Local well-posedness of the capillary-gravity water waves with acute contact angles (Q6615224)

The manuscript contributes significantly to understanding water wave dynamics involving complex boundary interactions, a subject of considerable mathematical and physical interest. The research addresses the capillary-gravity water wave equations involving acute contact angles between the free water surface and a fixed boundary. Classical water wave studies have focused primarily on smooth boundaries, but recent research has recognised the complexities introduced by non-smooth boundaries, such as acute contact points. This study expands upon prior work by the authors [SIAM J. Math. Anal. 52, No. 5, 4861--4899 (2020; Zbl 1450.35219); Commun. Pure Appl. Math. 74, No. 2, 225--285 (2021; Zbl 1471.35239)], which considered very small contact angles (up to \(\pi/16\)), by analysing cases where the contact angle can be as large as \(\pi/2\). Increasing the angle range introduces considerable analytical complexity, particularly due to the additional singularities present in the elliptic systems that describe the fluid's behaviour near the contact points. This issue demands advanced mathematical techniques to adequately manage the influence of these singularities on the regularity of the solution.\N\NTo investigate this problem, the authors employ a combination of geometric reformulation and singular decomposition techniques to manage the elliptic problem's singular nature near the corners. By rephrasing the water wave equations geometrically, they derive a priori energy estimates for the system. The primary method involves decomposing the solution into singular and regular parts, enabling the derivation of energy bounds despite the challenging boundary conditions. The authors extend previous results by establishing energy estimates not only for the velocity field but also for the curvature of the free surface. A key component of their methodology involves using the maximal regularity available for the velocity and singular decomposition techniques to isolate and handle the most problematic aspects of the solution near the contact points. Through rigorous analysis, they also construct an iterative scheme to verify that their energy estimates lead to a locally well-posed system.\N\NThe main findings of this study demonstrate that, under certain conditions, the capillary-gravity water wave system with acute contact angles is locally well-posed in a geometric framework. Specifically, the authors show that their a priori energy estimates allow the existence of a solution within a limited time interval, which depends on the initial data. These results extend the local well-posedness of water wave systems to cases where the contact angles are acute, significantly broadening the theoretical foundation of capillary-gravity waves and laying the groundwork for further exploration of more complex boundary geometries.