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On the rigorous derivation of the wave kinetic equation for NLS Hani, Zaher


Wave turbulence theory conjectures that the long-time behavior of â generic" solutions of nonlinear dispersive equations is governed (at least over certain long timescales) by the so-called wave kinetic equation (WKE). This approximation is supposed to hold in the limit when the size L of the domain goes to infinity, and the strength \alpha of the nonlinearity goes to 0. We will discuss some recent progress towards settling this conjecture, focusing on a recent joint work with Yu Deng (USC), in which we show that the answer seems to depend on the â scaling lawâ with which the limit is taken. More precisely, we identify two favorable scaling laws for which we justify rigorously this kinetic picture for very large times that are arbitrarily close to the kinetic time scale (i.e. within $L^\epsilon$ for arbitrarily small $\epsilon$). This is similar to how the Boltzmann-Grad scaling law is imposed in the derivation of Boltzmann's equation. We also give counterexamples showing divergences for the complementary scaling laws.

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