On the persistence of decorrelation in the theory of wave turbulence

We study the statistical properties of the solutions of the KadomstevPetviashvili equations (KP-I and KP-II) on the torus when the initial datum is a random variable. We give ourselves a random variable u0 with values in the Sobolev space Hs 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 eiθ for all θ ∈ R. We investigate about the persistence of the decorrelation between the Fourier coefficients (un(t))n of the solutions of KP-I or KP-II with initial datum u0 in the sense that we estimate the expectations E(unum) 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 ωk the dispersion relation, whether ωk + ωl − ωk+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(unum) 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.