We summarize some recent results on the Cauchy problem for the Kirchhoff equation partial derivative(tt)u - triangle u (1 + integral Td |del u|(2) = 0 on the d-dimensional torus Td, with initial data u(0, x), 8tu(0, x) of size 6 in Sobolev class. While the standard local theory gives an existence time of order 6-2, a quasilinear normal form allows to give a lower bound on the existence time of the order of 6-4 for all initial data, improved to 6-6 for initial data satisfying a suitable nonresonance condition. We also use such a normal form in an ongoing work with F. Giuliani and M. Guardia to prove existence of chaotic-like motions for the Kirchhoff equation.

Baldi, P., Haus, E. (In corso di stampa). Normal form and dynamics of the Kirchhoff equation. BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA [10.1007/s40574-022-00344-6].

Normal form and dynamics of the Kirchhoff equation

Baldi, P;Haus, E
In corso di stampa

Abstract

We summarize some recent results on the Cauchy problem for the Kirchhoff equation partial derivative(tt)u - triangle u (1 + integral Td |del u|(2) = 0 on the d-dimensional torus Td, with initial data u(0, x), 8tu(0, x) of size 6 in Sobolev class. While the standard local theory gives an existence time of order 6-2, a quasilinear normal form allows to give a lower bound on the existence time of the order of 6-4 for all initial data, improved to 6-6 for initial data satisfying a suitable nonresonance condition. We also use such a normal form in an ongoing work with F. Giuliani and M. Guardia to prove existence of chaotic-like motions for the Kirchhoff equation.
In corso di stampa
Baldi, P., Haus, E. (In corso di stampa). Normal form and dynamics of the Kirchhoff equation. BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA [10.1007/s40574-022-00344-6].
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/432137
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? 0
social impact