5. Analysis of the kinetic wave equation

Wellposedness theory
The kinetic wave equation reads : $$i\partial_t n_k =\int n_{k_1}n_{k_2}n_{k_3}n_k\left( \frac 1 {n_{k_1}} +\frac 1 {n_{k_2}}\frac 1 {n_{k_3}} +\frac 1 {n_k} \right) \delta (k_1k_2+k_3k) \delta (\Omega (k_1,k_2,k_3,k) ,$$ where $k\in\mathbb{Z}^d$ is a wave number, and $\Omega (k,l,m,n)=k^2l^2+m^2n^2$ encodes the resonances of the underlying linear system.Problem 5.1.
[Laure SaintRaymond] Find the optimal space(s) in which the kinetic wave equation is locally wellposed, or at least locally wellposed around the Zakharov solution.
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 selfsimilar mechanism to be understood ?
Remark. [Laure SaintRaymond] 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.