# Galerkin averaging method and Poincaré normal form for some quasilinear PDEs

Dario Bambusi^{[1]}

- [1] Dipartimento di Matematica Via Saldini 50 20133 Milano, Italy

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2005)

- Volume: 4, Issue: 4, page 669-702
- ISSN: 0391-173X

## Access Full Article

top## Abstract

top## How to cite

topBambusi, Dario. "Galerkin averaging method and Poincaré normal form for some quasilinear PDEs." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 4.4 (2005): 669-702. <http://eudml.org/doc/84576>.

@article{Bambusi2005,

abstract = {We use the Galerkin averaging method to construct a coordinate transformation putting a nonlinear PDE in Poincaré normal form up to finite order. We also give a rigorous estimate of the remainder showing that it is small as a differential operator of very high order. The abstract theorem is then applied to a quasilinear wave equation, to the water wave problem and to a nonlinear heat equation. The normal form is then used to construct approximate solutions whose difference from true solutions is estimated. In the case of hyperbolic equations we obtain an estimate of the error valid over time scales of order $\epsilon ^\{-1\}$ ($\epsilon $ being the norm of the initial datum), as in averaging theorems. For parabolic equations we obtain an estimate of the error valid over infinite time.},

affiliation = {Dipartimento di Matematica Via Saldini 50 20133 Milano, Italy},

author = {Bambusi, Dario},

journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},

language = {eng},

number = {4},

pages = {669-702},

publisher = {Scuola Normale Superiore, Pisa},

title = {Galerkin averaging method and Poincaré normal form for some quasilinear PDEs},

url = {http://eudml.org/doc/84576},

volume = {4},

year = {2005},

}

TY - JOUR

AU - Bambusi, Dario

TI - Galerkin averaging method and Poincaré normal form for some quasilinear PDEs

JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

PY - 2005

PB - Scuola Normale Superiore, Pisa

VL - 4

IS - 4

SP - 669

EP - 702

AB - We use the Galerkin averaging method to construct a coordinate transformation putting a nonlinear PDE in Poincaré normal form up to finite order. We also give a rigorous estimate of the remainder showing that it is small as a differential operator of very high order. The abstract theorem is then applied to a quasilinear wave equation, to the water wave problem and to a nonlinear heat equation. The normal form is then used to construct approximate solutions whose difference from true solutions is estimated. In the case of hyperbolic equations we obtain an estimate of the error valid over time scales of order $\epsilon ^{-1}$ ($\epsilon $ being the norm of the initial datum), as in averaging theorems. For parabolic equations we obtain an estimate of the error valid over infinite time.

LA - eng

UR - http://eudml.org/doc/84576

ER -

## References

top- [Bam03a] D. Bambusi, An averaging theorem for quasilinear Hamiltonian PDEs, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (2003), 685–712. Zbl1031.37056MR2015427
- [Bam03b] D. Bambusi, Birkhoff normal form for some nonlinear PDEs, Commun. Math. Phys. 234 (2003), 253–285. Zbl1032.37051MR1962462
- [Bam03c] D. Bambusi, Birkhoff normal form for some quasilinear Hamiltonian PDEs, preprint, 2003. MR2227839
- [BCP02] D. Bambusi, A. Carati and A. Ponno, The nonlinear Schrödinger equation as a resonant normal form, Discrete Contin. Dyn. Syst. Ser. B 2 (2002), 109–128. Zbl1068.37056MR1877043
- [BG03] D. Bambusi and B. Grébert, Forme normale pour NLS en dimension quelconque, C. R. Math. Acad. Sci. Paris 337 (2003), 409–414. Zbl1030.35143MR2015085
- [BG04] D. Bambusi and B. Grébert, Birkhoff normal form for PDEs with tame modulus, Duke Math. J. (2004), to appear. Zbl1110.37057MR2272975
- [BN98] D. Bambusi and N. N. Nekhoroshev, A property of exponential stability in nonlinear wave equations near the fundamental linear mode, Phys. D 122 (1998), 73–104. Zbl0937.35010MR1650123
- [Bou96] J. Bourgain, Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations, Geom. Funct. Anal. 6 (1996), 201–230. Zbl0872.35007MR1384610
- [Cra96] W. Craig, Birkhoff normal forms for water waves, In: “Mathematical problems in the theory of water waves” (Luminy, 1995), Vol. 200, Contemp. Math., Amer. Math. Soc., Providence, RI, 1996, 57–74. Zbl0953.76009MR1410500
- [CS93] W. Craig and C. SulemNumerical simulation of gravity waves, J. Comput. Phys. 108 (1993), 73–83. Zbl0778.76072MR1239970
- [CSS97] W. Craig, U. Schanz and C. Sulem, The modulational regime of three-dimensional water waves and the Davey-Stewartson system, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), 615–667. Zbl0892.76008MR1470784
- [CW95] W. Craig and P. A. Worfolk, An integrable normal form for water waves in infinite depth, Phys. D 84 (1995), 513–531. Zbl0883.35092MR1336546
- [DZ94] A. I. Dʼyachenko and V. E. Zakharov, Is free-surface hydrodynamics an integrable system? Phys. Lett. A 190 (1994), 144–148. Zbl0961.76511MR1283779
- [FS87] C. Foias and J.-C. Saut, Linearization and normal form of the Navier-Stokes equations with potential forces, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), 1–47. Zbl0635.35075MR877990
- [FS91] C. Foias and J.-C. Saut, Asymptotic integration of Navier-Stokes equations with potential forces. I, Indiana Univ. Math. J. 40 (1991), 305–320. Zbl0739.35066MR1101233
- [GP88] A. Giorgilli and A. Posilicano, Estimates for normal forms of differential equations near an equilibrium point, Z. Angew. Math. Phys. 39 (1988), 713–732. Zbl0685.58028MR963640
- [Kat75] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, In: “Spectral Theory and Differential Equations” (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Lecture Notes in Math., Vol. 448, Springer, Berlin, 1975, 25–70. Zbl0315.35077MR407477
- [Kro89] M. S. Krol, On a Galerkin-averaging method for weakly nonlinear wave equations, Math. Methods Appl. Sci. 11 (1989), 649–664. Zbl0707.35095MR1011811
- [Mat01] K. Matthies, Time-averaging under fast periodic forcing of parabolic partial differential equations: exponential estimates, J. Differential Equations 174 (2001), 133–180. Zbl1023.35055MR1844527
- [MS03] K. Matthies and A. Scheel, Exponential averaging for Hamiltonian evolution equations, Trans. Amer. Math. Soc. 355 (2003), 747–773. Zbl1008.37043MR1932724
- [Nik86] N. V. Nikolenko, The method of Poincaré normal forms in problems of integrability of equations of evolution type, Uspekhi Mat. Nauk 41 (1986), 109–152, 263. Zbl0632.35026MR878327
- [Pal96] H. Pals, The Galerkin-averaging method for the Klein-Gordon equation in two space dimensions, Nonlinear Anal. 27 (1996), 841–856. Zbl0861.35101MR1402170
- [PB05] A. Ponno and D. Bambusi, Korteweg-de Vries equation and energy sharing in Fermi-Pasta-Ulam, Chaos 15 (2005), 015107, 5. Zbl1080.37073MR2133458
- [Sha85] J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math. 38 (1985), 685–696. Zbl0597.35101MR803256
- [SV87] A. C. J. Stroucken and F. Verhulst, The Galerkin-averaging method for nonlinear, undamped continuous systems, Math. Methods Appl. Sci. 9 (1987), 520–549. Zbl0638.35057MR1200364
- [SW00] G. Schneider and C. E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model In: “International Conference on Differential Equations”, Vol. 1, 2 (Berlin, 1999), World Sci. Publishing, River Edge, NJ, 2000, 390–404. Zbl0970.35126MR1870156
- [Zak68] V. E. ZakharovStability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys. 2 (1968), 190–194.
- [Zeh78] E. Zehnder, C. L. Siegel’s linearization theorem in infinite dimensions, Manuscripta Math. 23 (1977/78), 363–371. Zbl0374.47037MR501144

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.