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

TL;DR

This paper investigates the long-term behavior of solutions to a kinetic equation related to weak turbulence in the nonlinear Schrödinger equation, revealing phenomena like condensate formation and energy transfer to infinity.

Contribution

It provides a rigorous analysis of weak solutions, proves global existence, characterizes stationary solutions, and constructs solutions demonstrating energy transfer to infinity.

Findings

Existence of global weak solutions for all finite mass initial data.

Instantaneous formation of a condensate at the origin.

Construction of solutions with energy transferred to infinity in a self-similar way.

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 Schr\"odinger 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.