Quasi-periodic solutions of forced Kirchhoff equation

Abstract

In this paper we prove the existence and the stability of small-amplitude quasi-periodic solutions with Sobolev regularity, for the 1-dimensional forced Kirchoff equation with periodic boundary conditions. This is the first KAM result for a quasi-linear wave-type equation. The main difficulties are: (i) the presence of the highest order derivative in the nonlinearity which does not allow to apply the classical KAM scheme, (ii) the presence of double resonances, due to the double multiplicity of the eigenvalues of −∂ xx . The proof is based on a Nash-Moser scheme in Sobolev class. The main point concerns the invertibility of the linearized operator at any approximate solution and the proof of tame estimates for its inverse in high Sobolev norm. To this aim, we conjugate the linearized operator to a 2 × 2, time independent, block-diagonal operator. This is achieved by using changes of variables induced by diffeomorphisms of the torus, pseudo-differential operators and a KAM reducibility scheme in Sobolev class.We consider the Kirchoff equation in 1-dimension with periodic boundary conditionsOur aim is to prove the existence and the linear stability of small-amplitude quasi-periodic solutions with Sobolev regularity, for δ small enough and for ω in a suitable Cantor like set of parameters with asymptotically full Lebesgue measure.The Kirchoff equation has been introduced for the first time in 1876 by Kirchoff, in dimension 1, without forcing term and with Dirichlet boundary conditions, namelyto describe the transversal free vibrations of a clamped string in which the dependence of the tension on the deformation cannot be neglected. It is a quasi-linear PDE, namely the nonlinear part of the equation contains as many derivatives as the linear differential operator. The Cauchy problem for the Kirchoff equation (also in higher dimension) has been extensively studied, starting from the pioneering paper of Bernstein [9]. Both local and global existence results have been established for initial data in Sobolev and analytic class, see [2], [3], [23], [24], [32], [40], [41]. Concernig the existence of periodic solutions, Kirchoff himself observed that the equation (1.2) admits a sequence of normal modes, namely solutions of the form v(t, xUnder the presence of the forcing term f (t, x) the normal modes do not persist, 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 in the integral term T |∂ x v| 2 dx and the equation (1.2) is equivalent to the infinitely many coupled ODEs v ′′ j (t) + j 2 v j (t) 1 + k k 2 |v k (t)| 2 = f j (t) , j = 1, 2, . . . .The existence of periodic solutions for the Kirchoff equation, also in higher dimension, have been proved by Baldi in [4], both for Dirichlet boundary conditions (v = 0 on ∂Ω) and for periodic boundary conditions (Ω = T d ). This result is proven via Nash-Moser method and thanks to the special structure of the nonlinearity (it is diagonal in space), the linearized operator at any approximate solution can be inverted ...

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

“…We mention the results of Eliasson-Kuksin [13] which proved the reducibility of the Schrödinger equation on T d with a small, quasi-periodic in time analytic potential and Grebert-Paturel [18] which proved the reducibility of the quantum harmonic oscillator on R d . Concerning KAM-reducibility with unbounded perturbations, we mention Bambusi [4], [5] for the reducibility of the quantum harmonic oscillator with unbounded perturbations (see also [6] in any dimension), [1], [2], [17] for fully non-linear KdV-type equations, [14], [15] for fully-nonlinear Schrödinger equations, [8], [9] for the water waves system and [20] for the Kirchhoff equation. 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.…”

“…Concerning KAM-reducibility with unbounded perturbations, we mention Bambusi [4], [5] for the reducibility of the quantum harmonic oscillator with unbounded perturbations (see also [6] in any dimension), [1], [2], [17] for fully non-linear KdV-type equations, [14], [15] for fully-nonlinear Schrödinger equations, [8], [9] for the water waves system and [20] for the Kirchhoff equation. 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.…”

A note on KAM for gravity-capillary water waves

“…These techniques have been applied by Feola-Procesi [18] also to quasi-linear perturbations of 1-d Schrödinger equations and by Montalto [29] to the Kirchhoff equation.…”

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

“…We mention [22,15] for the case of double eigenvalues and [25,26] in higher space dimension.All the aforementioned results concern semi-linear PDEs, namely PDEs in which the order of the nonlinearity is strictly smaller than the order of the linear part. For quasi-linear (either fully nonlinear) PDEs, the first KAM results have been proved by the Italian team in [4,5,6,31,28,27,43,16,7].To the best of our knowledge all the results for quasi-linear and fully nonlinear PDEs are only in one space dimension. The result proved in this paper is the first one concerning the existence of quasi-periodic solutions for a quasi-linear PDE in higher space dimension.The reason why we achieve our result, whereas for other PDEs this is not possible (at least at the present time), is not merely technical and can be roughly explained as follows.Almost all the literature about the existence of quasi-periodic solutions for dynamical systems in both finite and infinite dimension is ultimately related to a functional Newton scheme.…”

“…This is the first result of this type for a quasi-linear equations in high dimension. The proof is based on a Nash-Moser scheme in Sobolev class and a regularization procedure combined with a multiscale analysis in order to solve the linearized problem at any approximate solution.The existence of periodic solutions for the forced Kirchhoff equation in any dimension has been proved by Baldi in [2], while the existence of quasi-periodic solutions in one space dimension under periodic boundary conditions has been proved in [43].Note that equation (1.5) is a quasi-linear PDE and it is well known that the existence of global solutions (even not periodic or quasi-periodic) for quasi-linear PDEs is not guaranteed, see for instance the nonexistence results in [36,39] The existence of periodic solutions for wave-type equations with unbounded nonlinearities has been proved for instance in [46,20,19]. For the water waves equations, which are fully nonlinear PDEs, we mention [32,33,34,1]; see also [3] for fully non-linear Benjamin-Ono equations.The methods developed in the above mentioned papers do not work for proving the existence of quasiperiodic solutions.The existence of quasi-periodic solutions for PDEs with unbounded nonlinearities has been developed by Kuksin [37] for KdV and then .…”