We use the Galerkin averaging method to construct a coordinate transformation putting a nonlinear PDE in Poincar\'e 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 equation we obtain an estimate of the error valid over time scales of order $\epsilon^{-1}$ ($\epsilon$ the norm of the initial datum), as in averaging theorems. In parabolic equations we obtain an estimate of the error valid over infinite times.

Galerkin averaging method and Poincare' normal form for some quasilinear PDEs / D. Bambusi. - In: ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE. - ISSN 0391-173X. - 4:4(2005), pp. 669-702.

Galerkin averaging method and Poincare' normal form for some quasilinear PDEs

D. Bambusi
Primo
2005

Abstract

We use the Galerkin averaging method to construct a coordinate transformation putting a nonlinear PDE in Poincar\'e 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 equation we obtain an estimate of the error valid over time scales of order $\epsilon^{-1}$ ($\epsilon$ the norm of the initial datum), as in averaging theorems. In parabolic equations we obtain an estimate of the error valid over infinite times.
Settore MAT/07 - Fisica Matematica
2005
Article (author)
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/7504
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