Linearized Wave Turbulence Convergence Results for Three-Wave Systems
Linearized Wave Turbulence Convergence Results for Three-Wave Systems
复制标题
三波系统的线性波湍流收敛结果
DOI：
10.1007/s00220-020-03799-w
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发表时间：
2018
影响因子：
2.4
通讯作者：
E. Faou
中科院分区：
文献类型：
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作者：
E. Faou
研究具有频率有界且几乎连续的随机确定性三波半线性系统。这种系统可以通过考虑大周期域上的非线性点阵动力学或截断偏微分方程得到。我们假设非线性很小，并且噪声很小或空洞，并且仅在傅里叶模式的角度（随机相位强迫）中起作用。我们考虑随机初始数据，并假设这些系统具有与波浪湍流理论中出现的波浪动力学方程的某些瑞利-琼斯平稳解相对应的自然不变分布。我们考虑用这些不变分布的扰动的概率律绘制的随机初始模态。在随机情况下，我们证明了在渐近极限（小非线性，连续频率集和小噪声）下，傅里叶模振幅的重归一化波动在弱意义上收敛于围绕这些瑞利-琼斯谱的线性化波动动力学方程的解。此外，我们证明了在没有噪声的情况下，具有相同随机初始条件的确定性方程在概率意义上满足一般的Birkhoff约简，至少在某些参数域中没有动力学描述。
We consider stochastic and deterministic three-wave semi-linear systems with bounded and almost continuous set of frequencies. Such systems can be obtained by considering nonlinear lattice dynamics or truncated partial differential equations on large periodic domains. We assume that the nonlinearity is small and that the noise is small or void and acting only in the angles of the Fourier modes (random phase forcing). We consider random initial data and assume that these systems possess natural invariant distributions corresponding to some Rayleigh–Jeans stationary solutions of the wave kinetic equation appearing in wave turbulence theory. We consider random initial modes drawn with probability laws that are perturbations of theses invariant distributions. In the stochastic case, we prove that in the asymptotic limit (small nonlinearity, continuous set of frequency and small noise), the renormalized fluctuations of the amplitudes of the Fourier modes converge in a weak sense towards the solution of the linearized wave kinetic equation around these Rayleigh–Jeans spectra. Moreover, we show that in absence of noise, the deterministic equation with the same random initial condition satisfies a generic Birkhoff reduction in a probabilistic sense, without kinetic description at least in some regime of parameters.