A New Structure for the 2D water wave equation: Energy stability and Global well-posedness

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: $$(D_t^2-iA{\partial }_{\alpha })\theta =i\frac{t}{\alpha }|D_t^2\zeta {|}^2{D}_t\theta +R$$ 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.