On the persistence of decorrelation in the theory of wave turbulence

Anne-Sophie de Suzzoni[1]

  • [1] Université de Paris 13 Sorbonne Paris Cité F-93430 LAGA, UMR 7539 du CNRS, France

Journées Équations aux dérivées partielles (2013)

  • Volume: 38, Issue: 1-3, page 1-15
  • ISSN: 0752-0360

Abstract

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We study the statistical properties of the solutions of the Kadomstev-Petviashvili equations (KP-I and KP-II) on the torus when the initial datum is a random variable. We give ourselves a random variable u 0 with values in the Sobolev space H s with s big enough such that its Fourier coefficients are independent from each other. We assume that the laws of these Fourier coefficients are invariant under multiplication by e i θ for all θ ∈ ℝ . We investigate about the persistence of the decorrelation between the Fourier coefficients ( u n ( t ) ) n of the solutions of KP-I or KP-II with initial datum u 0 in the sense that we estimate the expectations E ( u n u m ¯ ) in function of time and the size ε of the initial datum. These estimates are sensitive to the presence or not of resonances in the three waves interaction, that is, denoting ω k the dispersion relation, whether ω k + ω l - ω k + l can be null (resonant model, KP-I) or not (non-resonant model, KP-II). In the case of a resonant equation, the expectations E ( u n u m ¯ ) remain small up to times of order o ( ε - 1 ) whereas in the case of a non-resonant equation, they do up to times of order o ( ε - 5 / 3 ) . The techniques used are different depending on the cases, we use Gronwall lemma and Gaussian large deviation estimates for the resonant case, and the normal form structure of KP-II in the other one.

How to cite

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de Suzzoni, Anne-Sophie. "On the persistence of decorrelation in the theory of wave turbulence." Journées Équations aux dérivées partielles 38.1-3 (2013): 1-15. <http://eudml.org/doc/275524>.

@article{deSuzzoni2013,
abstract = {We study the statistical properties of the solutions of the Kadomstev-Petviashvili equations (KP-I and KP-II) on the torus when the initial datum is a random variable. We give ourselves a random variable $u_0$ with values in the Sobolev space $H^s$ with $s$ big enough such that its Fourier coefficients are independent from each other. We assume that the laws of these Fourier coefficients are invariant under multiplication by $e^\{i\theta \}$ for all $\theta \in \mathbb\{R\}$. We investigate about the persistence of the decorrelation between the Fourier coefficients $(u_n(t))_n$ of the solutions of KP-I or KP-II with initial datum $u_0$ in the sense that we estimate the expectations $E(u_n \overline\{u_m\})$ in function of time and the size $\varepsilon $ of the initial datum. These estimates are sensitive to the presence or not of resonances in the three waves interaction, that is, denoting $\omega _k$ the dispersion relation, whether $\omega _k+\omega _l -\omega _\{k+l\}$ can be null (resonant model, KP-I) or not (non-resonant model, KP-II). In the case of a resonant equation, the expectations $E(u_n \overline\{u_m\})$ remain small up to times of order $o(\varepsilon ^\{-1\})$ whereas in the case of a non-resonant equation, they do up to times of order $o(\varepsilon ^\{-5/3\})$. The techniques used are different depending on the cases, we use Gronwall lemma and Gaussian large deviation estimates for the resonant case, and the normal form structure of KP-II in the other one.},
affiliation = {Université de Paris 13 Sorbonne Paris Cité F-93430 LAGA, UMR 7539 du CNRS, France},
author = {de Suzzoni, Anne-Sophie},
journal = {Journées Équations aux dérivées partielles},
keywords = {Wave turbulence; statistical equilibrium; random initial datum; invariant measures; nonlinear wave equation; Penrose transform},
language = {eng},
number = {1-3},
pages = {1-15},
publisher = {Groupement de recherche 2434 du CNRS},
title = {On the persistence of decorrelation in the theory of wave turbulence},
url = {http://eudml.org/doc/275524},
volume = {38},
year = {2013},
}

TY - JOUR
AU - de Suzzoni, Anne-Sophie
TI - On the persistence of decorrelation in the theory of wave turbulence
JO - Journées Équations aux dérivées partielles
PY - 2013
PB - Groupement de recherche 2434 du CNRS
VL - 38
IS - 1-3
SP - 1
EP - 15
AB - We study the statistical properties of the solutions of the Kadomstev-Petviashvili equations (KP-I and KP-II) on the torus when the initial datum is a random variable. We give ourselves a random variable $u_0$ with values in the Sobolev space $H^s$ with $s$ big enough such that its Fourier coefficients are independent from each other. We assume that the laws of these Fourier coefficients are invariant under multiplication by $e^{i\theta }$ for all $\theta \in \mathbb{R}$. We investigate about the persistence of the decorrelation between the Fourier coefficients $(u_n(t))_n$ of the solutions of KP-I or KP-II with initial datum $u_0$ in the sense that we estimate the expectations $E(u_n \overline{u_m})$ in function of time and the size $\varepsilon $ of the initial datum. These estimates are sensitive to the presence or not of resonances in the three waves interaction, that is, denoting $\omega _k$ the dispersion relation, whether $\omega _k+\omega _l -\omega _{k+l}$ can be null (resonant model, KP-I) or not (non-resonant model, KP-II). In the case of a resonant equation, the expectations $E(u_n \overline{u_m})$ remain small up to times of order $o(\varepsilon ^{-1})$ whereas in the case of a non-resonant equation, they do up to times of order $o(\varepsilon ^{-5/3})$. The techniques used are different depending on the cases, we use Gronwall lemma and Gaussian large deviation estimates for the resonant case, and the normal form structure of KP-II in the other one.
LA - eng
KW - Wave turbulence; statistical equilibrium; random initial datum; invariant measures; nonlinear wave equation; Penrose transform
UR - http://eudml.org/doc/275524
ER -

References

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  1. A.-S. de Suzzoni and N. Tzvetkov, On the propagation of weakly nonlinear random dispersive waves, Arch. Ration. Mech. Anal., 212, (2014), 849–874 Zbl1293.35287
  2. Anne-Sophie de Suzzoni, On the use of normal forms in the propagation of random waves, ArXiv e-prints (2013). Zbl1267.35130
  3. Jean-Marc Delort, Existence globale et comportement asymptotique pour l’équation de Klein-Gordon quasi linéaire à données petites en dimension 1, Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 1, 1–61. Zbl0990.35119MR1833089
  4. B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl. 15 (1970), 539-541. Zbl0217.25004
  5. A. J. Majda, D. W. McLaughlin, and E. G. Tabak, A one-dimensional model for dispersive wave turbulence, J. Nonlinear Sci. 7 (1997), no. 1, 9–44. Zbl0882.76035MR1431687
  6. Jalal Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math. 38 (1985), no. 5, 685–696. Zbl0597.35101MR803256
  7. N. Tzvetkov, Long time bounds for the periodic KP-II equation, Int. Math. Res. Not. (2004), no. 46, 2485–2496. Zbl1073.35195MR2078309
  8. V.E. Zakharov and N.N. Filonenko, Weak turbulence of capillary waves, Journal of Applied Mechanics and Technical Physics 8 (1967), no. 5, 37–40. 
  9. Vladimir Zakharov, Frédéric Dias, and Andrei Pushkarev, One-dimensional wave turbulence, Phys. Rep. 398 (2004), no. 1, 1–65. Zbl1003.76040MR2073490

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