Harmonic Analysis and Differential Equations Student Seminar: On the Cauchy problem for degenerate dispersive equations

Seminar | January 21 | 3:40-5 p.m. | 740 Evans Hall

 Sung-Jin Oh, Berkeley

 Department of Mathematics

In plasma physics or fluid dynamics, one sometimes encounters a degenerate dispersive equation, i.e., a nonlinear dispersive equation whose dispersion relation is degenerate (i.e., vanishes at some points). A satisfactory understanding of the Cauchy problem for such equations is still missing, largely due to the appearance of challenging (and interesting!) phenomena from degenerate dispersion, such as the strong focusing of bicharacteristics near the degeneracy.

The purpose of this talk is to provide an introduction to this topic, by focusing on simple examples. In the first part of my talk, I'll work with simple linear models, namely linear degenerate Schrödinger equations on the line, to demonstrate some key phenomena related to degenerate dispersion. Then in the second part of my talk, I'll describe some nonlinear illposedness results for a quasilinear degenerate Schrödinger equation on the line, whose proof builds off of the understanding of the linear models. This talk is based on joint work with In-Jee Jeong.