Two Problems in Liquid-Solid Interaction

近几十年来，粘性液体与固体相互作用的运动研究已成为应用研究的主要热点之一。该项目的目标是解决这一领域的两个基本方面。第一个考虑的是刚体在粘性液体中的振动引起的运动，当振动是由刚体内部的时间周期性位移的质量产生的。这一现象的广泛应用领域包括生物医学工程和微纳米技术设备的设计。第二个方面涉及流动引起的振荡对弹簧安装结构的影响。这些类型的问题在悬索桥的平衡和稳定性研究中是最相关的，在某些情况下，它们的倒塌，就像臭名昭著的塔科马海峡大桥灾难。该奖项还将为研究生参与该项目的研究提供机会。这两个项目都属于非线性分析领域，它们的研究需要使用新的数学思想，这些思想可能也会被应用于非线性PDE的其他领域。首先研究了存在相关线性化算子连续谱的液固相互作用问题的Hopf分岔问题（局部和全局）。到目前为止，这个问题是一个完全未知的领域。本文提出的研究的另一个重要的数学特征是详细研究了Fredholm负指标的非线性椭圆算子（与Navier-Stokes-like边值问题有关）的范围。有趣的是，这种分析可能会导致对振动球体如何以及为什么能在粘性液体中运动的严格定量理解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Over the past decades, the study of the motion of a viscous liquid interacting with solid bodies has become one of the main focuses of applied research. The objective of this project is to address two fundamental aspects in this area. The first regards the vibration-induced motion of a rigid body in a viscous liquid, when the vibration is produced by a time-periodically displaced mass inside the body. The wide area of applications of this phenomenon includes biomedical engineering and design of micro- and nano-technological equipment. The second aspect concerns the effect of flow-induced oscillations on a spring-mounted structure. These type of questions are of the utmost relevance in the study of the equilibrium and stability of suspension bridges and, in some cases, of their collapse, like in the notorious Tacoma Narrows Bridge disaster. This award will also provide opportunities for the involvement of graduate students in the project's research.Both projects fall into the realm of non-linear analysis and their study requires the use of fresh mathematical ideas that, possibly, will be employed also in other areas of applied nonlinear PDE's. First and foremost is the investigation of (local and global) Hopf bifurcation in a liquid-solid interaction problem in presence of a continuum spectrum of the relevant linearized operator. To date, this question represents an entirely uncharted territory. Another and not less important mathematical feature of the proposed research consists in the detailed study of the range of nonlinear elliptic operators (related to Navier-Stokes-like boundary-value problems) that are Fredholm of negative index. Interestingly enough, this kind of analysis may lead to the rigorous quantitative understanding of how and why a vibrating sphere can move in a viscous liquid.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.