Long-time behavior of resonant time-dependent perturbations of periodic transport equations on $\mathbb{R}$
Maria Teresa Rotolo
TL;DR
The paper analyzes the long-time behavior of a linear, time-dependent transport equation on the 1D torus under resonant, skew-adjoint perturbations. It introduces the resonant average $\langle V \rangle_m$ and classifies perturbations by the zeros of this average, proving a dichotomy: resonantly stable perturbations yield uniform Sobolev bounds on long times and resonantly unstable perturbations induce exponential Sobolev growth; the instability result relies on constructing an escape function for the Hamiltonian flow of $h(x,\xi)=\xi\langle V\rangle_m(x)$ and applying a positive commutator argument in microlocal analysis. The stability analysis uses a resonant normal form and a sequence of unitary conjugations to reduce the dynamics to a constant-coefficient transport up to arbitrarily small remainders, enabling energy estimates on time scales $O(\varepsilon^{-(N+1)})$. The results hold generically (open and dense) in the resonant class, complementing higher-dimensional instability results and non-resonant KAM-type reducibility, and they integrate pseudodifferential techniques with dynamical systems insights to describe energy transfer mechanisms in this linear setting.
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.