Control estimates for 0th order pseudodifferential operators

Abstract

We introduce the control conditions for 0th order pseudodifferential operators $\mathbf{P}$ whose real parts satisfy the Morse--Smale dynamical condition. We obtain microlocal control estimates under the control conditions. As a result, we show that there are no singular profiles in the solution to the evolution equation $(i{\partial }_t-\mathbf{P})u=f$ when $\mathbf{P}$ has a damping term that satisfies the control condition and $f\in C^{\infty }$. This is motivated by the study of a microlocal model for the damped internal waves.

20 pages, 3 figures. To appear in Int. Math. Res. Not. IMRN