(Six lectures: 10:00-11:30 am, 06/19, 06/20, 06/26, 06/27, 07/03, 07/04, 2023)

Description:

Internal waves are gravity waves in density stratified fluids. In 2D aquaria, Maas et al (1997) predicted and observed the velocity attractors of internal waves.

To reveal the mathematical mechanism for the formation of the attractors, Colin de Verdìere–Saint-Raymond (2018) introduced 0th order psedodifferential operators with Morse–Smale dynamical assumptions as microlocal models for internal waves. Many aspects of 0th order order has been studied since then: the scattering matrix (Wang 2019), the resonances (Galkowski–Zworski 2022), and the microlocal control estimates (Christianson–Wang–Wang 2023), etc.

The formation of internal wave attractor in 2D aquaria was studied by Dyatlov–Wang–Zworski (2021). The results connect internal waves with spectral theory and homeomorphism of the circle: it shows the appearance of attractors when the “chess billiard” satisfies the Morse–Smale condition. In the case when the chess billiard is ergodic, Li (2023) proved the absence of attractors.

In this mini-course, I will introduce the key ingredients in the analysis of internal wave attractors. No further background in analysis except for calculus is required.