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

Gary Moon

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arXiv · 2201.04713

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

Gary Moon

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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.

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Gary Moon. 2022-01-12. Local Well-Posedness of the Gravity-Capillary Water Waves System in the Presence of Geometry and Damping. https://arxiv.org/abs/2201.04713

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