On the Zakharov–L’vov Stochastic Model for Wave Turbulence

Dymov, Andrey; Kuksin, Sergei

doi:10.1134/s1064562420020106

Cited by 16 publications

(12 citation statements)

References 8 publications

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“…The setting. In this paper we continue the study of the Zakharov-L'vov stochastic model for wave turbulence (WT), initiated in [6,7]; see also a survey [8]. We start by recalling the classical and the Zakharov-L'vov stochastic settings of WT.…”

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confidence: 99%

The large-period limit for equations of discrete turbulence

Andrey¹,

Kuksin²,

Maiocchi³

et al. 2021

Preprint

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We consider the damped/driven cubic NLS equation on the torus of a large period L with a small nonlinearity of size λ, a properly scaled random forcing and dissipation. We examine its solutions under the subsequent limit when first λ → 0 and then L → ∞. The first limit, called the limit of discrete turbulence, is known to exist, and in this work we study the second limit L → ∞ for solutions to the equations of discrete turbulence. Namely, we decompose the solutions to formal series in amplitude and study the second order truncation of this series. We prove that the energy spectrum of the truncated solutions becomes close to solutions of a damped/driven nonlinear wave kinetic equation. Kinetic nonlinearity of the latter is similar to that which usually appears in works on wave turbulence, but is different from it (in particular, it is non-autonomous). Apart from tools from analysis and stochastic analysis, our work uses two powerful results from the number theory.

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confidence: 99%

The large-period limit for equations of discrete turbulence

Andrey¹,

Kuksin²,

Maiocchi³

et al. 2021

Preprint

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“…(39) If, in addition, we make the molecular chaos assumption g(t, x, s, r, v, r, ṽ) = g(t, x, r, v)g(t, s, r, ṽ), (40) then,…”

Section: Collective Behavior Kinetic Models Of Discrete Non-local Wav...mentioning

confidence: 99%

“…During the last few years, there has been a growing interests in rigorously understanding those kinetic equations. Starting with the pioneering work of Lukkarinen and Spohn [63], there have been a lot of recent works in in rigorously deriving WK equations (see, for instance [5,18,19,25,26,34,35,38,39,40,41,79] and the references therein). The analysis of WK and QK equations is also a topic of current interest.…”

Section: Introductionmentioning

confidence: 99%

Wave turbulence and collective behavior models for wave equations with short- and long-range interactions

Aceves¹,

Alonso²,

Tran³

2021

Preprint

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In this work, we discuss a situation which could lead to both wave turbulence and collective behavior kinetic equations. The wave turbulence kinetic models appear in the kinetic limit when the wave equations have local differential operators. Viewing wave equations on the lattice as chains of anharmonic oscillators and replacing the local differential operators (shortrange interactions) by non-local ones (long-range interactions), we arrive at a new Vlasov-type kinetic model in the mean field limit under the molecular chaos assumption reminiscent of models for collective behavior in which anharmonic oscillators replace individual particles. Dedicated to the 85th birthday of Professor Roland GlowinskiContents 1. Introduction 1 2. Wave turbulence kinetic models for discrete nonlinear wave equations with short-range interactions 4 3. Collective behavior kinetic models of discrete non-local wave equations with long-range interactions 7 References 10

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“…Works that rigorously derives the 4-wave kinetic equations out of statistical equilibrium from the cubic NLS equation with random initial data have been carried out by Buckmaster-Germain-Hani-Shatah [7,8], Deng-Hani [16,17], and Collot-Germain [13,14]. Works that try to derive the 4-wave kinetic equation from the stochastic cubic nonlinear Schrödinger equation (NLS) have been written by Dymov, Kuksin and collaborators in [18,19,20,21].…”

Section: Introductionmentioning

confidence: 99%

On the wave turbulence theory: ergodicity for the elastic beam wave equation

Rumpf¹,

Soffer²,

Tran³

2021

Preprint

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Inspiring by a recent work [56], we analyse a 3-wave kinetic equation, derived from the elastic beam wave equation on the lattice. The ergodicity condition states that two distinct wavevectors are supposed to be connected by a finite number of collisions. In this work, we prove that once the ergodicity condition is violated, the domain is broken into disconnected domains, called no-collision and collisional invariant regions. If one starts with a general initial condition, whose energy is finite, then in the long-time limit, the solutions of the 3-wave kinetic equation remain unchanged on the no-collision region and relax to local equilibria on the disjoint collisional invariant regions. This behavior of 3-wave systems was first described by Spohn in [54], without a detailed rigorous proof. Our proof follows Spohn's physically intuitive arguments.

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On the Zakharov–L’vov Stochastic Model for Wave Turbulence

Cited by 16 publications

References 8 publications

The large-period limit for equations of discrete turbulence

The large-period limit for equations of discrete turbulence

Wave turbulence and collective behavior models for wave equations with short- and long-range interactions

On the wave turbulence theory: ergodicity for the elastic beam wave equation

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