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

2022/01/12 by Gary Moon, Moon, Gary
Earth and Planetary Sciences · Mathematics · #35Q31 #35Q35 #76B03 #76B15 #76B45 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Ocean Waves and Remote Sensing

paper · pdf · doi:10.48550/arxiv.2201.04713

openalex publication_date 2022/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the gravity-capillary water waves problem in a domain Ω_t ⊂ \mathbbT × 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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