非线性浅水模型在具有时变气压的振荡水柱中的适定性
Well-posedness of a nonlinear shallow water model for an oscillating water column with time-dependent air pressure
AI总结：
本文提出含时变气压的振荡水柱非线性浅水模型，将其化为传输问题并证明局部适定性。
摘要：
本文提出了一种新的振荡水柱（OWC）非线性数学模型。在该装置存在的情况下，一维浅水方程被重新表述为一个与波浪和固定部分浸没结构相互作用相关的传输问题。通过施加无阻尼情形下流体-OWC总能量守恒，我们能够推导出一个传输条件，该条件涉及装置腔室内随时间变化的气压，而非如\cite{bocchihevergara2021}中那样的恒定大气压。随后我们证明，该传输问题可归结为一个拟线性双曲型初边值问题，其边界条件为半线性，由一个依赖于PDE解在边界上的迹的ODE决定。通过使用PDE的线性估计和ODE的非线性估计的迭代格式，我们建立了此类一般问题的局部适定性。
英文摘要：
We propose in this paper a new nonlinear mathematical model of an oscillating water column (OWC). The one-dimensional shallow water equations in the presence of this device is reformulated as a transmission problem related to the interaction between waves and a fixed partially-immersed structure. By imposing the conservation of the total fluid-OWC energy in the non-damped scenario, we are able to derive a transmission condition that involves a time-dependent air pressure inside the chamber of the device, instead of a constant atmospheric pressure as in \cite{bocchihevergara2021}. We then show that the transmission problem can be reduced to a quasilinear hyperbolic initial boundary value problem with a semi-linear boundary condition determined by an ODE depending on the trace of the solution to the PDE at the boundary. Local well-posedness for general problems of this type is established via an iterative scheme by using linear estimates for the PDE and nonlinear estimates for the ODE.
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