On the derivation of the wave kinetic equation for NLS
On the derivation of the wave kinetic equation for NLS
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DOI：
10.1017/fmp.2021.6
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发表时间：
2019-12
期刊：
影响因子：
--
通讯作者：
Yu Deng;Z. Hani
中科院分区：
文献类型：
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作者：
Yu Deng;Z. Hani
波浪湍流理论中的一个基本问题是了解波动运动方程如何描述其相关的非线性色散方程的长时间动力学。追溯到1928年Peierls的工作，物理学文献中的形式推导表明，这样的动力学描述(对于准备充分的随机数据)应该在大的动力学时间尺度$T{\mathm{kin}}\gg1$并且在区域的大小L到无穷大而非线性的强度$\α$到$0$(弱非线性)的极限区域内成立。对于三次非线性薛定谔方程，$T{mathm{kin}}=O\Left(\α^{-2}\right)$和$\α$通过$α=\lambda^2 L^{-d}$与解的守恒质量$\lambda$有关.在这篇文章中，我们研究了这一不朽论断的严格合理性，并表明答案似乎取决于取$(\α，L)$极限的特定标度律，其精神类似于在推导玻尔兹曼方程时如何强加玻尔兹曼-格拉德标度律。特别地，存在两个有利的标度律：当$α$逼近$0时，如$L^{-\varepsilon+}$或$L^{-1-\varepsilon}{2}+}$(对于任意小的$\varepsilon$)，我们证明了直到时间尺度$O(T_{mathm{kin}}L^{-\varepsilon})的波动动力学方程是绝对收敛的(作为成对树上的和)。对于其他标度律，我们证明了动力学描述在时间尺度$T_*\ll T_{\mathm{kin}}$的开始，并确定了在超过$T_*$的时间内变得非常大的特定相互作用。特别是，相关的树木扩张在那里绝对是不同的。鉴于这些相互作用，将这种标度定律的动力学描述从$T_*$扩展到$T_{\mathm{kin}}$似乎需要新的方法和想法。
Abstract 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.