Microlocal analysis of forced waves

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.