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Abstract In this article we show that the Euler equations, when linearized around a low frequency perturbation to Couette flow, exhibit norm inflation in Gevrey-type spaces as time tends to infinity. Thus, echo chains are shown to be a (secondary) linear instability mechanism. Furthermore, we develop a more precise analysis of cancellations in the resonance mechanism, which yields a modified exponent in the high frequency regime. This allows us, in addition, to remove a logarithmic constraint on the perturbations present in prior works by Bedrossian, Deng and Masmoudi, and to construct solutions which are initially in a Gevrey class for which the velocity asymptotically converges in Sobolev regularity but diverges in Gevrey regularity. 

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.