Sobolev instability for perturbations of periodic transport equations

TL;DR

This paper demonstrates that generic perturbations of periodic transport equations lead to exponential Sobolev norm divergence, revealing instability phenomena in both two-dimensional and higher-dimensional settings.

Contribution

It introduces a normal form approach to establish Sobolev instabilities for perturbed transport equations on compact manifolds, extending previous understanding of stability.

Findings

Sobolev norms diverge exponentially for generic perturbations

Higher-dimensional cases exhibit similar instability under Morse-Smale assumptions

Uses microlocal analysis and hyperbolic dynamics techniques

Abstract

We consider linear and time-dependent perturbations of periodic transport equations on the two-dimensional torus. For generic perturbations, we prove the existence of a large class of initial data whose Sobolev norms diverge exponentially fast. In higher dimensions, this remains true under a Morse-Smale assumption on the resonant part of the perturbation. In both cases, this is achieved by a normal form procedure and by studying Sobolev instabilities for time-dependent perturbations of Morse-Smale transport equations. The latter are analyzed on general compact manifolds using techniques from microlocal analysis and hyperbolic dynamics.