INTERNAL WAVES IN A 2D AQUARIUM

Internal waves describe perturbations of a stable-stratified fluid. In an effectively 2D aquarium $\Omega \subset \mathbb{R}^2$, internal waves can be modeled by the equation
$$
(\partial_t^2 \Delta + \partial_{x_2}^2)u(x, t) = f(x) \cos(\lambda t), \quad t \ge 0, \quad x \in \Omega
$$
with Dirichlet boundary and homogeneous initial 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.