Nonlinear Partial Differential Equations

Date: January 19, 2021

Time: 3:00PM - 4:00PM

Location: Zoom

Speaker: Zaher Hani, University of Michigan

Title: On the rigorous derivation of the wave kinetic equations

Abstract: Wave turbulence theory conjectures that the behavior of “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 “scaling law” 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 $\epsilon$). These two scaling laws are similar to how the Boltzmann-Grad scaling law is imposed in the derivation of Boltzmann's equation. We also give counterexamples showing certain divergences for other scaling laws. If time permits, we will also discuss some work in progress with J. Shatah (Courant) and S. Meyerson (Warwick) extending these results to more general dispersive relations and nonlinearities.