Local-in-Time Existence of $L^1$ solutions to the Gravity Water Wave Kinetic Equation

TL;DR

This paper proves the local-in-time existence of strong solutions in $L^1$ for the gravity water wave kinetic equation, by analyzing the collision kernel's behavior and establishing bounds that facilitate rigorous mathematical treatment.

Contribution

It provides a new upper bound on the collision kernel in a specific regime, improving previous estimates, and constructs $L^1$ strong solutions with preserved physical properties.

Findings

Established a rigorous upper bound of $oldsymbol{ ext{O}(|k||k_3|)}$ for the collision kernel.

Proved local-in-time existence of $L^1$ strong solutions for initial data in weighted $L^2 igcap L^ ext{infty}$ spaces.

Demonstrated conservation of physical properties such as energy and mass during solution evolution.

Abstract

In this paper, we study the Cauchy problem for the four-wave kinetic equation describing the weak turbulence of gravity water waves. The mathematical challenges of this analysis stem primarily from two interrelated aspects: (1) the extreme algebraic complexity of the collision kernel, where controlling its growth in the highly non-local regime constitutes the primary analytical bottleneck, and (2) the construction of strong solutions under the resulting singular integral operators. First, we re-analyze the interaction kernel in this precise regime, where the interacting wave numbers satisfy $|k|, |k_3| \gg |k_1|, |k_2|$. We establish a rigorous upper bound of $\mathcal{O}(|k||k_3|)$, which rigorously verifies the asymptotic smallness of the interaction coefficient anticipated in the physics literature \cite{zakharov2010energy, geogjaev2017numerical, geogjaev2025properties}. Furthermore, this result improves upon the recent $\mathcal{O}\big((|k||k_3|)^{3/2}\big)$ estimate proposed in \cite{waterkernel2024}, demonstrating a strictly milder singularity of wave interactions in this limit. Physically, this regime governs the energy exchange between disparate scales, such as the modulation of short gravity waves by long ocean swells. Second, leveraging this crucial integrability gain alongside a refined structural decomposition of the collision operator, we establish the local-in-time existence of $L^1$ strong solutions to the gravity water kinetic equation for initial data in a suitably weighted $L^2 \cap L^\infty$ space. Specifically, we prove that for any initial data in this class, the resulting $L^1$ strong solution strictly propagates the weighted $L^2 \cap L^\infty$ regularity and conserves the fundamental physical properties of the kinetic model.