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

Yulin Pan, Xiaoxu Wu

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.

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

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 . We establish a rigorous upper bound of , 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 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 strong solutions to the gravity water kinetic equation for initial data in a suitably weighted space. Specifically, we prove that for any initial data in this class, the resulting strong solution strictly propagates the weighted regularity and conserves the fundamental physical properties of the kinetic model.
Paper Structure (10 sections, 30 theorems, 278 equations, 1 figure)

This paper contains 10 sections, 30 theorems, 278 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mathematical Setup and Main Result
  3. Comparison with the NLS Kinetic Model and our approach
  4. Outline of the proof.
  5. Structure of the Collision Kernel for Gravity Water Waves
  6. The T_k1,kk̂2,k3 + T_k2,k3k̂1,k contribution
  7. The T_k,k1k̂2,k3 + T_k3,k2k̂1,k contribution
  8. Dissipativity and Operator Bounds
  9. Proof of the Main Theorem
  10. Auxiliary Estimates

Key Result

Theorem 1.2

Assume the dimension is $d \geq 2$. For any initial data $f_0 \in \mathcal{B}$, there exists a local existence time $T = T(\|f_0\|_{2,{22+5d}},\|f_0\|_{\infty,{12+4d}}) > 0$, depending strictly on the weighted norms of the initial data, such that the gravity water wave kinetic equation 4wave admits

Figures (1)

  • Figure 1: Decomposition scheme and a priori estimates for the collision kernel.

Theorems & Definitions (62)

  • Remark 1
  • Definition 1.1: Strong $L^1$ Solutions
  • Theorem 1.2
  • Definition 1.3
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 52 more