2D internal waves in an ergodic setting

TL;DR

This paper investigates 2D internal waves in a controlled setting, demonstrating that under certain irrational dynamics, wave energy remains bounded over time, contrasting with known behaviors in periodic attractor scenarios.

Contribution

It establishes bounded energy solutions for internal waves in an ergodic setting, extending understanding of wave behavior under irrational classical dynamics.

Findings

Energy remains bounded in ergodic dynamics

Spectral measure near forcing frequency is very small

Contrasts with behavior in periodic attractor cases

Abstract

We study a model of internal waves under periodic forcing in an effectively 2-dimensional aquarium. When the underlying classical dynamics has sufficiently irrational rotation number, we prove that the solution to the internal waves equation remains bounded in energy space in time. This result is in contrast with the works by Colin de Verdi\`ere--Saint-Raymond and Dyatlov--Wang--Zworski, which gave a description of singular profiles when the underlying dynamics has periodic attractors. Our result is proved by showing the spectral measure near the forcing frequency is very small.