Global Stability of the Kolmogorov--Zakharov Spectrum for the Wave Kinetic Equation
Abstract
We prove global nonlinear stability near the mass Kolmogorov--Zakharov spectrum fM(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 fM for all time in a weighted norm. This norm allows f(t,ω)∼ a0(t)ω−7/6 as ω→0 and f(t,ω)∼ a∞(t)ω−7/6 as ω→∞. 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 a0(t)fM at low frequencies and a∞(t)fM 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 a0(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 Ua0,a∞ 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.
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BibTeX
@misc{wang2026global,
title = {{Global Stability of the Kolmogorov--Zakharov Spectrum for the Wave Kinetic Equation}},
author = {Jiajun Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34359},
eprint = {2610.34359},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34359},
}
RIS
TY - UNPB
TI - Global Stability of the Kolmogorov--Zakharov Spectrum for the Wave Kinetic Equation
AU - Jiajun Wang
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34359
UR - https://arxiv.org/abs/2610.34359
AB - 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)∼ a_0(t)ω^−7/6 as ω→0 and f(t,ω)∼ a_∞(t)ω^−7/6 as ω→∞. 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_∞(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_∞ 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.
ER -