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

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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

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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

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