the formal topological boundary condition |\phi|\rightarrow 1 as |x|\rightarrow +\infty. To facilitate our analysis, we develop a Littlewood-Paley theory based on the covariant heat equation in caloric gauge, which we use to perform a paradifferential decomposition of the Chern-Simons-Schrodinger equation in a geometric fashion.
