March 3, 2025

About the speaker

Dr. Fangchi Yan is a postdoctoral researcher in the Department of Mathematics at Virginia Tech. He received his PhD from the University of Notre Dame under the supervision of Professor Alex Himonas. His research focuses on the well- posedness of nonlinear partial differential equations, particularly the Korteweg–de Vries (KdV) and nonlinear Schrödinger (NLS) equations. Utilizing techniques from harmonic analysis and general analysis, he investigates both initial value problems and initial-boundary value problems.

Radial solutions of initial-boundary value problems of nonlinear Schrödinger equations in R^n

This talk presents recent work, in collaboration with Professor Shu-Ming Sun, on the well-posedness of initial-boundary value problems (ibvps) for the nonlinear Schrödinger (NLS) equation in radially symmetric domains of R . We consider three types of domains: a ball centered at the origin, the exterior of a ball, and an $n$- dimensional annulus. A key challenge in analyzing these problems lies in the proper characterization of boundary data in appropriate Sobolev spaces. We establish optimal regularity conditions required for well-posedness in each of these domains and highlight the differences in boundary regularity requirements between them. Additionally, we discuss the role of Strichartz estimates and their limitations in bounded domains. These results provide the first known well- posedness results for ibvps of NLS equations in higher-dimensional bounded regions.