
5. Analysis of the kinetic wave equation

1. Well-posedness theory

The kinetic wave equation reads : $$i\partial_t n_k =\int n_{k_1}n_{k_2}n_{k_3}n_k\left( -\frac 1 {n_{k_1}} +\frac 1 {n_{k_2}}-\frac 1 {n_{k_3}} +\frac 1 {n_k} \right) \delta (k_1-k_2+k_3-k) \delta (\Omega (k_1,k_2,k_3,k) ,$$ where $k\in\mathbb{Z}^d$ is a wave number, and $\Omega (k,l,m,n)=|k|^2-|l|^2+|m|^2-|n|^2$ encodes the resonances of the underlying linear system.

Problem 5.1.

[Laure Saint-Raymond] Find the optimal space(s) in which the kinetic wave equation is locally well-posed, or at least locally well-posed around the Zakharov solution.
1. Remark. [Pierre Germain] Rely on the work by Escobedo and Velazquez.
• Large time behavior

Problem 5.2.

What can be said about the formation of condensates ? Is there a self-similar mechanism to be understood ?
1. Remark. [Laure Saint-Raymond] See the 2001 paper by Pomeau, Rica et al., in Physica D : “Dynamical formation of a Bose–Einstein condensate”.

Cite this as: AimPL: Mathematical questions in wave turbulence theory, available at http://aimpl.org/waveturb.