MIT PDE/Analysis Seminar
Tuesday, September 19, 2023 at 4:00pm to 5:00pm
MIT, 2-136 182 Memorial Drive, Cambridge, MA
Title: Advances in Spherical Capillary Water Waves System
Abstract: The speaker aims to report some advances in the study of irrotational oscillation of a water droplet under zero gravity. The governing physical laws, resembling the well-studied capillary water waves equation, are converted to a quasilinear dispersive para-differential system defined on the 2-sphere. Regarding this conversion, a coordinate-independent, global para-differential calculus defined on compact Lie groups and homogeneous spaces is developed as a toolbox. After discussing Cauchy theory under this novel formalism, the speaker will propose several unsolved problems concerning existence of periodic solutions, normal form reduction and generic lifespan estimates. It is pointed out that all of these problems are closely related to certain Diophantine equations.