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Reducibility of Klein-Gordon equations with maximal order perturbations

Massimiliano Berti, Roberto Feola, Michela Procesi, Shulamit Terracina

Abstract

We prove that all the solutions of a quasi-periodically forced linear Klein-Gordon equation $ψ_{tt}-ψ_{xx}+\mathtt{m}ψ+Q(ωt)ψ=0 $ where $ Q(ωt) := a^{(2)}(ωt, x) \partial_{xx} + a^{(1)}(ωt, x)\partial_x + a^{(0)}(ωt, x) $ is a differential operator of order $ 2 $, parity preserving and reversible, are almost periodic in time and uniformly bounded for all times, provided that the coefficients $ a^{(2) }, a^{(1) }, a^{(0) } $ are small enough and the forcing frequency $ω\in {\mathbb R}^ν$ belongs to a Borel set of asymptotically full measure. This result is obtained by reducing the Klein-Gordon equation to a diagonal constant coefficient system with purely imaginary eigenvalues. The main difficulty is the presence in the perturbation $ Q (ωt) $ of the second order differential operator $ a^{(2)}(ωt, x)\partial_{xx} $. In suitable coordinates the Klein-Gordon equation is the composition of two backward/forward quasi-periodic in time perturbed transport equations with non-constant coefficients, up to lower order pseudo-differential remainders. A key idea is to straighten this first order pseudo-differential operator with bi-characteristics through a novel quantitative Egorov analysis.

Reducibility of Klein-Gordon equations with maximal order perturbations

Abstract

We prove that all the solutions of a quasi-periodically forced linear Klein-Gordon equation where is a differential operator of order , parity preserving and reversible, are almost periodic in time and uniformly bounded for all times, provided that the coefficients are small enough and the forcing frequency belongs to a Borel set of asymptotically full measure. This result is obtained by reducing the Klein-Gordon equation to a diagonal constant coefficient system with purely imaginary eigenvalues. The main difficulty is the presence in the perturbation of the second order differential operator . In suitable coordinates the Klein-Gordon equation is the composition of two backward/forward quasi-periodic in time perturbed transport equations with non-constant coefficients, up to lower order pseudo-differential remainders. A key idea is to straighten this first order pseudo-differential operator with bi-characteristics through a novel quantitative Egorov analysis.
Paper Structure (22 sections, 56 theorems, 563 equations)

This paper contains 22 sections, 56 theorems, 563 equations.

Table of Contents

  1. Introduction and main results
  2. Ideas of the proof.
  3. A quantitative Egorov result and a straightening theorem for bi-charcteristics
  4. Functional Setting
  5. $2\times 2$ block decomposition
  6. Pseudo-differential operators
  7. Algebraic properties of pseudo-differential operators.
  8. Bony-smoothing couples
  9. The torus diffeomorphism
  10. A quantitative Egorov theorem
  11. Straightening of a first order operator with bi-characteristics
  12. Reducibility of the Klein-Gordon equation
  13. Symmetrization up to smoothing remainders
  14. Symmetrization at order 1
  15. Symmetrization at lower orders
  16. ...and 7 more sections

Key Result

Theorem 1.1

(Sobolev stability). Assume oddness, revers and fix any $\mathtt{m} >0$. Let $\nu\in{\mathbb N}$ and fix ${{s}_0}>(\nu+7)/2$. There is $\bar{s}:=\bar{s}(\nu)$ and for any $s>{{s}_0}$ there exists $\mathfrak{d}_0(s) > 0$ such that, assuming the smallness condition then there exists a Borel set of frequencies $\mathcal{C}_{\infty} \subset \Lambda$ of asymptotically full measure as $\mathfrak{d}\to

Theorems & Definitions (126)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof : Proof of the stability result.
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • ...and 116 more