Microlocal analysis of forced waves

TL;DR

This paper applies microlocal analysis and pseudodifferential operator estimates to understand the long-term behavior of solutions to a forced wave equation with a self-adjoint operator, inspired by models of internal waves.

Contribution

It introduces a microlocal framework using radial estimates for pseudodifferential operators to analyze forced wave evolution under hyperbolic dynamical assumptions.

Findings

Describes long-time evolution of solutions to forced wave equations.

Connects microlocal analysis with models of internal stratified fluid waves.

Provides tools for analyzing wave propagation in complex media.

Abstract

We use radial estimates for pseudodifferential operators to describe long time evolution of solutions to $ i u_t - P u = f $ where $ P $ is a self-adjoint 0th order pseudodifferential operator satisfying hyperbolic dynamical assumptions and where $ f $ is smooth. This is motivated by recent results of Colin de Verdi\`ere and Saint-Raymond [arXiv:1801.05582] concerning a microlocal model of internal waves in stratified fluids.