Friday, January 21, 2022 2:00 PM
Semyon Dyatlov (MIT)

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

(∂t2 Δ+∂x22)u(t,x)=f(x)cos (λt),   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.