On the derivation of the wave kinetic equation for NLS. (23rd July 2021) - Deng, Yu; Hani, Zaher
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_kin 1 and in a limiting regime where the size L of the domain goes to infinity and the strength α of the nonlinearity goes to 0 (weak nonlinearity). For the cubic nonlinear Schrödinger equation, T_kin=O(α ^-2) and α is related to the conserved mass λ of the solution via α =λ ^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 (α, 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 α approaches 0 like L^-ε + or like L^-1-ε 2+ (for arbitrary small ε ), we exhibit the wave kinetic equation up to time scales O(T_kinL^-ε ), 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_*T_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_kin for such scaling laws seems to require new methods and ideas.