Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schrödinger equations on &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ h\mathbb{Z} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;

Bernier, Joackim

doi:10.3934/dcds.2019131

Cited by 5 publications

References 14 publications

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

et al. 2021

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We consider the semi-linear beam equation on the d dimensional irrational torus with smooth nonlinearity of order n − 1 with n ≥ 3 and d ≥ 2. If ε ≪ 1 is the size of the initial datum, we prove that the lifespan Tε of solutions is O(ε −A(n−2) −) where A ≡ A(d, n) = 1 + 3 d−1 when n is even and A = 1 + 3 d−1 + max(4−d d−1 , 0) when n is odd. For instance for d = 2 and n = 3 (quadratic nonlinearity) we obtain Tε = O(ε −6 −), much better than O(ε −1), the time given by the local existence theory. The irrationality of the torus makes the set of differences between two eigenvalues of √ ∆ 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. Contents 1. Introduction 1 2. Small divisors 9 3. The Birkhoff normal form step 19 4. The modified energy step 30 5. Proof of Theorem 1 33 References 36 2010 Mathematics Subject Classification. 35Q35, 35Q53, 37K55. Key words and phrases. Lifespan for semi-linear PDEs, Birkhoff normal forms, modified energy, irrational torus. 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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