On the Normal Form of the Kirchhoff Equation

Author:

Baldi Pietro,Haus EmanueleORCID

Abstract

AbstractConsider the Kirchhoff equation $$\begin{aligned} \partial _{tt} u - \Delta u \Big ( 1 + \int _{\mathbb {T}^d} |\nabla u|^2 \Big ) = 0 \end{aligned}$$ tt u - Δ u ( 1 + T d | u | 2 ) = 0 on the d-dimensional torus $$\mathbb {T}^d$$ T d . In a previous paper we proved that, after a first step of quasilinear normal form, the resonant cubic terms show an integrable behavior, namely they give no contribution to the energy estimates. This leads to the question whether the same structure also emerges at the next steps of normal form. In this paper, we perform the second step and give a negative answer to the previous question: the quintic resonant terms give a nonzero contribution to the energy estimates. This is not only a formal calculation, as we prove that the normal form transformation is bounded between Sobolev spaces.

Funder

Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni

Publisher

Springer Science and Business Media LLC

Subject

Analysis

Reference35 articles.

1. Arosio, A.: Averaged evolution equations. The Kirchhoff string and its treatment in scales of Banach spaces. In: 2nd Workshop on Functional-Analytic Methods in Complex Analysis (Trieste, 1993). World Scientific, Singapore

2. Arosio, A., Panizzi, S.: On the well-posedness of the Kirchhoff string. Trans. Am. Math. Soc. 348, 305–330 (1996)

3. Baldi, P.: Periodic solutions of forced Kirchhoff equations. Ann. Sci. Norm. Sup. Pisa, Cl. Sci. (5) 8, 117–141 (2009)

4. Baldi, P., Haus, E.: On the existence time for the Kirchhoff equation with periodic boundary conditions. Nonlinearity 33(1), 196–223 (2020)

5. Baldi, P., Haus, E.: Longer lifespan for many solutions of the Kirchhoff equation, preprint (arXiv:2007.03543)

Cited by 2 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Normal form and dynamics of the Kirchhoff equation;Bollettino dell'Unione Matematica Italiana;2022-12-29

2. Longer Lifespan for Many Solutions of the Kirchhoff Equation;SIAM Journal on Mathematical Analysis;2022-01-06

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