Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

One dimensional energy cascades in a fractional quasilinear NLS

Alberto Maspero, Federico Murgante

TL;DR

This work addresses energy transfer and Sobolev-norm inflation for a 1D quasilinear dispersive PDE with fractional dispersion, modeled by $\partial_t u = -\mathrm{i} |D|^\alpha u + |u|^2 u_x$, $\alpha\in(0,1)$. A novel mechanism is developed via a paradifferential normal form that yields a transport operator with non-constant coefficients, and a weak-$\Lambda$ normal form isolates two tangential modes to produce an effective equation; a paradifferential Mourre argument then provides a positive commutator that enforces energy cascade to high frequencies, yielding exponential growth of high-frequency energy for long-time controlled, two-mode data. The main result proves the existence of smooth initial data with arbitrarily small $H^s$-norm (for large $s$) that explode to arbitrarily large $H^s$-norm at a later time, while remaining small in low norms and conserving mass. This work supplies a robust framework for energy transfer in quasilinear dispersive PDEs on compact manifolds and offers a foundational paradigm for studying turbulence-like dynamics in fluids through rigorous, constructive instability mechanisms.

Abstract

We consider the problem of transfer of energy to high frequencies in a quasilinear Schrödinger equation with sublinear dispersion, on the one dimensional torus. We exhibit initial data undergoing finite but arbitrary large Sobolev norm explosion: their initial norm is arbitrary small in Sobolev spaces of high regularity, but at a later time becomes arbitrary large. We develop a novel mechanism producing instability, which is based on extracting, via paradifferential normal forms, an effective equation driving the dynamics whose leading term is a non-trivial transport operator with non-constant coefficients. We prove that such operator is responsible for energy cascades via a positive commutator estimate inspired by Mourre's commutator theory.

One dimensional energy cascades in a fractional quasilinear NLS

TL;DR

This work addresses energy transfer and Sobolev-norm inflation for a 1D quasilinear dispersive PDE with fractional dispersion, modeled by , . A novel mechanism is developed via a paradifferential normal form that yields a transport operator with non-constant coefficients, and a weak- normal form isolates two tangential modes to produce an effective equation; a paradifferential Mourre argument then provides a positive commutator that enforces energy cascade to high frequencies, yielding exponential growth of high-frequency energy for long-time controlled, two-mode data. The main result proves the existence of smooth initial data with arbitrarily small -norm (for large ) that explode to arbitrarily large -norm at a later time, while remaining small in low norms and conserving mass. This work supplies a robust framework for energy transfer in quasilinear dispersive PDEs on compact manifolds and offers a foundational paradigm for studying turbulence-like dynamics in fluids through rigorous, constructive instability mechanisms.

Abstract

We consider the problem of transfer of energy to high frequencies in a quasilinear Schrödinger equation with sublinear dispersion, on the one dimensional torus. We exhibit initial data undergoing finite but arbitrary large Sobolev norm explosion: their initial norm is arbitrary small in Sobolev spaces of high regularity, but at a later time becomes arbitrary large. We develop a novel mechanism producing instability, which is based on extracting, via paradifferential normal forms, an effective equation driving the dynamics whose leading term is a non-trivial transport operator with non-constant coefficients. We prove that such operator is responsible for energy cascades via a positive commutator estimate inspired by Mourre's commutator theory.
Paper Structure (19 sections, 36 theorems, 352 equations)

This paper contains 19 sections, 36 theorems, 352 equations.

Table of Contents

  1. Introduction
  2. Scheme of the proof
  3. Functional setting
  4. Paradifferential calculus
  5. Admissible transformations
  6. Analysis of weak resonances
  7. Paradifferential normal form
  8. Block diagonalization
  9. Reduction of the highest order
  10. The weak $\Lambda$-normal form
  11. Identification and proof of Theorem \ref{['thm:nf']}
  12. The effective equation
  13. Instability via paradifferential Mourre theory
  14. The Mourre operator
  15. Growth of Sobolev norms
  16. ...and 4 more sections

Key Result

Theorem 1.1

There exists $s_0> \frac{3}{2}$ such that given any $s > 3 s_0$, $0< \delta \leq 1$ and $K \geq 1$, there exists a solution $u(t) \in H^s({\mathbb T}, {\mathbb C})$ of eq:main and a time $T>0$ such that Moreover

Theorems & Definitions (82)

  • Theorem 1.1
  • Definition 2.1: Symbols
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5: Classes of $m$-operators
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Lemma 2.9
  • ...and 72 more