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Sobolev instability for perturbations of periodic transport equations

Gabriel Rivière, Maria Teresa Rotolo

TL;DR

The paper develops a microlocal framework to prove exponential growth of positive Sobolev norms for a broad class of linear, time-dependent transport equations with small, non-decaying perturbations on compact manifolds. Central to the argument is constructing an escape function for the lifted Morse–Smale dynamics on the cotangent bundle, enabling a positive commutator estimate that yields sharp Sobolev-instability results. The authors then apply a resonant normal form to periodic transport on the torus, reducing to a resonant-average problem governed by a Morse–Smale vector field, and show that generic potentials in dimension two induce instability; this uses Peixoto’s genericity results. The work connects dynamical systems, microlocal analysis, and partial differential equations to characterize energy-transfer mechanisms that drive high-frequency growth, with potential implications for nonlinear PDE dynamics. Overall, the paper provides a rigorous mechanism for exponential Sobolev-growth in resonant transport, clarifies the role of Morse–Smale resonances, and establishes genericity results in low dimensions via a resonant normal form construction.

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.

Sobolev instability for perturbations of periodic transport equations

TL;DR

The paper develops a microlocal framework to prove exponential growth of positive Sobolev norms for a broad class of linear, time-dependent transport equations with small, non-decaying perturbations on compact manifolds. Central to the argument is constructing an escape function for the lifted Morse–Smale dynamics on the cotangent bundle, enabling a positive commutator estimate that yields sharp Sobolev-instability results. The authors then apply a resonant normal form to periodic transport on the torus, reducing to a resonant-average problem governed by a Morse–Smale vector field, and show that generic potentials in dimension two induce instability; this uses Peixoto’s genericity results. The work connects dynamical systems, microlocal analysis, and partial differential equations to characterize energy-transfer mechanisms that drive high-frequency growth, with potential implications for nonlinear PDE dynamics. Overall, the paper provides a rigorous mechanism for exponential Sobolev-growth in resonant transport, clarifies the role of Morse–Smale resonances, and establishes genericity results in low dimensions via a resonant normal form construction.

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.
Paper Structure (34 sections, 25 theorems, 183 equations)

This paper contains 34 sections, 25 theorems, 183 equations.

Table of Contents

  1. Introduction and main results
  2. Morse-Smale transport equations on manifolds
  3. Comparison with the Schrödinger case
  4. Organization of the article.
  5. Acknowledgements
  6. Review of dynamical setting
  7. Morse-Smale flows
  8. Energy function.
  9. Morse-Smale vector fields on surfaces.
  10. Symplectic lift on the cotangent bundle
  11. Dynamics of Morse-Smale symplectic lifts
  12. Construction of the escape function
  13. Construction of the order function
  14. Notations.
  15. Item $(i)$.
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $n=2$, $\omega = 1$ and $\nu\in{\mathbb N}^2\setminus\{0\}$. Then there exists an open and dense subset $\mathcal{O}_\nu$ of The set is endowed with its standard Fréchet topology.$C^{\infty}({\mathbb T}^3;{\mathbb R}^2)$ such that, for every $V\in \mathcal{O}_\nu$, the following holds. There exi where $u_{\varepsilon}(t)$ is the solution to equation transport.torus with initial datum $u_{\vare

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 2.1
  • Definition 2.2: Morse-Smale vector fields
  • Lemma 2.3
  • Lemma 2.4: Meyer Energy function
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7: Symplectic lift
  • ...and 51 more