气象和滑坡海啸的数学研究：Proudman共振
A mathematical study of meteo and landslide tsunamis : The Proudman resonance
摘要：
在本文中，我们想了解Proudman共振。当大气扰动的传播速度接近于典型的水波传播速度时，水体在浅水中对大气扰动的响应是一种共振响应。我们在这里表明，同样的共振存在滑坡海啸，我们提出了一种数学方法来调查这些现象的基础上的推导，理由和分析相关的渐近模型。这种方法使我们能够研究更复杂的现象，没有处理在物理学文献，如一个变量的底部或普劳德曼共振在更深的水域的推广的影响。首先，考虑小物理参数的依赖性，我们证明了具有移动底部和表面非定常压力的水波方程的局部适定性，并证明了这些方程是一个哈密顿系统（推广了Zakharov [33]的结果）.然后，我们证明了一些线性渐近模型，以研究Proudman共振和海底滑坡海啸，我们研究了线性水波方程和色散估计，使我们能够调查海平面的振幅。为了完成这些渐近模型，我们增加了一些数值模拟。
英文摘要：
In this paper, we want to understand the Proudman resonance. It is a resonant respond in shallow waters of a water body on a traveling atmospheric disturbance when the speed of the disturbance is close to the typical water wave velocity. We show here that the same kind of resonance exists for landslide tsunamis and we propose a mathematical approach to investigate these phenomena based on the derivation, justification and analysis of relevant asymptotic models. This approach allows us to investigate more complex phenomena that are not dealt with in the physics literature such as the influence of a variable bottom or the generalization of the Proudman resonance in deeper waters. First, we prove a local well-posedness of the water waves equations with a moving bottom and a non constant pressure at the surface taking into account the dependence of small physical parameters and we show that these equations are a Hamiltonian system (which extend the result of Zakharov [33]). Then, we justify some linear asymptotic models in order to study the Proudman resonance and submarine landslide tsunamis; we study the linear water waves equations and dispersion estimates allow us to investigate the amplitude of the sea level. To complete these asymptotic models, we add some numerical simulations.
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