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

Abstract

top
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 ϵ - 1 ( ϵ 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.

How to cite

top

Bambusi, 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
  1. [Bam03a] D. Bambusi, An averaging theorem for quasilinear Hamiltonian PDEs, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (2003), 685–712. Zbl1031.37056MR2015427
  2. [Bam03b] D. Bambusi, Birkhoff normal form for some nonlinear PDEs, Commun. Math. Phys. 234 (2003), 253–285. Zbl1032.37051MR1962462
  3. [Bam03c] D. Bambusi, Birkhoff normal form for some quasilinear Hamiltonian PDEs, preprint, 2003. MR2227839
  4. [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
  5. [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
  6. [BG04] D. Bambusi and B. Grébert, Birkhoff normal form for PDEs with tame modulus, Duke Math. J. (2004), to appear. Zbl1110.37057MR2272975
  7. [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
  8. [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
  9. [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
  10. [CS93] W. Craig and C. SulemNumerical simulation of gravity waves, J. Comput. Phys. 108 (1993), 73–83. Zbl0778.76072MR1239970
  11. [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
  12. [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
  13. [DZ94] A. I. Dʼyachenko and V. E. Zakharov, Is free-surface hydrodynamics an integrable system? Phys. Lett. A 190 (1994), 144–148. Zbl0961.76511MR1283779
  14. [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
  15. [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
  16. [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
  17. [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
  18. [Kro89] M. S. Krol, On a Galerkin-averaging method for weakly nonlinear wave equations, Math. Methods Appl. Sci. 11 (1989), 649–664. Zbl0707.35095MR1011811
  19. [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
  20. [MS03] K. Matthies and A. Scheel, Exponential averaging for Hamiltonian evolution equations, Trans. Amer. Math. Soc. 355 (2003), 747–773. Zbl1008.37043MR1932724
  21. [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
  22. [Pal96] H. Pals, The Galerkin-averaging method for the Klein-Gordon equation in two space dimensions, Nonlinear Anal. 27 (1996), 841–856. Zbl0861.35101MR1402170
  23. [PB05] A. Ponno and D. Bambusi, Korteweg-de Vries equation and energy sharing in Fermi-Pasta-Ulam, Chaos 15 (2005), 015107, 5. Zbl1080.37073MR2133458
  24. [Sha85] J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math. 38 (1985), 685–696. Zbl0597.35101MR803256
  25. [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
  26. [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
  27. [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. 
  28. [Zeh78] E. Zehnder, C. L. Siegel’s linearization theorem in infinite dimensions, Manuscripta Math. 23 (1977/78), 363–371. Zbl0374.47037MR501144

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.