Global Stability of the Kolmogorov--Zakharov Spectrum for the Wave Kinetic Equation

Abstract

We prove global nonlinear stability near the mass Kolmogorov--Zakharov spectrum $f_M(ω)=ω^{-7/6}$ for the isotropic cubic wave kinetic equation, including small time-dependent external forces. For sufficiently small initial perturbations and forcing, the solution remains uniformly close to $f_M$ for all time in a weighted norm. This norm allows $f(t,ω)\sim a_0(t)ω^{-7/6}$ as $ω\to0$ and $f(t,ω)\sim a_\infty(t)ω^{-7/6}$ as $ω\to\infty$. When the external force $ϕ$ is independent of time, the solution converges at an algebraic rate to a stationary solution in weaker weighted norms. We also give a necessary and sufficient condition on the initial perturbation for convergence in the original weighted norm. These results resolve, for small perturbations of the Kolmogorov--Zakharov spectrum, the question of convergence to a stationary solution under time-independent forcing raised by Collot, Dietert, and Germain (arXiv:2208.00947, Section 1.3). To establish this stability, the collision integral is expanded around $a_0(t)f_M$ at low frequencies and $a_\infty(t)f_M$ at high frequencies, separating it into a linear term with time-dependent coefficients and a nonlinear remainder. This decomposition yields improved control of the nonlinear remainder. The main difficulty is that $a_0(t)$ may vary in time, so the linear operator itself changes with time and its evolution cannot be represented by the semigroup of a single fixed operator. To address this difficulty, we use a parametrix construction to explicitly construct the evolution operator $U_{a_0,a_\infty}$ and establish the required uniform bounds. Moreover, our arguments provide a general framework for proving global stability that may also apply to other evolution equations.