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 \(-\partial _{xx}\). 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 \(2 \times 2\), 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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alazard, T., Baldi, P.: Gravity capillary standing water waves. Arch. Rat. Mech. Anal. 217(3), 741–830 (2015)
Arosio, A., Spagnolo, S.: Global solutions of the Cauchy problem for a nonlinear hyperbolic equation. In: Brezis, H., Lions, J.L. (eds.) Nonlinear PDEs and Their Applications, Collége de France Seminar, vol. VI, 126, Research Notes on Mathematics 109. Pitman, Boston (1984)
Arosio, A., Panizzi, S.: On the well-posedness of the Kirchhoff string. Trans. Am. Math. Soc. 348(1), 305–330 (1996)
Baldi, P.: Periodic solutions of forced Kirchhoff equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 8, 117–141 (2009)
Baldi, P.: Periodic solutions of fully nonlinear autonomous equations of Benjamin–Ono type. Ann. Inst. H. Poincaré (C) Anal. Non Linéaire 30(1), 33–77 (2013)
Baldi, P., Berti, M., Montalto, R.: KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation. Math. Ann. 359(1–2), 471–536 (2014)
Baldi, P., Berti, M., Montalto, R.: KAM for autonomous quasi-linear perturbations of KdV. Ann. Inst. H. Poincaré (C) Anal. Non Linéaire 33, 1589–1638 (2016)
Baldi, P., Berti, M., Montalto, R.: KAM for autonomous quasi-linear perturbations of mKdV. Boll. Unione Mat. Ital. 9, 143–188 (2016)
Bernstein, S.N.: Sur une classe d’ équations fonctionelles aux dérivées partielles. Izv. Akad. Nauk SSSR Ser. Mat. 4, 17–26 (1940)
Berti, M., Biasco, P., Procesi, M.: KAM theory for the Hamiltonian DNLW. Ann. Sci. Éc. Norm. Supér. (4) 46(fascicule 2), 301–373 (2013)
Berti, M., Biasco, P., Procesi, M.: KAM theory for the reversible derivative wave equation. Arch. Ration. Mech. Anal. 212, 905–955 (2014)
Berti, M., Bolle, P.: Sobolev quasi periodic solutions of multidimensional wave equations with a multiplicative potential. Nonlinearity 25, 2579–2613 (2012)
Berti, M., Bolle, P.: Quasi-periodic solutions with Sobolev regularity of NLS on \( \mathbb{T}^d \) with a multiplicative potential. Eur. J. Math. 15, 229–286 (2013)
Berti, M., Corsi, L., Procesi, M.: An abstract Nash–Moser theorem and quasi-periodic solutions for NLW and NLS on compact Lie groups and homogeneous manifolds. Commun. Math. Phys. 334(3), 1413–1454 (2015)
Berti, M., Montalto, R.: Quasi-periodic Standing Wave Solutions of Gravity Capillary Standing Water Waves. arXiv:1602.02411 (2016)
Bourgain J.: Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE. Int. Math. Res. Not. (11) (1994)
Bourgain, J.: Quasi-periodic solutions of Hamiltonian perturbations of \(2D\) linear Schrödinger equations. Ann. Math. 148, 363–439 (1998)
Bourgain, J.: Periodic solutions of nonlinear wave equations. In: Harmonic Analysis and Partial Differential Equations. Chicago Lectures in Mathematics, pp. 69–97. University of Chicago Press (1999)
Bourgain, J.: Green’s Function Estimates for Lattice Schrödinger Operators and Applications, Annals of Mathematics Studies 158. Princeton University Press, Princeton (2005)
Craig, W.: Problèmes de petits diviseurs dans les équations aux dérivées partielles, Panoramas et Synthèses, 9. Société Mathématique de France, Paris (2000)
Craig, W., Wayne, C.E.: Newton’s method and periodic solutions of nonlinear wave equation. Commun. Pure Appl. Math. 46, 1409–1498 (1993)
Chierchia, L., You, J.: KAM Tori for 1D nonlinear wave equations with periodic boundary conditions. Commun. Math. Phys 211, 497–525 (2000)
D’Ancona, P., Spagnolo, S.: Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math. 108, 247–262 (1992)
Dickey, R.W.: Infinite systems of nonlinear oscillation equations related to the string. Proc. Am. Math. Soc. 23(3), 459–468 (1969)
Eliasson, L.H., Kuksin, S.: On reducibility of Schrödinger equations with quasiperiodic in time potentials. Commun. Math. Phys. 286, 125–135 (2009)
Eliasson, L.H., Kuksin, S.: KAM for non-linear Schrödinger equation. Ann. Math. 172, 371–435 (2010)
Feola, R.: KAM for Quasi-linear Forced Hamiltonian NLS. arXiv:1602.01341 (2016)
Feola, R., Procesi, M.: Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations. J. Differ. Equ. 259(7), 3389–3447 (2015)
Iooss, G., Plotnikov, P.I., Toland, J.F.: Standing waves on an infinitely deep perfect fluid under gravity. Arch. Ration. Mech. Anal. 177(3), 367–478 (2005)
Iooss, G., Plotnikov, P.I.: Small divisor problem in the theory of three-dimensional water gravity waves. Mem. Am. Math. Soc. 200(940) (2009)
Iooss, G., Plotnikov, P.I.: Asymmetrical three-dimensional travelling gravity waves. Arch. Ration. Mech. Anal. 200(3), 789–880 (2011)
Lions, J.L.: On some questions in boundary value problems of mathematical physics. In: de la Penha, G.M., Medeiros, L.A. (eds.) Contemporary developments in continuum mechanics and PDEs. North-Holland, Amsterdam (1978)
Kappeler, T., Pöschel, J.: KAM and KdV. Springer, Berlin (2003)
Klainermann, S., Majda, A.: Formation of singularities for wave equations including the nonlinear vibrating string. Commun. Pure Appl. Math. 33, 241–263 (1980)
Kuksin, S.: A KAM theorem for equations of the Korteweg–de Vries type. Rev. Math. Math. Phys. 10(3), 1–64 (1998)
Kuksin, S., Pöschel, J.: Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation. Ann. Math. 2(143), 149–179 (1996)
Lax, P.: Development of singularities of solutions of nonlinear hyperbolic partial differential equations. J. Math. Phys. 5, 611–613 (1964)
Liu, J., Yuan, X.: Spectrum for quantum Duffing oscillator and small-divisor equation with large-variable coefficient. Commun. Pure Appl. Math. 63(9), 1145–1172 (2010)
Liu, J., Yuan, X.: A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations. Commun. Math. Phys. 307, 629–673 (2011)
Manfrin, R.: On the global solvability of Kirchhoff equation for non-analytic initial data. J. Differ. Equ. 211, 38–60 (2005)
Matsuyama, T., Ruzhansky, M.: Global well-posedness of the Kirchhoff equation and Kirchhoff systems. Springer proceedings in mathematics and statistics. Anal. Methods Interdiscip. Appl. 116, 81–96 (2014)
Pokhozhaev, S.I.: On a class of quasilinear hyperbolic equations, Mat. Sbornik 96, 152–166 (1975) (English transl.: Mat. USSR Sbornik 25, 145–158 (1975))
Procesi, M., Xu, X.: Quasi-Töplitz functions in KAM theorem. SIAM J. Math. Anal. 45(4), 2148–2181 (2013)
Rabinowitz, P.H.: Periodic solutions of nonlinear hyperbolic partial differential equations, part I and II. Commun. Pure Appl. Math. 20, 145–205 (1967)
Rabinowitz, P.H.: Periodic solutions of nonlinear hyperbolic partial differential equations, part I and II. Commun. Pure Appl. Math. 22, 15–39 (1969)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported in part by the Swiss National Science Foundation.
Rights and permissions
About this article
Cite this article
Montalto, R. Quasi-periodic solutions of forced Kirchhoff equation. Nonlinear Differ. Equ. Appl. 24, 9 (2017). https://doi.org/10.1007/s00030-017-0432-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-017-0432-3
Keywords
- Kirchhoff equation
- Quasi-linear PDEs
- Quasi-periodic solutions
- Infinite-dimensional dynamical systems
- KAM for PDEs
- Nash–Moser theory