On the transfer of energy towards infinity in the theory of weak turbulence for the nonlinear Schrödinger equation

Abstract

We study the mathematical properties of a kinetic equation 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, i.e. a Dirac mass at the origin 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, which is transferred to infinity in a self-similar manner.

References in corpus (3)

• On the Boltzmann equation for diffusively excited granular media
• Weakly nonlinear Schrödinger equation with random initial data
• Self-similar solutions with fat tails for Smoluchowski's coagulation equation with locally bounded kernels

Cited by in corpus (6)

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• On the energy cascade of 3-wave kinetic equations: Beyond Kolmogorov-Zakharov solutions
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• Exact solutions for a coherent phenomenon of condensation in conservative Hamiltonian systems
• On self-similar solutions to a kinetic equation arising in weak turbulence theory for the nonlinear Schrödinger equation
• Multicomponent coagulation systems: existence and non-existence of stationary non-equilibrium solutions