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Long-Time Existence for Semi-linear Beam Equations on Irrational Tori

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Abstract

We consider the semi-linear beam equation on the d dimensional irrational torus with smooth nonlinearity of order \(n-1\) with \(n\ge 3\) and \(d\ge 2\). If \(\varepsilon \ll 1\) is the size of the initial datum, we prove that the lifespan \(T_\varepsilon \) of solutions is \(O(\varepsilon ^{-A(n-2)^-})\) where \(A\equiv A(d,n)= 1+\frac{3}{d-1}\) when n is even and \(A= 1+\frac{3}{d-1}+\max (\frac{4-d}{d-1},0)\) when n is odd. For instance for \(d=2\) and \(n=3\) (quadratic nonlinearity) we obtain \(T_\varepsilon =O(\varepsilon ^{-6^-})\), much better than \(O(\varepsilon ^{-1})\), the time given by the local existence theory. The irrationality of the torus makes the set of differences between two eigenvalues of \(\sqrt{\Delta ^2+1}\) accumulate to zero, facilitating the exchange between the high Fourier modes and complicating the control of the solutions over long times. Our result is obtained by combining a Birkhoff normal form step and a modified energy step.

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Notes

  1. Actually there are papers in which such procedure is iterated. We quote for instance [15] and reference therein.

  2. When you have a control of the small divisors involving only \(\mu _3(j)\) then you can solve the homological equation at any order and you obtain an almost global existence result in the spirit of [2]. This would be the case if we consider the semi-linear beam equation on the squared torus \({\mathbb {T}}^d\).

  3. Notice that there is no resonant term of odd order by Proposition 2.2, in other words \(Z_3=Z_5=0\).

  4. In fact in Sect. 4, for the sake of simplicity, we prefer to apply a modified energy strategy to all the terms of \(Q_6\) (see also Remark 1.2).

  5. i.e. \(\forall k,\ell \in \llbracket 1,r \rrbracket , \ k\ne \ell \Rightarrow \ j_{k}\ne j_{\ell }\).

  6. see Definition 2.1.

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Authors and Affiliations

  1. Laboratoire de Mathématiques Jean Leray, Université de Nantes, UMR CNRS 6629, 2, rue de la Houssinière, 44322, Nantes Cedex 03, France

    Joackim Bernier, Roberto Feola & Benoît Grébert

  2. Laboratoire Jacques-Louis Lions, Sorbonne Université, UMR CNRS 7598, 4, Place Jussieu, 75005, Paris Cedex 05, France

    Felice Iandoli

Authors
  1. Joackim Bernier
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  2. Roberto Feola
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Correspondence to Benoît Grébert.

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Communicated by Yingfei Yi.

In memory of Walter Craig whose beautiful voice, always relevant and friendly, we miss.

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Felice Iandoli has been supported by ERC grant ANADEL 757996. Roberto Feola, Joackim Bernier and Benoit Grébert have been supported by the Centre Henri Lebesgue ANR-11-LABX- 0020-01 and by ANR-15-CE40-0001-02 “BEKAM” of the ANR.

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Bernier, J., Feola, R., Grébert, B. et al. Long-Time Existence for Semi-linear Beam Equations on Irrational Tori. J Dyn Diff Equat 33, 1363–1398 (2021). https://doi.org/10.1007/s10884-021-09959-3

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  • DOI: https://doi.org/10.1007/s10884-021-09959-3

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