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We study the mathematical properties of a kinetic equation which describesthe long time behaviour of solutions to the weak turbulence equation associatedto the cubic nonlinear Schr\"odinger equation. In particular, we give a precisedefinition of weak solutions and prove global existence of solutions for allinitial data with finite mass. We also prove that any nontrivial initial datumyields the instantaneous onset of a condensate, i.e. a Dirac mass at the originfor any positive time. Furthermore we show that the only stationary solutionswith finite total measure are Dirac masses at the origin. We finally constructsolutions with finite energy, which is transferred to infinity in aself-similar manner.

On the transfer of energy towards infinity in the theory of weak turbulence for the nonlinear Schr\"odinger equation