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On the wave turbulence theory of 2D gravity waves, I: deterministic energy estimates

Yu Deng, Alexandru D. Ionescu, Fabio Pusateri

TL;DR

This work lays the foundation for a rigorous wave turbulence program for two-dimensional gravity water waves by developping a deterministic framework that yields a quintic energy increment bounded by an $L^{\infty}$-based norm, overcoming derivative-loss obstacles inherent to quasilinear dynamics. It combines paralinearization, normal-form transforms, and a detailed treatment of quartic bulk terms via admissible symbols, with careful handling of resonances, including Benjamin-Feir sites. A long-term regularity theory is established for small data on large tori, using Strichartz-type decay within a $Z$-norm bootstrap to obtain lifespans up to $T_{R,\varepsilon} \gtrsim \varepsilon^{-3}\min(\varepsilon^{-3},R^{3/4})$. The results provide a rigorous stepping stone toward the derivation of wave kinetic equations in the large-box limit (to be pursued in a companion paper) and demonstrate how deterministic energy control and randomness propagation can be synergistically employed in a quasilinear dispersive setting. Collectively, the paper advances the mathematical understanding of wave turbulence for water waves by bridging high-regularity energy methods, paradifferential calculus, and long-time dispersive analysis.

Abstract

Our goal in this paper is to initiate the rigorous investigation of wave turbulence and derivation of wave kinetic equations (WKE) for water waves models. This problem has received intense attention in recent years in the context of semilinear models, such as semilinear Schrödinger equations or multi-dimensional KdV-type equations. However, our situation here is different since the water waves equations are quasilinear and the solutions cannot be constructed by iteration of the Duhamel formula due to unavoidable derivative loss. This is the first of two papers in which we design a new strategy to address this issue, in the context of 2D gravity waves.

On the wave turbulence theory of 2D gravity waves, I: deterministic energy estimates

TL;DR

This work lays the foundation for a rigorous wave turbulence program for two-dimensional gravity water waves by developping a deterministic framework that yields a quintic energy increment bounded by an -based norm, overcoming derivative-loss obstacles inherent to quasilinear dynamics. It combines paralinearization, normal-form transforms, and a detailed treatment of quartic bulk terms via admissible symbols, with careful handling of resonances, including Benjamin-Feir sites. A long-term regularity theory is established for small data on large tori, using Strichartz-type decay within a -norm bootstrap to obtain lifespans up to . The results provide a rigorous stepping stone toward the derivation of wave kinetic equations in the large-box limit (to be pursued in a companion paper) and demonstrate how deterministic energy control and randomness propagation can be synergistically employed in a quasilinear dispersive setting. Collectively, the paper advances the mathematical understanding of wave turbulence for water waves by bridging high-regularity energy methods, paradifferential calculus, and long-time dispersive analysis.

Abstract

Our goal in this paper is to initiate the rigorous investigation of wave turbulence and derivation of wave kinetic equations (WKE) for water waves models. This problem has received intense attention in recent years in the context of semilinear models, such as semilinear Schrödinger equations or multi-dimensional KdV-type equations. However, our situation here is different since the water waves equations are quasilinear and the solutions cannot be constructed by iteration of the Duhamel formula due to unavoidable derivative loss. This is the first of two papers in which we design a new strategy to address this issue, in the context of 2D gravity waves.
Paper Structure (38 sections, 22 theorems, 792 equations)

This paper contains 38 sections, 22 theorems, 792 equations.

Table of Contents

  1. Introduction
  2. Free boundary Euler equations and water waves
  3. Motivation and main ideas
  4. An acceptable quintic energy inequality
  5. Remarks
  6. Small data long-term regularity
  7. Remarks and main ideas
  8. Organization
  9. Preliminaries
  10. The Fourier transform and Littlewood-Paley projections
  11. Multipliers and associated operators
  12. Function spaces
  13. Elements of paradifferential calculus
  14. Paralinearization of the water waves system
  15. The Dirichlet-Neumann operator and the good variable
  16. ...and 23 more sections

Key Result

Theorem 1.1

Assume that $N_0\geq 6$, $N_1\in[4,N_0-2]$, $T_1\leq T_2\in\mathbb{R}$, and $(h,\phi)\in C([T_1,T_2]:H^{N_0}\times \dot{H}^{N_0,1/2})$ is a solution of the system gWW. Moreover, assume thatSee HW for the definition of the spaces $\dot{W}^{N,b}$ and Lemma DNmainpro for the definition of the good unkn for any $t\in[T_1,T_2]$, where $A\in (0,\infty)$ and $\varepsilon:[T_1,T_2]\to [0,\infty)$ is a con

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3: Hörmander-Mikhlin norm
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 31 more