For a bounded smooth planar domain Ω we study the forced evolution problem for the 4th order PDE
(1) ( ∂_t^2 Δ + ∂_{x_2}^2 ) u(t,x) = f(x) cos (λ t), t ≥ 0, x ∈ Ω
with homogeneous initial conditions and Dirichlet boundary conditions 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 (1) 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.