Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition (complete version)

Dymov, Andrey; Kuksin, Sergei

doi:10.48550/arxiv.1907.02279

Cited by 15 publications

(45 citation statements)

References 9 publications

(30 reference statements)

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“…The proof of Theorems 1.2, 1.3 and 1.4 are based on the framework introduced in [9] to prove Theorem 1.1. The latter work came as a culmination of an extensive research effort over the past years to provide a rigorous justification of the wave kinetic equation starting from the corresponding nonlinear dispersive PDE as a first principle [21,3,14,11,12,13,8,6,7]. This is Hilbert's sixth problem for waves; its particle analog is the rigorous derivation of the Boltzmann equation from Newtonian mechanics (see [17,20,15] and references therein).…”

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confidence: 99%

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Propagation of chaos and the higher order statistics in the wave kinetic theory

Deng¹,

Hani²

2021

Preprint

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This manuscript continues and extends in various directions the result in [9], which gave a full derivation of the wave kinetic equation (WKE) from the nonlinear Schrödinger (NLS) equation in dimensions d ≥ 3. The wave kinetic equation describes the effective dynamics of the second moments of the Fourier modes of the NLS solution at the kinetic timescale, and in the kinetic limit in which the size of the system diverges to infinity and the strength of the nonlinearity vanishes asymptotically according to a specified scaling law.Here, we investigate the behavior of the joint distribution of these Fourier modes and derive their effective limit dynamics at the kinetic timescale. In particular, we prove propagation of chaos in the wave setting: initially independent Fourier modes retain this independence in the kinetic limit. Such statements are central to the formal derivations of all kinetic theories, dating back to the work of Boltzmann (Stosszahlansatz). We obtain this by deriving the asymptotics of the higher Fourier moments, which are shown to be given by solutions of the so-called wave kinetic heirarchy (WKH) with factorized initial data, hence giving a rigorous justification of the latter system. We treat both Gaussian and non-Gaussian initial distributions. In the Gaussian setting, we prove propagation of Gaussianity as we show that the asymptotic distribution retains the Gaussianity of the initial data in the limit. In the non-Gaussian setting, we derive the limiting equations for the higher order moments, as well as for the density function (PDF) of the solution. Some of the results we prove were conjectured in the physics literature, others appear to be new. This gives a complete description of the statistics of the solutions in the kinetic limit.

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confidence: 99%

“…Note that the works [3,6,7,8,9,11,12,13,21] concern cubic nonlinearities or 4-wave interactions, while the works [1,14,26] concern quadratic nonlinearities or 3-wave interactions. Both models represent a lot of important physical scenarios.…”

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confidence: 99%

Propagation of chaos and the higher order statistics in the wave kinetic theory

Deng¹,

Hani²

2021

Preprint

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“…T d ds |s − x| d+2α−2 (∆ s g)(t, x, s, r, v, r, ṽ) = c β,d g(t, x, r, v) dr dṽ (r − r)(−∆ x ) α g(t, x, r, ṽ) =: −1 d,α Σ g (t, x, r) g(t, x, r, v) , and deduce from (39) the Vlasov-type equation ∂ t g(t, x, r, v) + v∂ r g(t, x, r, v) + C 1 [g, g](t, x, r, v) = 0, (…”

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

confidence: 84%

“…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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“…In addition, to tackle the long standing open conjecture about the long time behavior of high Sobolev norms H s ps ą 1q of solutions of dispersive equations on the torus, discussed in the works of Bourgain [1] and Colliander, Staffilani, Keel, Takaoka, Tao [8], the rigorous justification of wave turbulence theory has been revisited during the last few years in [2,6,7,9,10,15,16,17,18,19,20,22,26,27,48].…”

Section: Introductionmentioning

confidence: 99%

A Numerical Scheme for Wave Turbulence: 3-Wave Kinetic Equations

Walton¹,

Tran²

2021

Preprint

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We introduce a finite volume scheme to solve isotropic 3-wave kinetic equations. We test our numerical solution against theoretical results concerning the long time behavior of the energy and observe that our solutions verify the energy cascade phenomenon. Up to our knowledge, this is the first numerical scheme that could capture the long time asymptotic behavior of solutions to isotropic 3-wave kinetic equations, where the energy cascade can be observed. Our numerical energy cascade rates are in good agreement with the theoretical one obtained by . Our finite volume algorithm relies on a new identity, that allows one to reduce the number of terms needed to be approximated in the collision operators.

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Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition (complete version)

Cited by 15 publications

References 9 publications

Propagation of chaos and the higher order statistics in the wave kinetic theory

Propagation of chaos and the higher order statistics in the wave kinetic theory

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

A Numerical Scheme for Wave Turbulence: 3-Wave Kinetic Equations

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