# 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

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topde 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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