In this paper we prove the existence and the stability of small-amplitude quasi-periodic solutions with Sobolev regularity, for the 1-dimensional forced Kirchhoff 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 - ∂x x. 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
Quasi-periodic solutions of forced Kirchhoff equation / R. Montalto. - In: NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS. - ISSN 1021-9722. - 24:1(2017 Feb), pp. 9.1-9.71. [10.1007/s00030-017-0432-3]
Quasi-periodic solutions of forced Kirchhoff equation
R. Montalto
2017
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 Kirchhoff 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 - ∂x x. 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 classFile | Dimensione | Formato | |
---|---|---|---|
Montalto2017_Article_Quasi-periodicSolutionsOfForce.pdf
accesso riservato
Tipologia:
Publisher's version/PDF
Dimensione
1.42 MB
Formato
Adobe PDF
|
1.42 MB | Adobe PDF | Visualizza/Apri Richiedi una copia |
2434:607464.pdf
accesso aperto
Tipologia:
Post-print, accepted manuscript ecc. (versione accettata dall'editore)
Dimensione
672.48 kB
Formato
Adobe PDF
|
672.48 kB | Adobe PDF | Visualizza/Apri |
Pubblicazioni consigliate
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.