A fundamental question in wave turbulence theory is to understand how the wave kinetic equation describes the long-time dynamics of its associated nonlinear dispersive equation. Formal derivations in the physics literature, dating back to the work of Peierls in 1928, suggest that such a kinetic description should hold (for well-prepared random data) at a large kinetic time scale $T_{\mathrm {kin}} \gg 1$ and in a limiting regime where the size L of the domain goes to infinity and the strength $\alpha $ of the nonlinearity goes to $0$ (weak nonlinearity). For the cubic nonlinear Schrödinger equation, $T_{\mathrm {kin}}=O\left (\alpha ^{-2}\right )$ and $\alpha $ is related to the conserved mass $\lambda $ of the solution via $\alpha =\lambda ^2 L^{-d}$ . In this paper, we study the rigorous justification of this monumental statement and show that the answer seems to depend on the particular scaling law in which the $(\alpha , L)$ limit is taken, in a spirit similar to how the Boltzmann–Grad scaling law is imposed in the derivation of Boltzmann’s equation. In particular, there appear to be two favourable scaling laws: when $\alpha $ approaches $0$ like $L^{-\varepsilon +}$ or like $L^{-1-\frac {\varepsilon }{2}+}$ (for arbitrary small $\varepsilon $ ), we exhibit the wave kinetic equation up to time scales $O(T_{\mathrm {kin}}L^{-\varepsilon })$ , by showing that the relevant Feynman-diagram expansions converge absolutely (as a sum over paired trees). For the other scaling laws, we justify the onset of the kinetic description at time scales $T_*\ll T_{\mathrm {kin}}$ and identify specific interactions that become very large for times beyond $T_*$ . In particular, the relevant tree expansion diverges absolutely there. In light of those interactions, extending the kinetic description beyond $T_*$ toward $T_{\mathrm {kin}}$ for such scaling laws seems to require new methods and ideas.

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NLS波动动力学方程的推导

波浪湍流理论的一个基本问题是理解波浪动力学方程如何描述其相关非线性色散方程的长期动力学。物理学文献中的形式推导可以追溯到 Peierls 在 1928 年的工作，表明这种动力学描述应该（对于精心准备的随机数据）在很大程度上适用动力学时间尺度 $T_{\mathrm {kin}} \gg 1$ 并且在一个限制机制中，其中的大小大号领域的无限和力量 $\阿尔法$ 的非线性去 $0$ （弱非线性）。对于三次非线性薛定谔方程， $T_{\mathrm {kin}}=O\left (\alpha ^{-2}\right )$ 和 $\阿尔法$ 与守恒质量有关 $\λ$ 通过解决方案 $\alpha =\lambda ^2 L^{-d}$ . 在本文中，我们研究了这一具有里程碑意义的陈述的严格论证，并表明答案似乎取决于特定的比例律其中 $(\alpha , L)$ 本着类似于玻尔兹曼方程的推导中施加玻尔兹曼-格雷德标度律的精神，采用了极限。特别是，似乎有二有利的比例定律：当 $\阿尔法$ 方法 $0$ 喜欢 $L^{-\varepsilon +}$ 或喜欢 $L^{-1-\frac {\varepsilon }{2}+}$ （对于任意小 $\伐普西隆$ )，我们展示了时间尺度上的波动动力学方程 $O(T_{\mathrm {kin}}L^{-\varepsilon })$ ，通过显示相关的费曼图展开绝对收敛（作为配对树的总和）。对于其他比例定律，我们证明动力学描述在时间尺度上的开始是正确的 $T_*\ll T_{\mathrm {kin}}$ 并确定在以后的时间里变得非常大的特定交互 $T_*$ . 特别是，相关的树扩展在那里绝对发散。鉴于这些相互作用，将动力学描述扩展到 $T_*$ 朝向 $T_{\mathrm {kin}}$ 因为这样的比例定律似乎需要新的方法和想法。