# Microlocal analysis of internal waves in 2D aquaria

## Location

For a bounded smooth planar domain Ω, we study the forced evolution problem for the 4th order PDE

(∂*t*2 Δ+∂*x*22)*u*(*t*,*x*)=*f*(*x*)cos (λ*t*), *t*≥ 0, *x*∈Ω

with homogeneous initial conditions and Dirichlet boundary condition on ∂Ω. This is motivated by concentration of fluid velocity on attractors for stratified fluids in effectively 2-dimensional aquaria, first observed experimentally in 1997.

The behavior of solutions to the problem above is intimately tied to the *chess billiard* map on the boundary ∂Ω, which depends on the forcing frequency λ. Under the natural assumption that the chess billiard *b* has the Morse–Smale property, we show that as *t*→∞ the singular part of the solution *u* concentrates on the attractive cycle of *b*. The proof combines various tools from microlocal analysis, scattering theory, and hyperbolic dynamics. Joint work with Jian Wang and Maciej Zworski.