On the Occurrence of Resonance in Elastic-Dissipative Coupled Systems

在过去的二十年里，研究粘性液体在刚体或可变形物体中的运动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。第一个问题是关于粘性液体与弹性结构相互作用时共振现象的发生。例如，在动脉血流建模的背景下，最重要的是要找出，对于给定的模型，心脏泵血的脉动作用是否会产生不现实的动脉壁高振幅振荡。第二种方法处理所谓的粘性液体均匀流中结构的涡激振荡。这种现象被认为是烟囱和桥梁在风荷载作用下倒塌的主要原因。在这个意义上，最著名的事件可能是塔科马海峡大桥的失败。从技术角度出发，通过研究模拟流固耦合作用的非线性双曲-抛物方程耦合系统的时间周期解的存在性和Hopf分岔的存在性来解决上述问题。该项目的主要重点之一是了解耗散机制（系统的抛物线分量）是否可以防止弹性结构（双曲线分量）发生共振。其中一些问题需要在无界的空间域中表述，比如无限的管道系统或有限数量物体外的区域。后者可能增加进一步的复杂性，因为在这种情况下，线性化的相关运算符可能具有非空的基本谱。

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.