The HADES seminar on Tuesday, October 22th, will be at 2:00pm in Room 740.
Speaker: Zhenhao Li
Abstract: The internal waves equation describes perturbations of a stable-stratified fluid. In an effectively 2D aquarium $\Omega \subset \mathbb{R}^2$, the equation is given by
$ (\partial_t^2 \Delta + \partial_{x_2}^2)u(x, t) = f(x) \cos(\lambda t), \quad t \ge 0, \quad x \in \Omega$
with Dirichlet boundary conditions. The behavior of the equation is intimately related to the underlying classical dynamics, and Dyatlov-Wang-Zworski proved that for $\Omega$ with smooth boundary, strong singularities form along the periodic trajectories of the underlying dynamics. Such phenomenon was first experimentally observed in 1997 by Maas-Lam in an aquarium with corners. We will discuss some recent work proving that corners contribute additional mild singularities that propagate according to the dynamics, matching the experimental observations.
-smoothing estimates for the wave equation with harmonic potential. For the proof, we linearize an FIO parametrix, which yields Klein-Gordon propagation with variable mass parameter. We obtain decoupling and square function estimates depending on the mass parameter, which yields local smoothing estimates with sharp loss of derivatives. The obtained range is sharp in 1D, and partial results are obtained in higher dimensions.