We consider linear, time-dependent perturbations of periodic transport equations on tori of arbitrary dimension. We identify a class of perturbations that admit a large set of unstable solutions: to detect such behavior, we prove that their positive Sobolev norms grow exponentially fast in time. Moreover, we prove that this class of perturbations is generic in dimension two. 

The proof relies on a normal form procedure combined with the study of dynamical properties of Morse-Smale transport systems. 

In this talk, after a general introduction to the topic, we describe the result and ideas of the proof, focusing on the interplay between pseudodifferential theory and hyperbolic dynamics. This is a joint work with Gabriel Rivière.