A Reducibility Result for a Class of Linear Wave Equations on ${\mathbb T}^d$

Abstract

We prove a reducibility result for a class of quasi-periodically forced linear wave equations on the d-dimensional torus T d of the formwhere the perturbation P(ωt) is a second order operator of the form P(ωt) = −a(ωt)∆ − R(ωt), the frequency ω ∈ R ν is in some Borel set of large Lebesgue measure, the function a : T ν → R (independent of the space variable) is sufficiently smooth and 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 T d . As a corollary, we get that the linearized Kirchhoff equation at a smooth and sufficiently small quasi-periodic function is reducible.Keywords: reducibility of linear wave equations on T d , KAM for PDEs, unbounded perturbations.

Reducibility of 1-d Schrödinger equation with unbounded time quasiperiodic perturbations. III

“…The problem of reducibility of equations of the form of (1.1) has a long history, and the main results have been obtained in [Com87, DŠ96, DLŠV02, Kuk93, BG01, LY10, Bam18, Bam17] (see [Bam17] for a more detailed history). We also mention that our result is limited to the one dimensional case, while some results on this problems in more then one dimension have been recently obtained [GP16,BGMR18,Mon17b,FGMP18]. We also recall that related techniques have been used in order to get a control on the growth of Sobolev norms in [BGMR17,Mon18].…”

Growth of Sobolev norms for time dependent periodic Schrödinger equations with sublinear dispersion

“…Note that in [1], [2], [17], [8], [9], [20] the reducibility of the linearized equations is obtained as a consequence of the KAM theorems proved for the corresponding nonlinear equations. In the case of sublinear growth of the eigenvalues, the first KAM-reducibility result is proved in [3] for the pure gravity water waves equations and the technique has been extended in [22] to deal with a class of linear wave equations on T d with smoothing quasi-periodic in time perturbations. We now state in a precise way the main results of this paper.…”

Quasi-periodic solutions for the forced Kirchhoff equation on $ \newcommand{\m}{\mu} \newcommand{\T}{\mathbb T} \T^d$

“…Under the presence of the forcing term f (t, x) the normal modes do not persist 1 , since, expanding v(t, x) = j v j (t) sin(jx), f (t, x) = j f j (t) sin(jx), all the components v j (t) are coupled. to follow the above scheme and reduce completely the linearized operator (this is done in [44]), one obtains a bound on the inverse of the linearized operator L(u) of the form L(u) −1 h s s h s+σ + u 2s+σ h s0+σ for s ≥ s 0 , where σ is a constant depending only on ν and d. It is well known that a bound of this type is not enough for making the Newton scheme convergent; see [42].…”

“…Indeed in these cases the complete reduction to constant coefficients is achieved. However the three papers [44,29,8] deal only with linear equations, whereas in the nonlinear case one has to fit the reducibility of the linearized operator with the Newton scheme. For instance, if in our case one tries…”