On the wave turbulence theory of 2D gravity waves, II: propagation of randomness

TL;DR

The paper proves long-time existence for 2D gravity water waves with randomized initial data on large tori by integrating energy methods with a probabilistic propagation framework. It introduces a two-tier normal-form/renormalization scheme and a sophisticated tree/couple/molecule counting apparatus to overcome derivative loss and resonances, achieving a time window $T_1\sim\epsilon^{-8/3+}$ with controlled Sobolev and $L^{\infty}$ bounds. This work solidifies a rigorous foundation for wave turbulence in a quasilinear, large-energy setting and outlines a blueprint for extending the approach to related dispersive-fluid models. The combination of deterministic energy control, probabilistic expansion, and combinatorial resonance analysis represents a substantial advance toward deriving wave-kinetic behavior from first principles in water waves.

Abstract

This is the second part of our work initiating the rigorous study of wave turbulence for water waves equations. We combine energy estimates, normal forms, and probabilistic and combinatorial arguments to complete the construction of long-time solutions with random initial data for the 2d (1d interface) gravity water waves system on large tori. This is the first long-time regularity result for solutions of water waves systems with large energy (but small local energy), which is the correct setup for applications to wave turbulence. Such a result is only possible in the presence of randomness.

Figures (5)

• Figure 1: An example of an irregular chain (Definition \ref{['defirrechain']}).
• Figure 2: An example of two congruent irregular chains (Definition \ref{['defcong']}) which are twists of each other. Here the signs of the involved nodes are shown in the picture.
• Figure 3: An example of a cut as in Definition \ref{['defcut']}.
• Figure 4: An example of a problematic scenario with $q=2$ and $\pi=(3,4,1,2)$. Here the leaves of the same color are paired (with the bracket after the leaf labels representing the number of its pair); for illustration, we also draw separate pictures that include the $\mathfrak m_j$ and $\mathfrak p_j$ for even and odd $j$ separately.
• Figure 5: The molecule $\mathbb M$ corresponding to the structure $\mathcal{R}$ in Figure \ref{['fig:counter']}. Here each atom $v_j$ corresponds to the I-branching node $\mathfrak n_j$, with each edge marked by the corresponding vector in the decoration. Each of the two double bonds is understood as consisting of two bonds of opposite direction.

Theorems & Definitions (86)

• Theorem 1.1
• Proposition 2.1: Normal form reduction
• Proposition 2.2
• proof
• Theorem 2.3: Energy increment
• Proposition 2.4: Propagation of randomness
• Proposition 2.5: Main Bootstrap
• proof
• Proposition 3.1
• proof : Proof of Proposition \ref{['properror1']}