A Toy Model for Damped Water Waves

Models for damped water waves

In this paper we derive some new weakly nonlinear asymptotic models describing viscous waves in deep water with or without surface tension effects. These asymptotic models take into account several different dissipative effects and are obtained from the free boundary problems formulated in the works of Dias, Dyachenko and Zakharov (Physics Letters A, 2008), Jiang, Ting, Perlin and Schultz (Journal of Fluid Mechanics,1996) and Wu, Liu and Yue (Journal of Fluid Mechanics, 2006).

Turbulent cascades for a family of damped Szegö equations

In this paper, we study the transfer of energy from low to high frequencies for a family of damped Szegö equations. The cubic Szegö equation has been introduced as a toy model for a totally non-dispersive degenerate Hamiltonian equation. It is a completely integrable system which develops growth of high Sobolev norms, detecting transfer of energy and hence cascades phenomena. Here, we consider a two-parameter family of variants of the cubic Szegö equation and prove that adding a damping term unexpectedly promotes the existence of turbulent cascades. Furthermore, we give a panorama of the dynam…

Long-Time Asymptotic Behavior of Dissipative Boussinesq System

In this paper, we study various dissipative mechanics associated with the Boussinesq systems which model two-dimensional small amplitude long wavelength water waves. We will show that the decay rate for the damped one-directional model equations, such as the KdV and BBM equations, holds for some of the damped Boussinesq systems which model two-directional waves.

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

We consider the gravity-capillary water waves problem in a domain $Ω_t \subset \mathbb{T} \times \mathbb{R}$ 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 a…

浅水方程的行波问题：适定性与粘性及表面张力消失极限

The traveling wave problem for the shallow water equations: well-posedness and the limits of vanishing viscosity and surface tension

本文研究带阻尼浅水系统的孤立行波解，发展了小数据适定性理论，并利用 Nash-Moser 隐函数定理证明行波解在有无粘性或表面张力及所有非平凡速度下均持续存在，且在消失极限下保持强范数连续性。

分数阶波动方程的阻尼及其在水波中的应用

Damping for fractional wave equations and applications to water waves

本文研究阻尼分数阶水波线性模型，针对足够正则初值建立并显式给出能量衰减率，在部分模型上首次证明表面波能量衰减。