On the Occurrence of Resonance in Elastic-Dissipative Coupled Systems

GaldiDMS-1614011在过去的二十年里，粘性液体在刚性或变形体存在下的运动研究已经成为应用研究的主要焦点之一。该项目的目的是解决上述现象的重要方面，并旨在实现以下目标。第一种是关于粘性液体与弹性结构相互作用中的共振现象。例如，在动脉血流建模的背景下，最重要的是找出对于给定的模型，心脏泵血的脉动动作是否会产生不切实际的动脉壁高幅度振荡。第二个是关于粘性液体均匀流动中结构的涡激振动问题。这一现象被认为是风荷载作用下烟囱和桥梁倒塌的主要原因。从这个意义上说，最著名的事件可能是塔科马窄行桥的失败。从技术的角度出发，通过研究非线性双曲-抛物型方程耦合系统的时间周期解的存在性和Hopf分支的发生来解决上述问题。本项目的主要重点之一是了解耗散机制(系统的抛物线部分)是否能够防止弹性结构(双曲部分)发生共振。其中一些问题需要在无界的空间域中表达，如无限管道系统或有限数量物体外部的区域。后者可能会增加更多的复杂性，因为在这种情况下，线性化的相关算子可能具有非空的本质谱。

GaldiDMS-1614011 Over the past two decades, the study of the motion of a viscous liquid in the presence of rigid or deformable bodies has become one of the main focuses of applied research. The aim of this project is to address significant aspects of the above type of phenomenon and is directed toward the accomplishment of the following objectives. The first one regards occurrence of resonance phenomena in the interaction of a viscous liquid with an elastic structure. For example, in the context of modeling of arterial blood flow it is of the utmost importance to find out whether, for a given model, the pulsatile action of the heart pumping blood would produce an unrealistic high-amplitude oscillation of the arterial wall. The second one deals with the so-called vortex-induced oscillations of a structure in the uniform stream of a viscous liquid. This phenomenon is considered to play a major role in the collapse of chimneys and bridges under wind load. The most famous event in this sense is probably the failure of the Tacoma Narrows bridge. From a technical viewpoint, the above questions are addressed by studying the existence of time-periodic solutions and the occurrence of Hopf bifurcation in coupled systems of nonlinear hyperbolic-parabolic equations that model fluid-structure interactions. One of the main focuses of this project is to understand whether the dissipative mechanism (the parabolic component of the system) can prevent the occurrence of resonance in the elastic structure (the hyperbolic component). Some of these problems need to be formulated in unbounded spatial domains, like infinite pipeline systems or regions exterior to a finite number of bodies. The latter may add an even further complication, in that the linearized relevant operators may possess, in such cases, a non-empty essential spectrum.