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LONG TIME BEHAVIOR OF THE SOLUTIONS OF NLW ON THE $d$-DIMENSIONAL TORUS

Published online by Cambridge University Press:  06 March 2020

JOACKIM BERNIER
Affiliation:
Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, Université Paul Sabatier, F-31062Toulouse Cedex 9, France; joackim.bernier@math.univ-toulouse.fr
ERWAN FAOU
Affiliation:
Univ Rennes, INRIA, CNRS, IRMAR - UMR 6625, F-35000Rennes, France; Erwan.Faou@inria.fr
BENOÎT GRÉBERT
Affiliation:
Laboratoire de Mathématiques Jean Leray, Université de Nantes, UMR CNRS 6629, 2, rue de la Houssinière, 44322Nantes Cedex 03, France; benoit.grebert@univ-nantes.fr

Abstract

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We consider the nonlinear wave equation (NLW) on the $d$-dimensional torus $\mathbb{T}^{d}$ with a smooth nonlinearity of order at least 2 at the origin. We prove that, for almost any mass, small and smooth solutions of high Sobolev indices are stable up to arbitrary long times with respect to the size of the initial data. To prove this result, we use a normal form transformation decomposing the dynamics into low and high frequencies with weak interactions. While the low part of the dynamics can be put under classical Birkhoff normal form, the high modes evolve according to a time-dependent linear Hamiltonian system. We then control the global dynamics by using polynomial growth estimates for high modes and the preservation of Sobolev norms for the low modes. Our general strategy applies to any semilinear Hamiltonian Partial Differential Equations (PDEs) whose linear frequencies satisfy a very general nonresonance condition. The (NLW) equation on $\mathbb{T}^{d}$ is a good example since the standard Birkhoff normal form applies only when $d=1$ while our strategy applies in any dimension.

Type
Mathematical Physics
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

References

Bernier, J., Faou, E. and Grébert, B., ‘Rational normal forms and stability of small solutions to nonlinear Schrödinger equations’. Preprint, 2018, arXiv:1812.11414.Google Scholar
Bambusi, D., ‘Birkhoff normal form for some nonlinear PDEs’, Comm. Math. Phys. 234 (2003), 253283.CrossRefGoogle Scholar
Bambusi, D., ‘A Birkhoff normal form theorem for some semilinear PDEs’, inHamiltonian Dynamical Systems and Applications (Springer, 2007), 213247.Google Scholar
Bambusi, D., Delort, J.-M., Grébert, B. and Szeftel, J., ‘Almost global existence for Hamiltonian semilinear Klein–Gordon equations with small Cauchy data on Zoll manifolds’, Comm. Pure Appl. Math. 60(11) (2007), 16651690.CrossRefGoogle Scholar
Bambusi, D. and Grébert, B., ‘Forme normale pour NLS en dimension quelconque’, C. R. Acad. Sci. Paris, Ser. I 337 (2003), 409414.CrossRefGoogle Scholar
Bambusi, D. and Grébert, B., ‘Birkhoff normal form for PDE’s with tame modulus’, Duke Math. J. 135(3) (2006), 507567.CrossRefGoogle Scholar
Bourgain, J., ‘Construction of periodic solutions of nonlinear wave equations in higher dimension’, Geom. Funct. Anal. 5 (1995), 629639.CrossRefGoogle Scholar
Bourgain, J., ‘Construction of approximative and almost-periodic solutions of perturbed linear Schrödinger and wave equations’, Geom. Funct. Anal. 6 (1996), 201230.CrossRefGoogle Scholar
Bourgain, J., ‘On Melnikov’s persistency problem’, Math. Res. Lett. 4 (1997), 445458.CrossRefGoogle Scholar
Cazenave, T., Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10 (New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003).CrossRefGoogle Scholar
Cohen, D., Hairer, E. and Lubich, C., ‘Conservation of energy, momentum and actions in numerical discretizations of nonlinear wave equations’, Numer. Math. 110 (2008), 113143.CrossRefGoogle Scholar
Faou, E. and Debussche, A., ‘Modified energy for split-step methods applied to the linear Schrödinger equation’, SIAM J. Numer. Anal. 47 (2009), 37053719.Google Scholar
Delort, J.-M., ‘On long time existence for small solutions of semi-linear Klein–Gordon equations on the torus’, J. Anal. Math. 107 161194.CrossRefGoogle Scholar
Delort, J.-M. and Szeftel, J., ‘Long-time existence for semi-linear Klein–Gordon equations with small Cauchy data on Zoll manifolds’, Amer. J. Math. 128(5) (2006), 11871218.CrossRefGoogle Scholar
Dujardin, G. and Faou, E., ‘Normal form and long time analysis of splitting schemes for the linear Schrödinger equation with small potential’, Numer. Math. 106 (2007), 223262.CrossRefGoogle Scholar
Eliasson, L. H., ‘Perturbations of stable invariant tori for Hamiltonian systems’, Ann. Sc. Norm. Super. Pisa Cl. Sci. 15 (1988), 115147.Google Scholar
Eliasson, L. H., Grébert, B. and Kuksin, S. B., ‘KAM for non-linear beam equation’, Geom. Funct. Anal. 26 (2016), 15881715.CrossRefGoogle Scholar
Faou, E., Geometric Numerical Integration and Schrödinger Equations, Zurich Lectures in Advanced Mathematics (European Mathematical Society, 2012), 146.CrossRefGoogle Scholar
Faou, E. and Grébert, B., ‘A Nekhoroshev-type theorem for the nonlinear Schrödinger equation on the torus’, Anal. PDE 6 (2013), 12431262.CrossRefGoogle Scholar
Faou, E., Grébert, B. and Paturel, E., ‘Birkhoff normal form for splitting methods applied to semi linear Hamiltonian PDEs. Part I: finite dimensional discretization’, Numer. Math. 114 (2010), 429458.CrossRefGoogle Scholar
Faou, E., Grébert, B. and Paturel, E., ‘Birkhoff normal form for splitting methods applied to semi linear Hamiltonian PDEs. Part II: abstract splitting’, Numer. Math. 114 (2010), 459490.CrossRefGoogle Scholar
Faou, E., Gauckler, L. and Lubich, C., ‘Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus’, Comm. PDE 38 (2013), 11231140.CrossRefGoogle Scholar
Faou, E., Gauckler, L. and Lubich, C., ‘Plane wave stability of the split-step Fourier method for the nonlinear Schrödinger equation’, Forum Math. Sigma 2 (2014), 45.CrossRefGoogle Scholar
Grébert, B., ‘Birkhoff normal form and Hamiltonian PDEs’, Sémin. Congr. 15 (2007), 146.Google Scholar
Grébert, B., Imekraz, R. and Paturel, É., ‘Normal forms for semilinear quantum harmonic oscillators’, Commun. Math. Phys. 291 (2009), 763798.CrossRefGoogle Scholar
Hairer, E., Lubich, C. and Wanner, G., Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, (Springer, 2006).Google Scholar
Hofmanová, M. and Schratz, K., ‘An exponential-type integrator for the KdV equation’, Numer. Math. 136 (2017), 11171137.CrossRefGoogle Scholar
Kuksin, S. B., Nearly Integrable Infinite Dimensional Hamiltonian Systems, Lecture Notes in Mathematics, 1556 (Springer, Berlin, 1993).CrossRefGoogle Scholar
Mel’nikov, V. K., ‘On some cases of conservation of conditionally periodic motions under a small change of the Hamiltonian function’, Soviet Math. Dokl. 6 (1965), 15921596.Google Scholar
Mel’nikov, V. K., ‘A family of conditionally periodic solutions of a Hamiltonian system’, Soviet Math. Dokl. 9(4) (1968), 882885.Google Scholar
Moser, J., ‘Convergent series expansions for quasiperiodic motions’, Math. Ann. 169 (1967), 136176.CrossRefGoogle Scholar
Ostermann, A. and Schratz, K., ‘Low regularity exponential-type integrators for semilinear Schrödinger equations’, Found. Comput. Math. 18 (2018), 731755.CrossRefGoogle Scholar
Pöschel, J., ‘A KAM-theorem for some nonlinear partial differential equations’, Ann. Sc. Norm. Super. Pisa Cl. Sci. 23 (1996), 119148.Google Scholar
Shang, Z., ‘Resonant and Diophantine step sizes in computing invariant tori of Hamiltonian systems’, Nonlinearity 13 (2000), 299308.CrossRefGoogle Scholar
Xu, J. and You, J., ‘Persistence of lower-dimensional tori under the first Melnikov’s non-resonance condition’, J. Math. Pures Appl. 80 (2001), 10471067.CrossRefGoogle Scholar