Local Well-Posedness of the Gravity-Capillary Water Waves System in the Presence of Geometry and Damping

TL;DR

This paper proves local well-posedness of the gravity-capillary water waves system with complex geometry and damping, extending previous models and analyzing solution lifespan using energy methods.

Contribution

It establishes local well-posedness and lifespan results for water waves with variable bottom, obstacles, background current, and damping, in a geometric setting.

Findings

Water waves system is locally well-posed in complex geometric domains.

Damping does not affect the local well-posedness and lifespan results.

Energy methods effectively analyze the system's behavior.

Abstract

We consider the gravity-capillary water waves problem in a domain $\Omega_t \subset \mathbb{T} \times \mathbb{R}$ with substantial geometric features. Namely, we consider a variable bottom, smooth obstacles in the flow and a constant background current. We utilize a vortex sheet model introduced by Ambrose, et. al. in arXiv:2108.01786. We show that the water waves problem is locally-in-time well-posed in this geometric setting and study the lifespan of solutions. We then add a damping term and derive evolution equations that account for the damper. Ultimately, we show that the same well-posedness and lifespan results apply to the damped system. We primarily utilize energy methods.