On the derivation of the wave kinetic equation for NLS

Abstract

AbstractA 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 largekinetic time scale$T_{\mathrm {kin}} \gg 1$and in a limiting regime where the sizeLof 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}$.In this paper, we study the rigorous justification of this monumental statement and show that the answer seems to depend on the particularscaling lawin which the$(\alpha , L)$limit is taken, in a spirit similar to how the Boltzmann–Grad scaling law is imposed in the derivation of Boltzmann’s equation. In particular, there appear to betwofavourable scaling laws: when$\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.