Analysis and Applied Mathematics Seminar
Anudeep Kumar Arora
UIC
Singularities and global solutions in the Schrödinger-Hartree equation
Abstract: In 1924, Louis De Broglie proposed in his PhD thesis the theory of matter waves. On January 26, 1926, Erwin Schrödinger published his work in which he constructed the wave equation for de Broglie's matter waves, now known as the Schrödinger equation. Schrödinger equations are partial differential equations of dispersive category.
In this work, we study the generalized Hartree (gHartree) equation, which is a nonlinear Schrödinger type equation with a nonlocal nonlinearity, of a convolution type. We first, in the energy-subcritical regime, classify the behavior of finite energy solutions under the mass-energy assumption, identifying the sharp threshold for global (scattering) versus finite time (blow-up) solutions. We exhibit two methods of obtaining scattering: one via Kenig-Merle concentration - compactness and another one is using Dodson-Murphy approach via Morawetz estimate and Tao's scattering criteria. Next, we are interested in the phenomenon of wave collapse (blow-up), for which, we investigate stable singularity formations in the mass-critical gHartree equation and rigorously prove a stable blow-up formation in dimension 3.
Monday November 16, 2020 at 4:00 PM in Zoom