The APDE seminar on Monday, 4/5, will be given by Maciej Zworski online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis (gmoschidis@berkeley.edu) or Federico Pasqualotto (fpasqualotto@berkeley.edu).

Title: Internal waves and homeomorphism of the circle.

Abstract: The connection between the formation of internal waves in fluids and
the dynamics of homeomorphisms of the circle was investigated by
oceanographers in the 90s and resulted in novel experimental
observations (Maas et al, 1997). The specific homeomorphism is given
by a chess billiard” and has been considered by many authors (John
1941, Arnold 1957, Ralston 1973, … , Lenci et al 2021). The relation
between the nonlinear dynamics of this homeomorphism and linearized
internal waves provides a striking example of classical/quantum
correspondence (in a classical and surprising setting of fluids!) and,
using a model of tori and of zeroth order pseudodifferential
operators, it has been a subject of recent research, first by Colin de
Verdière-Saint Raymond 2020 and then by Dyatlov, Galkowski, Wang and
the speaker. In these works, many facets of the relationship between
hyperbolic sources and sinks for the classical dynamics and internal
waves in fluids were explained. I will present some of these results
as well as some numerical discoveries (including those of
Almonacid-Nigam 2020). I will also describe various open problems.

The APDE seminar on Monday, 3/29, will be given by Larry Guth online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis (gmoschidis@berkeley.edu) or Federico Pasqualotto (fpasqualotto@berkeley.edu).

Title: Local smoothing for the wave equation.

Abstract: The local smoothing problem asks about how much solutions to the wave equation can focus. It was formulated by Chris Sogge in the early 90s. Hong Wang, Ruixiang Zhang, and I recently proved the conjecture in two dimensions.
In this talk, we will build up some intuition about waves to motivate the conjecture, and then discuss some of the obstacles and some ideas from the proof.

The APDE seminar on Monday, 3/8, will be given by John Anderson online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis (gmoschidis@berkeley.edu) or Federico Pasqualotto (fpasqualotto@berkeley.edu).

Title: Stability results for anisotropic systems of wave equations

Abstract: In this talk, I will describe a global stability result for a nonlinear anisotropic system of wave equations. This is motivated by studying phenomena involving characteristics with multiple sheets. For the proof, I will describe a strategy for controlling the solution based on bilinear energy estimates. Through a duality argument, this will allow us to prove decay in physical space using decay estimates for the homogeneous wave equation as a black box. The final proof will also require us to exploit a certain null condition that is present when the anisotropic system of wave equations satisfies a structural property involving the light cones of the equations.