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

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Abstract

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.

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Acknowledgments

We thank B. Niethammer for comments that helped to clarify the structure of self-similar 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 3.4. 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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  1. Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    A. H. M. Kierkels & J. J. L. Velázquez

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  1. A. H. M. Kierkels
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  2. J. J. L. Velázquez
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Correspondence to A. H. M. Kierkels.

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Kierkels, A.H.M., Velázquez, J.J.L. On the Transfer of Energy Towards Infinity in the Theory of Weak Turbulence for the Nonlinear Schrödinger Equation. J Stat Phys 159, 668–712 (2015). https://doi.org/10.1007/s10955-015-1194-0

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  • DOI: https://doi.org/10.1007/s10955-015-1194-0

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