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

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: $$(D_t^2-iA\partial_{\alpha})\theta=i\frac{t}{\alpha}|D_t^2\zeta|^2D_t\theta+R$$ where $R$ behaves well in the energy estimate. As a key consequence, we derive the uniform bound $$\sup_{t\geq 0}\Big(\norm{D_t\zeta(\cdot,t)}_{H^{s+1/2}}+\norm{\zeta_{\alpha}(\cdot,t)-1}_{H^s}\Big)\leq C\epsilon,$$ 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.