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Title of Journal: J Stat Phys

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Abbravation: Journal of Statistical Physics

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Springer US

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DOI

10.1002/pssa.2210540246

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1572-9613

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On the Transfer of Energy Towards Infinity in the

Authors: A H M Kierkels J J L Velázquez
Publish Date: 2015/01/31
Volume: 159, Issue: 3, Pages: 668-712
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Abstract

We study the mathematical properties of a kinetic equation derived in Escobedo and Velázquez arXiv13055746v1 mathph 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 selfsimilar mannerWe thank B Niethammer for comments that helped to clarify the structure of selfsimilar solutions to problems with multiple conserved quantities and for remarks concerning the final form of this paper We also thank the anonymous reviewer who pointed out the continuity statement for the measure of the origin which is contained in Proposition 34 The authors acknowledge support through the CRC 1060 The mathematics of emergent effects at the University of Bonn that is funded through the German Science Foundation DFG

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