KAM for the nonlinear beam equation

详细信息    查看全文

• 作者：L. Hakan Eliasson ; Benoît Grébert ; Sergei B. Kuksin
• 关键词：and phrasesBeam equation ; KAM theory ; Hamiltonian systems
• 刊名：Geometric and Functional Analysis
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：26
• 期：6
• 页码：1588-1715
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis;
• 出版者：Springer International Publishing
• ISSN：1420-8970
• 卷排序：26

文摘

In this paper we prove a KAM theorem for small-amplitude solutions of the non linear beam equation on the d-dimensional torus$$u_{tt}+\Delta^2 u+m u + \partial_u G(x,u)=0, \quad t \in {\mathbb{R}}, x \in {\mathbb{T}^d}, \quad (*)$$where \({G(x,u)=u^4+ O(u^5)}\). Namely, we show that, for generic m, many of the small amplitude invariant finite dimensional tori of the linear equation \({(*)_{G=0}}\), written as the system$$u_t=-v,\quad v_t=\Delta^2 u+mu,$$persist as invariant tori of the nonlinear equation \({(*)}\), re-written similarly. The persisted tori are filled in with time-quasiperiodic solutions of \({(*)}\). If \({d\ge2}\), then not all the persisted tori are linearly stable, and we construct explicit examples of partially hyperbolic invariant tori. The unstable invariant tori, situated in the vicinity of the origin, create around them some local instabilities, in agreement with the popular belief in the nonlinear physics that small-amplitude solutions of space-multidimensional Hamiltonian PDEs behave in a chaotic way.Keywords and phrasesBeam equationKAM theoryHamiltonian systemsReferencesArn06.V.I. Arnold. Mathematical Methods in Classical Mechanics, 3rd edn. Springer, Berlin (2006).Bam03.Bambusi D.: Birkhoff normal form for some nonlinear PDEs. Communications in Mathematical Physics 234, 253–283 (2003)MathSciNetCrossRefMATHGoogle ScholarBG06.Bambusi D., Grébert B.: Birkhoff normal form for PDE’s with tame modulus. Duke Mathematical Journal 135(3), 507–567 (2006)MathSciNetCrossRefMATHGoogle ScholarBB12.Berti M., Bolle P.: Sobolev quasi periodic solutions of multidimensional wave equations with a multiplicative potential. Nonlinearity 25, 2579–2613 (2012)MathSciNetCrossRefMATHGoogle ScholarBB13.Berti M., Bolle P.: Quasi-periodic solutions with Sobolev regularity of NLS on \({\mathbb{T}^d}\) and a multiplicative potential. Journal of the European Mathematical Society 15, 229–286 (2013)MathSciNetCrossRefMATHGoogle ScholarBK95.Bobenko A. I., Kuksin S. B.: The nonlinear Klein–Gordon equation on an interval as a perturbed Sine–Gordon equation. Commentarii Mathematici Helvetici 70, 63–112 (1995)MathSciNetCrossRefMATHGoogle ScholarBou95.Bourgain J.: Construction of approximative and almost-periodic solutions of perturbed linear Schrödinger and wave equations. GAFA 6, 201–235 (1995)MATHGoogle ScholarBou98.Bourgain J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Shödinger equation. Annals of Mathematics 148, 363–439 (1998)MathSciNetCrossRefMATHGoogle ScholarBou04.J. Bourgain. Green’s function estimates for lattice Schrödinger operators and applications. Annals of Mathematical Studies, Princeton (2004).CKSTT10.Colliander J., Keel M., Staffilani G., Takaoka H., Tao T.: Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation. Inventiones Mathematicae 181, 31–113 (2010)MathSciNetCrossRefMATHGoogle ScholarCra00.W. Craig. Problèmes de Petits Diviseurs dans les Équations aux Dérivées Partielles. Panoramas et Synthèses, Société Mathématique de France (2000).Eli88.Eliasson L.H.: Perturbations of stable invariant tori for Hamiltonian systems. Annali della Scoula Normale Superiore di Pisa 15, 115–147 (1988)MathSciNetMATHGoogle ScholarEli98.L.H Eliasson. Perturbations of Linear Quasi-Periodic Systems. In: Dynamical Systems and Small Divisors (Cetraro, Italy, 1998), 1–60, Lect. Notes Math. 1784, Springer (2002).Eli01.L.H Eliasson. Almost reducibility of linear quasi-periodic systems. In: Smooth Ergodic Theory and its Applications (Seattle, WA, 1999), 679–705, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI (2001).EGK14.L.H. Eliasson, B. Grébert and S.B. Kuksin. KAM for the non-linear Beam equation 1: small-amplitude solutions. arXiv:1412.2803v3.EGK15.L.H. Eliasson, B. Grébert and S.B. Kuksin. KAM for the nonlinear beam equation 2: a normal form theorem. arXiv:1502.02262.EK08.Eliasson L.H., Kuksin S.B.: Infinite Töplitz–Lipschitz matrices and operators. Zeitschrift für angewandte Mathematik und Physik 59, 24–50 (2008)MathSciNetCrossRefMATHGoogle ScholarEK09.Eliasson L.H., Kuksin S.B.: On reducibility of Schrödinger equations with quasiperiodic in time potentials. Communications in Mathematical Physics 286(1), 125–135 (2009)MathSciNetCrossRefMATHGoogle ScholarEK10.Eliasson L.H., Kuksin S.B.: KAM for the nonlinear Schrödinger equation. Annals of Mathematics 172, 371–435 (2010)MathSciNetCrossRefMATHGoogle ScholarGP16.B. Grébert and É. Paturel. KAM for the Klein Gordon equation on \({\mathbb{S}^d}\), Bollettino dellUnione Matematica Italiana 9 (2016), 237–288.GXY11.Geng J., Xu X., You J.: An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation. Advances in Mathematics 226, 5361–5402 (2011)MathSciNetCrossRefMATHGoogle ScholarGY06.Geng J., You J.: A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces. Communications in Mathematical Physics 262, 343–372 (2006)MathSciNetCrossRefMATHGoogle ScholarGY06.Geng J., You J.: KAM tori for higher dimensional beam equations with constant potentials. Nonlinearity 19, 2405–2423 (2006)MathSciNetCrossRefMATHGoogle ScholarHW08.G. H. Hardy and E. M. Wright. An Introduction to the Theory of Number. Oxford University Press, Oxford (2008).Hör76.Hörmander L.: Note on Hölder Estimates. The boundary problem of physical geodesy. Archive for Rational Mechanics and Analysis 62, 1–52 (1976)MathSciNetCrossRefMATHGoogle ScholarKP02.S. Krantz and H. Parks. A Premier of Real Analytic Functions. Birkhäuser, Basel (2002).Kuk87.Kuksin S. B.: Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum. Functional Analysis and Its Applications 21, 192–205 (1987)MathSciNetCrossRefMATHGoogle ScholarKuk93.S. B. Kuksin. Nearly Integrable Infinite-dimensional Hamiltonian Systems. Lecture Notes in Mathematics, 1556. Springer, Berlin (1993).Kuk00.S. B. Kuksin. Analysis of Hamiltonian PDEs. Oxford University Press, Oxford (2000).KP96.Kuksin S. B., Pöschel J.: Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation. Annals of Mathematics 143, 149–179 (1996)MathSciNetCrossRefMATHGoogle ScholarMS71.J. Moser and C. L. Siegel. Lectures on Celestial Mechanics. Springer, Berlin (1971).Pös96.Pöschel J.: Quasi-periodic solutions for a nonlinear wave equation. Commentarii Mathematici Helvetici 71, 269–296 (1996)MathSciNetCrossRefMATHGoogle ScholarPP12.Procesi C., Procesi M.: A normal form of the nonlinear Schrdinger equation with analytic non–linearities. Communications in Mathematical Physics 312, 501–557 (2012)MathSciNetCrossRefMATHGoogle ScholarPP13.Procesi C., Procesi M.: A KAM Algorithm for the Resonant Nonlinear Schrödinger Equation. Advances in Mathematics 272, 399–470 (2015)MathSciNetCrossRefMATHGoogle ScholarVin55.I. M. Vinogradov. An Introduction to the Theory of Numbers. Pergamon Press, London (1955).Wan.Wang W.-M.: Energy supercritical nonlinear Schrödinger equations: Quasiperiodic solutions. Duke Mathematical Journal 165, 1129–1192 (2016)MathSciNetMATHGoogle ScholarYou99.You J.: Perturbations of lower dimensional tori for Hamiltonian systems. Journal of Differential Equations 152, 1–29 (1999)MathSciNetCrossRefMATHGoogle ScholarCopyright information© Springer International Publishing 2016Authors and AffiliationsL. Hakan Eliasson1Benoît Grébert2Sergei B. Kuksin3Email author1.Université Paris Diderot, Sorbonne Paris Cité, Institut de, Mathémathiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS, Sorbonne Universités, UPMC Univ. Paris 06ParisFrance2.Laboratoire de Mathématiques Jean Leray, Université de Nantes, UMR CNRS 6629Nantes Cedex 03France3.CNRS, Institut de Mathémathiques de Jussieu-Paris Rive Gauche, UMR 7586, Université Paris Diderot, Sorbonne Paris Cité, Sorbonne Universités, UPMC Univ. Paris 06ParisFrance About this article CrossMark Publisher Name Springer International Publishing Print ISSN 1016-443X Online ISSN 1420-8970 About this journal Reprints and Permissions Article actions .buybox { margin: 16px 0 0; position: relative; } .buybox { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; } .buybox { zoom: 1; } .buybox:after, .buybox:before { content: ''; display: table; } .buybox:after { clear: both; } /*---------------------------------*/ .buybox .buybox__header { border: 1px solid #b3b3b3; border-bottom: 0; padding: 8px 12px; position: relative; background-color: #f2f2f2; } .buybox__header .buybox__login { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; letter-spacing: .017em; display: inline-block; line-height: 1.2; padding: 0; } .buybox__header .buybox__login:before { position: absolute; top: 50%; -webkit-transform: perspective(1px) translateY(-50%); transform: perspective(1px) translateY(-50%); content: '\A'; width: 34px; height: 34px; left: 10px; } /*---------------------------------*/ .buybox .buybox__body { padding: 0; padding-bottom: 16px; position: relative; text-align: center; background-color: #fcfcfc; border: 1px solid #b3b3b3; } .buybox__body .buybox__section { padding: 16px 12px 0 12px; text-align: left; } .buybox__section .buybox__buttons { text-align: center; width: 100%; } /********** mycopy buybox specific **********/ .buybox.mycopy__buybox .buybox__section .buybox__buttons { border-top: 0; padding-top: 0; } /******/ .buybox__section:nth-child(2) .buybox__buttons { border-top: 1px solid #b3b3b3; padding-top: 20px; } .buybox__buttons .buybox__buy-button { display: inline-block; text-align: center; margin-bottom: 5px; padding: 6px 12px; } .buybox__buttons .buybox__price { white-space: nowrap; text-align: center; font-size: larger; padding-top: 6px; } .buybox__section .buybox__meta { letter-spacing: 0; padding-top: 12px; } .buybox__section .buybox__meta:only-of-type { padding-top: 0; position: relative; bottom: 6px; } /********** mycopy buybox specific **********/ .buybox.mycopy__buybox .buybox__section .buybox__meta { margin-top: 0; margin-bottom: 0; } /******/ .buybox__meta .buybox__product-title { display: inline; font-weight: bold; } .buybox__meta .buybox__list { line-height: 1.3; } .buybox__meta .buybox__list li { position: relative; padding-left: 1em; list-style: none; margin-bottom: 5px; } .buybox__meta .buybox__list li:before { font-size: 1em; content: '\2022'; float: left; position: relative; top: .1em; font-family: serif; font-weight: 600; text-align: center; line-height: inherit; color: #666; width: auto; margin-left: -1em; } .buybox__meta .buybox__list li:last-child { margin-bottom: 0; } /*---------------------------------*/ .buybox .buybox__footer { border: 1px solid #b3b3b3; border-top: 0; padding: 8px 12px; position: relative; border-style: dashed; } /*-----------------------------------------------------------------*/ @media screen and (min-width: 460px) and (max-width: 1074px) { .buybox__body .buybox__section { display: inline-block; vertical-align: top; padding: 12px 12px; padding-bottom: 0; text-align: left; width: 48%; } .buybox__body .buybox__section { padding-top: 16px; padding-left: 0; } .buybox__section:nth-of-type(2) .buybox__meta { border-left: 1px solid #d3d3d3; padding-left: 28px; } .buybox__section:nth-of-type(2) .buybox__buttons { border-top: 0; padding-top: 0; padding-left: 16px ; } .buybox__buttons .buybox__buy-button { } /********** article buybox specific **********/ .buybox.article__buybox .buybox__section:nth-of-type(2) { margin-top: 16px; padding-top: 0; } .buybox.article__buybox .buybox__section:nth-of-type(2) .buybox__meta { margin-top: 40px; padding-top: 0; padding-bottom: 45px; } .buybox.article__buybox .buybox__section:nth-of-type(2) .buybox__meta:only-of-type { margin-top: 8px; padding-top: 12px; padding-bottom: 12px; } /********** mycopy buybox specific **********/ .buybox.mycopy__buybox .buybox__section:first-child { width: 69%; } .buybox.mycopy__buybox .buybox__section:last-child { width: 29%; } /******/ } /*-----------------------------------------------------------------*/ @media screen and (max-width: 459px) { /********** mycopy buybox specific **********/ .buybox.mycopy__buybox .buybox__body { padding-bottom: 5px; } .buybox.mycopy__buybox .buybox__section:last-child { text-align: center; width: 100%; } .buybox.mycopy__buybox .buybox__buttons { display: inline-block; width: 150px ; } /******/ } /*-----------------------------------------------------------------*/ Log in to check access Buy (PDF) EUR 34,95 Unlimited access to the full article Instant download Include local sales tax if applicable Find out about institutional subscriptions (function () { var forEach = function (array, callback, scope) { for (var i = 0; i Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. More information Accept Over 10 million scientific documents at your fingertips