On the Transfer of Energy Towards Infinity in the Theory of Weak Turbulence for the Nonlinear Schrödinger Equation

• Published in: Journal of statistical physics 2015-05, Vol.159 (3), p.668-712
• Main Authors: Kierkels, A. H. M., Velázquez, J. J. L.
• Format: Article
• Language: English
• Subjects: Mathematical and Computational Physics; Physical Chemistry; Physics; Physics and Astronomy; Quantum Physics; Statistical Physics and Dynamical Systems; Theoretical
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-015-1194-0

We study the mathematical properties of a kinetic equation, derived in Escobedo and Velázquez ( arXiv:1305.5746v1 [math-ph]), which describes the long time behaviour of solutions to the weak turbulence equation associated to the cubic nonlinear Schrödinger equation. In particular, we give a precise definition of weak solutions and prove global existence of solutions for all initial data with finite mass. We also prove that any nontrivial initial datum yields the instantaneous onset of a condensate, by which we mean that for any nontrivial solution the mass of the origin is strictly positive for any positive time. Furthermore we show that the only stationary solutions with finite total measure are Dirac masses at the origin. We finally construct solutions with finite energy, where the energy is transferred to infinity in a self-similar manner.