Optimal local well-posedness theory for the kinetic wave equation

Germain, Pierre; Ionescu, Alexandru D.; Tran, Minh Binh

doi:10.1016/j.jfa.2020.108570

Cited by 32 publications

(38 citation statements)

References 69 publications

(73 reference statements)

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“…Besides the 3-wave kinetic equation, the rigorous study of 4-wave kinetic equation is a very important subject (see [20,12,11,25,26] and references therein). Finally, we note that wave-turbulence kinetic equations have very similar form with the quantum Boltzmann equations, used to describe the evolution of diluted bose gases at high and low temperature.…”

Section: Introductionmentioning

confidence: 99%

A reaction network approach to the theory of acoustic wave turbulence

Tran¹,

Crăciun²,

Smith³

et al. 2020

Journal of Differential Equations

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We propose a new approach to study the long time dynamics of the wave kinetic equation in the statistical description of acoustic turbulence. The approach is based on rewriting the discrete version of the wave kinetic equation in the form of a chemical reaction network, then employing techniques used to study the Global Attractor Conjecture to investigate the long time dynamics of the newly obtained chemical system. We show that the solution of the chemical system converges to an equilibrium exponentially in time. In addition, a resonance broadening modification of the acoustic wave kinetic equation is also studied with the same technique. For the near-resonance equation, if the resonance broadening frequency is larger than a threshold, the solution of the system goes to infinity as time evolves.

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Section: Introductionmentioning

confidence: 99%

A reaction network approach to the theory of acoustic wave turbulence

Tran¹,

Crăciun²,

Smith³

et al. 2020

Journal of Differential Equations

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“…A precise meaning can be given to the Dirac functions used above (cf. [11]), and W plays the role of B in the Boltzmann equation. In this context the variable v represents a wave vector, which is more usually denoted k or p, however to keep a uniform notation throughout the paper we stick with the letter v.…”

Section: Introductionmentioning

confidence: 99%

Rigorous Study of the Equilibria of Collision Kernels Appearing in the Theory of Weak Turbulence

Breden

Desvillettes

2019

Arch Rational Mech Anal

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In this paper, we rigorously obtain all the equilibria of collision kernels of type "two particles give two particles" appearing in weak turbulence theory under very general assumptions, thus completing the "equality case" in Boltzmann's H-theorem for those models. We also provide some rigorous results for collision kernels of type "two particles give one particle", under assumptions which include some of the most classical kernels of this type. The method of proof is inspired by the quantitative estimates obtained for the Landau equation in [8].

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“…During the last 10 years, quantum kinetic theory has also become as an important topic with a lot of interest (see [44,43,10,6,7,14,9,16,17,20,37,1,46,51,11,26,54,47] and references therein). According to the theory, the density function of the thermal cloud satisfies a quantum Boltzmann equation and the wave function of the condensate follows the nonlinear Gross-Pitaevskii equation.…”

Section: Introductionmentioning

confidence: 99%

On coupling kinetic and Schrödinger equations

Soffer

Tran

2018

Journal of Differential Equations

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We consider in this paper a system coupling a linear quantum Boltzmann equation and a defocusing cubic nonlinear Schrödinger equation. The Schrodinger equation reflects the dynamics of the wave function of the Bose-Einstein Condensate and the kinetic part of the system describes the evolution of the density function of the thermal cloud. An existence and uniqueness result for the system is supplied. We also prove the convergence to equilibrium of the density function of the thermal cloud and a scattering theory for the wave function of the condensate.

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Optimal local well-posedness theory for the kinetic wave equation

Cited by 32 publications

References 69 publications

A reaction network approach to the theory of acoustic wave turbulence

A reaction network approach to the theory of acoustic wave turbulence

Rigorous Study of the Equilibria of Collision Kernels Appearing in the Theory of Weak Turbulence

On coupling kinetic and Schrödinger equations

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