Long-time behavior of resonant time-dependent perturbations of periodic transport equations on $\mathbb{R}$

TL;DR

This paper analyzes the long-term behavior of solutions to resonant, time-dependent perturbations of periodic transport equations on the real line, revealing conditions for exponential instability or stability over long times.

Contribution

It introduces a novel combination of pseudodifferential and dynamical systems techniques to characterize stability and instability in resonant regimes of transport equations.

Findings

Existence of solutions with exponentially growing Sobolev norms.

Conditions under which solutions remain stable for long times.

Explicit construction of an escape function using microlocal analysis.

Abstract

We consider linear, time-dependent and skew-adjoint perturbations of periodic transport equations on the one-dimensional torus. We describe the long-time behavior of solutions for all non-degenerate perturbations in resonant regime, proving that either there exist solutions whose Sobolev norms explode exponentially fast, provoking energy transfer phenomena, or all solutions remain stable for arbitrarily long time scales. The proof combines pseudodifferential tools with dynamical systems results: we perform a resonant normal form procedure to reduce our analysis to the classical dynamics for the resonant equation. The main difficulty lies in the proof of the instability result, for which we explicitly construct an escape function associated to the dynamics. This is obtained by means of a positive commutator estimate on the operator associated with the escape function, exploiting microlocal analysis.