A New Structure for the 2D water wave equation: Energy stability and Global well-posedness
Qingtang Su, Siwei Wang
TL;DR
The paper gives a global-wellposedness result for 2D gravity water waves with infinite depth and a 1D interface under small, localized data by revealing a refined structure of the nonlinearity. It introduces two localization lemmas and a Transition-of-Derivatives method to recast the system into a form where the leading nonlinear term cancels in energy estimates, enabling uniform-in-time control of high-order energies. The authors establish an almost-conserved higher-order energy and derive decay in key norms, while allowing weaker low-frequency constraints on the initial data than prior works. This advances understanding of long-time behavior for water waves by combining a physical-space formulation with sharp harmonic-analytic tools. The approach leverages Wu’s coordinate framework, a cubic-type decomposition, and new derivative-transition mechanisms to achieve global well-posedness with quantitative energy and decay estimates.
Abstract
We study the two-dimensional gravity water waves with a one-dimensional interface with small initial data. Our main contributions include the development of two novel localization lemmas and a Transition-of-Derivatives method, which enable us to reformulate the water wave system into the following simplified structure: $$(D_t^2-iA\partial_α)θ=i\frac{t}α|D_t^2ζ|^2D_tθ+R$$ where $R$ behaves well in the energy estimate. As a key consequence, we derive the uniform bound $$ \sup_{t\geq 0}\Big(\norm{D_tζ(\cdot,t)}_{H^{s+1/2}}+\norm{ζ_α(\cdot,t)-1}_{H^s}\Big)\leq Cε, $$ which enhances existing global uniform energy estimates for 2D water waves by imposing less restrictive constraints on the low-frequency components of the initial data.