Long-time existence for multi-dimensional periodic water waves

Ionescu, Alexandru D.; Pusateri, Fabio

doi:10.1007/s00039-019-00490-8

Cited by 34 publications

(22 citation statements)

References 58 publications

(88 reference statements)

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“…Concerning the normal form theory, and limiting ourselves to quasilinear PDEs on compact manifolds, we mention, in addition to the aforementioned papers of Delort [16,17], the abstract result of Bambusi [6], the recent literature on water waves by Craig and Sulem [13], Ifrim and Tataru [28], Ionescu and Pusateri [29,30], Berti and Delort [8], Berti, Feola and Pusateri [9,10], and the work by Feola and Iandoli [19] on the quasilinear NLS on T.…”

Section: Related Literaturementioning

confidence: 99%

On the Normal Form of the Kirchhoff Equation

Baldi

Haus

2020

J Dyn Diff Equat

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Consider 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.

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Section: Related Literaturementioning

confidence: 99%

On the Normal Form of the Kirchhoff Equation

Baldi

Haus

2020

J Dyn Diff Equat

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“…In the case of (1.7) on T d for d ≥ 2, this local time has been extended by Delort [13] and then improved in different ways by Fang and Zhang [19], Zhang [29] and Feola-Grébert-Iandoli [20] (in this last case a quasi linear Klein Gordon equation is considered). We quote also the remarkable work on multidimensional periodic water wave by Ionescu-Pusateri [26].…”

Section: Introductionmentioning

confidence: 98%

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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“…Unfortunately we have also a power of the highest frequency µ 1 which represents, in principle, a loss of derivatives. However, this loss of derivatives may be transformed in a loss of length of the lifespan through partition of frequencies, as done for instance in [10,17,12,6].…”

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confidence: 99%

“…In this case the equation may be not recasted as a semi-linear Schrödinger equation. Being a quasi-linear system, we expect that a para-differential approach, in the spirit of [17,12] should be applied. However, in this case, the quasi-linear term is quadratic, hence big.…”

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confidence: 99%

Long-time stability of the quantum hydrodynamic system on irrational tori

Feola,

Iandoli,

Murgante

2021

Preprint

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We consider the quantum hydrodynamic system on a d-dimensional irrational torus with d = 2, 3. We discuss the behaviour, over a "non-trivial" time interval, of the H s -Sobolev norms of solutions. More precisely we prove that, for generic irrational tori, the solutions, evolving form ε-small initial conditions, remain bounded in H s for a time scale of order O(ε −1−1/(d−1)+ ), which is strictly larger with respect to the time-scale provided by local theory. We exploit a Madelung transformation to rewrite the system as a nonlinear Schrödinger equation. We therefore implement a Birkhoff normal form procedure involving small divisors arising form three waves interactions. The main difficulty is to control the loss of derivatives coming from the exchange of energy between high Fourier modes. This is due to the irrationality of the torus which prevent to have "good separation" properties of the eigenvalues of the linearized operator at zero. The main steps of the proof are: (i) to prove precise lower bounds on small divisors; (ii) to construct a modified energy by means of a suitable high/low frequencies analysis, which gives an a priori estimate on the solutions.

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Long-time existence for multi-dimensional periodic water waves

Cited by 34 publications

References 58 publications

On the Normal Form of the Kirchhoff Equation

On the Normal Form of the Kirchhoff Equation

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

Long-time stability of the quantum hydrodynamic system on irrational tori

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