论文标题
在存在几何形状和阻尼的情况下，重力毛细血管系统的局部适合性度
Local Well-Posedness of the Gravity-Capillary Water Waves System in the Presence of Geometry and Damping
论文作者
论文摘要
我们考虑域中的重力毛细血管问题$ω_t\ subset \ mathbb {t} \ times \ times \ mathbb {r} $具有实质性的几何特征。也就是说，我们考虑了可变的底部，流动的障碍物和恒定的背景电流。我们利用Ambrose等引入的涡流模型。 al。在Arxiv：2108.01786。我们表明，在这种几何环境中，水波问题是局部及时的，并研究了溶液的寿命。然后，我们添加一个阻尼项，并得出解释阻尼器的演化方程。最终，我们表明，相同的适应性和寿命结果适用于阻尼系统。我们主要利用能量方法。
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 and lifespan results apply to the damped system. We primarily utilize energy methods.
