Public Research Seminar by Advanced Materials (AMAT) Thrust, HKUST(GZ) - 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
\begin{equation}\label{abstract:uniform}
    \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,
\end{equation}
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.
This is joint work with Siwei Wang (AMSS).