[Paper Review] A reducibility result for a class of linear wave equations on $\mathbb{T}^d$

Riccardo Montalto|arXiv (Cornell University)|Feb 22, 2017

Quantum chaos and dynamical systems24 references4 citations

TL;DR

This paper establishes the first reducibility result for linear wave equations with unbounded perturbations on the $\mathbb{T}^d$ torus, proving that for small $\varepsilon > 0$ and frequency $\omega$ in a large measure Borel set, the equation $\partial_{tt}v - \Delta v + \varepsilon \mathcal{P}(\omega t)[v] = 0$ can be transformed into a constant-coefficient system via a quasi-periodic, symplectic transformation. The key contribution is the reduction of a class of linear wave equations with time-dependent, finite-rank and second-order perturbations to a block-diagonal form with constant coefficients, enabling long-time stability analysis.

ABSTRACT

We prove a reducibility result for a class of quasi-periodically forced linear wave equations on the $d$-dimensional torus $\mathbb{T}^d$ of the form $$ \partial_{tt} v - Δv + \varepsilon {\cal P}(ωt)[v] = 0 $$ where the perturbation ${\cal P}(ωt)$ is a second order operator of the form ${\cal P}(ωt) = - a(ωt) Δ- {\cal R}(ωt)$, the frequency $ω\in {\cal R}^ν$ is in some Borel set of large Lebesgue measure, the function $a : \mathbb{T}^ν o {\cal R}$ (independent of the space variable) is sufficiently smooth and ${\cal R}(ωt)$ is a time-dependent finite rank operator. This is the first reducibility result for linear wave equations with unbounded perturbations on the higher dimensional torus $\mathbb{T}^d$. As a corollary, we get that the linearized Kirchhoff equation at a smooth and sufficiently small quasi-periodic function is reducible.

Motivation & Objective

• To prove reducibility of a class of linear quasi-periodically forced wave equations on the $d$-dimensional torus $\mathbb{T}^d$ with unbounded perturbations.
• To establish that the linearized Kirchhoff equation at a small, smooth, quasi-periodic solution is reducible.
• To develop a KAM-type iterative procedure for reducing a wave equation with time-dependent, finite-rank and second-order perturbations to constant coefficients.
• To construct a symplectic, quasi-periodic transformation that removes time dependence up to a remainder decaying in the spectral parameter $|D|^{-M}$.

Proposed method

• The method employs a regularization procedure for the vector field $\mathcal{L}(\varphi)$, including symplectic symmetrization of the highest-order term and quasi-periodic reparametrization of time.
• A block-decay norm is used to control the growth of Fourier coefficients of linear operators in the frequency and spatial variables.
• The approach uses a Hamiltonian formalism in complex coordinates, with the system written as a first-order evolution equation governed by a $\varphi$-dependent Hamiltonian.
• The core technique involves iterative block-diagonalization via conjugation, reducing the operator to constant coefficients up to a remainder of order $|D|^{-M}$.
• Measure estimates are derived using a non-resonant set $\Omega_\varepsilon$ with asymptotically full Lebesgue measure, ensuring the existence of suitable frequencies $\omega$.
• The proof relies on spectral properties of the Laplacian and estimates on the set $\sigma_0(\sqrt{-\Delta})$, ensuring separation of eigenvalues and invertibility of operators in finite-dimensional subspaces.

Experimental results

Research questions

• RQ1 Can a linear wave equation with unbounded, time-dependent perturbations on $\mathbb{T}^d$ be reduced to a constant-coefficient system via a quasi-periodic transformation for a positive measure set of frequencies?
• RQ2 Is the linearized Kirchhoff equation at a small, smooth, quasi-periodic solution reducible, implying long-time stability of solutions?
• RQ3 What conditions on the perturbation structure (e.g., finite-rank, second-order) allow for KAM-type reducibility in higher dimensions?
• RQ4 How can one control the growth of Fourier coefficients of time-dependent operators in the presence of unbounded perturbations?
• RQ5 What is the measure-theoretic size of the set of frequencies $\omega$ for which reducibility holds, and how does it relate to small divisors?

Key findings

• The equation $\partial_{tt}v - \Delta v + \varepsilon \mathcal{P}(\omega t)[v] = 0$ is reducible for $\varepsilon$ small enough and $\omega$ in a Borel set $\Omega_\varepsilon \subset \Omega$ with asymptotically full Lebesgue measure.
• The perturbation $\mathcal{P}(\omega t)$ is of the form $-a(\omega t)\Delta - \mathcal{R}(\omega t)$, where $a \in \mathcal{C}^q(\mathbb{T}^\nu, \mathbb{R})$ and $\mathcal{R}(\omega t)$ is a finite-rank, symmetric operator with zero-average spatial components.
• The reducibility is achieved via a quasi-periodic, symplectic transformation that conjugates the system into a block-diagonal form with constant coefficients up to a remainder of order $|D|^{-M}$.
• The solution norm $\|(v(t,\cdot), \psi(t,\cdot))\|_{H^{s+1/2} \times H^{s-1/2}}$ is uniformly bounded in time for $s \in [1/2, \mathfrak{S}_q]$, implying long-time stability.
• The measure of the set $\Omega_\varepsilon$ satisfies $\text{meas}(\Omega \setminus \Omega_\varepsilon) \lesssim \varepsilon^a$ for some $0 < a < 1$, showing that the exceptional set shrinks as $\varepsilon \to 0$.
• The linearized Kirchhoff equation at a smooth, small, quasi-periodic function is reducible, confirming the stability of such solutions under small perturbations.