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A note on cascade flux laws for the stochastically-driven nonlinear Schrödinger equation

Jacob Bedrossian

Abstract

In this note we point out some simple sufficient (plausible) conditions for `turbulence' cascades in suitable limits of damped, stochastically-driven nonlinear Schrödinger equation in a $d$-dimensional periodic box. Simple characterizations of dissipation anomalies for the wave action and kinetic energy in rough analogy with those that arise for fully developed turbulence in the 2D Navier-Stokes equations are given and sufficient conditions are given which differentiate between a `weak' turbulence regime and a `strong' turbulence regime. The proofs are relatively straightforward once the statements are identified, but we hope that it might be useful for thinking about mathematically precise formulations of the statistically-stationary wave turbulence problem.

A note on cascade flux laws for the stochastically-driven nonlinear Schrödinger equation

Abstract

In this note we point out some simple sufficient (plausible) conditions for `turbulence' cascades in suitable limits of damped, stochastically-driven nonlinear Schrödinger equation in a -dimensional periodic box. Simple characterizations of dissipation anomalies for the wave action and kinetic energy in rough analogy with those that arise for fully developed turbulence in the 2D Navier-Stokes equations are given and sufficient conditions are given which differentiate between a `weak' turbulence regime and a `strong' turbulence regime. The proofs are relatively straightforward once the statements are identified, but we hope that it might be useful for thinking about mathematically precise formulations of the statistically-stationary wave turbulence problem.
Paper Structure (11 sections, 13 theorems, 121 equations)

This paper contains 11 sections, 13 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Necessary and sufficient conditions for full dissipation anomaly
  3. Preliminaries
  4. Proof of Theorem \ref{['thm:Inverse']}
  5. Proof of Theorem \ref{['thm:Direct']}
  6. Characterization of non-anomaly
  7. Proof of Theorems \ref{['thm:ICFail']} and \ref{['thm:DCFail']}
  8. Partial dissipation anomaly
  9. Remarks on the strong turbulence regime
  10. Discussion and conjectures
  11. Fourier analysis conventions and Littlewood-Paley

Key Result

Lemma 1.1

For all $\lambda,\sigma,\nu>$, all invariant measures $\mu$ of def:NLS satisfy the balance of wave-action dissipation and the balance of Hamiltonian energy dissipation In what follows we use the following terminologies

Theorems & Definitions (33)

  • Lemma 1.1: Balance of wave action and Hamiltonian dissipation KS04
  • Theorem 2.1: Inverse cascade
  • Theorem 2.2: Direct cascade of kinetic energy
  • Remark 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Remark 2.6
  • Remark 2.7
  • Example 2.8: Examples of admissible forcing
  • Definition 2.9
  • ...and 23 more