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Normal form and dynamics of the Kirchhoff equation

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Abstract

We summarize some recent results on the Cauchy problem for the Kirchhoff equation

$$\begin{aligned} \partial _{tt} u - \Delta u \Bigg ( 1 + \int _{{{\mathbb {T}}}^d} |\nabla u|^2 \Bigg ) = 0 \end{aligned}$$

on the d-dimensional torus \({{\mathbb {T}}}^d\), with initial data u(0, x), \(\partial _t u(0,x)\) of size \(\varepsilon \) in Sobolev class. While the standard local theory gives an existence time of order \(\varepsilon ^{-2}\), a quasilinear normal form allows to give a lower bound on the existence time of the order of \(\varepsilon ^{-4}\) for all initial data, improved to \(\varepsilon ^{-6}\) 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.

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Data sharing is not applicable to this article as no datasets were generated or analysed during the current study, which is entirely theoretical.

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Acknowledgements

We would like to thank Claudio Bonanno and the DinAmici group for the participation to the DAI days. This work has been supported by PRIN 2020XB3EFL “Hamiltonian and dispersive PDEs”.

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli Federico II, Via Cintia, Monte S. Angelo, 80126, Naples, Italy

    Pietro Baldi

  2. Dipartimento di Matematica e Fisica, Università Roma Tre, Largo San Leonardo Murialdo 1, 00146, Rome, Italy

    Emanuele Haus

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  1. Pietro Baldi
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Correspondence to Emanuele Haus.

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Baldi, P., Haus, E. Normal form and dynamics of the Kirchhoff equation. Boll Unione Mat Ital 16, 337–349 (2023). https://doi.org/10.1007/s40574-022-00344-6

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