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## On the Zakharov’s weak turbulence theory for capillary waves

In this paper with M.-B. Tran, we construct solutions to the following weak turbulence kinetic equation for capillary waves (cf. Hasselmann ’62, Zakharov ’67):

\displaystyle \begin{aligned} \partial_tf + 2 \nu |k|^2 f \ = \ Q[f] \end{aligned}

describing the dynamics of wave density ${f(t,k)}$ at wavenumber ${k \in \mathbb{R}^d}$, ${d \ge 2}$. Here, ${\nu}$ denotes the positive coefficient of fluid viscosity, and ${Q[f]}$ is the collision term, describing pure resonant three-wave interactions:

$\displaystyle Q[f](k) \ = \ \iint \Big[ R_{k,k_1,k_2}[f] - R_{k_1,k,k_2}[f] - R_{k_2,k,k_1}[f] \Big] dk_1dk_2$

with ${R_{k,k_1,k_2} [f]= 4\pi |V_{k,k_1,k_2}|^2(f_1f_2-ff_1-ff_2)}$, with ${V_{k,k_1,k_2}}$ being the collision kernel. The integration is taken over resonant manifolds (of ${\mathbb{R}^{d-1}}$ dimension), defined by the resonant conditions

$\displaystyle k = k_1 + k_2 , \qquad \mathcal{E}_k = \mathcal{E}_{k_1} + \mathcal{E}_{k_2}$

with ${\mathcal{E}_k}$ denoting the dispersion relation of the waves. For capillary waves, ${\mathcal{E}_k = \sqrt{\sigma |k|^3}}$, ${\sigma}$ the surface tension.

According to the Zahkarov’s weak turbulence theory, the kinetic equation admits nontrivial equilibria ${f_\infty}$ so that ${Q[f_\infty] =0}$, which resembles the Kolmogorov spectrum of hydrodynamic turbulence describing the energy cascade. Such a stationary solution is often referred to as Kolmogorov-Zakharov spectra; see, for instance, the book of Nazarenko ’11. Several efforts have been made ever since its derivation in the 60’s, the fundamental question of existence and uniqueness of solutions to the kinetic equation remains unsolved. The aim of this paper is to provide a (radial) solution to this very question.