Control Theory, Qualitative Analysis, and Approximation of Coupled Structure-Flow Interaction Systems

该项目旨在发展某些流体结构相互作用（FSI）偏微分方程（PDE）系统的定性和定量分析。这种耦合的PDE系统在数学上描述了各种生物现象，例如封装血管壁结构内血流的相互作用。特别是，哺乳动物的血管壁是由粘弹性材料组成的，由于血液运输过程中产生的血流动力学力，血管壁会发生很大的变形。这种动脉壁和不可压缩血流之间的相互作用在数学上通过复合（多层）FSI PDE来实现，正是本项目所考虑的类型。此外，该项目将重点关注FSI PDE模型，这些模型通常由弹性动力学与可压缩或不可压缩的Stokes或Navier Stokes流体流动耦合组成，已知用于描述土木工程中看到的各种现象：例如，流体或气体流动与位移弹性膜的相互作用，以及桥梁和高层建筑等结构的空气动力学。对于这样的FSI动力学，项目研究的重点将是开发一个连续的数值近似理论，该理论与流体流动PDE分量表现出Navier-Stokes非线性以及板PDE分量产生非线性的情况有关。该项目还将为本科生和研究生提供研究培训机会。此外，为了促进STEM领域的学习，该项目将包括K-12外展活动。这项研究需要开发新的方法来解决多层FSI系统的存在性、唯一性、长期行为和数值近似解的问题。项目研究的一部分旨在：（I.i）建立新的数学方法，以确定由多层弹性方程与不可压缩流体流动耦合组成的FSI解的存在性和唯一性；（I.ii）确定这些解的定性行为，包括随着时间的推移获得最优理性衰减率的可能性。（I.i）和（I.ii）的结果可以为哺乳动物血液运输过程中由动脉壁变形引起的动脉瘤的发生率和病理提供定性的认识。此外，该项目的研究目标是：（II.i）提供本质上新颖的混合变分公式，以获得耦合（可压缩和不可压缩）Navier stokes -完全非线性Kirchoff板FSI系统的解，该系统描述了土木工程中的某些现象；（II.ii）构建一个与上述FSI PDE系统的边界控制相关的数学控制理论，在物理相关的情况下，边界控制在板块组件中是活跃的。该项目预计，（II.i）中提到的适位性的混合变分方法将产生可实现的多层FSI系统解的数值近似方案，其收敛速度比现有文献中的更快，计算成本更低。此外，（二。二）中提到的边界可控性项目工作与某些流体力学应用是一致的；也就是说，边界控制律的目的是诱导三维腔室内流体速度的混合，最终达到非混沌或静止的流动状态。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。

The project aims to develop qualitative and quantitative analysis of certain fluid structure interaction (FSI) partial differential equation (PDE) systems. Such coupled PDE systems mathematically describe various biological phenomena, such as interactions of blood flow within encapsulating blood vessel wall structures. In particular, mammalian blood vascular walls, being composed of viscoelastic materials, undergo large deformations due to the hemodynamic forces generated during the blood transport process. This interaction between arterial walls and incompressible blood flow is mathematically realized by a composite (multilayered) FSI PDE precisely of the type under consideration in this project. In addition, this project will focus on those FSI PDE models, generally composed of elastic dynamics coupled to compressible or incompressible Stokes or Navier Stokes fluid flows, that are known to describe a variety of phenomena seen in civil engineering: for instance, the interaction of fluid or gas flows with displacing elastic membranes, and the aerodynamics of structures such as bridges and tall buildings. For such FSI dynamics, the focus of the project research will be the development of a continuous and numerical approximation theory, relevant to cases when the fluid flow PDE component manifests the Navier-Stokes nonlinearity, as well as when nonlinearities emanate from the plate PDE component. This project will also provide research training opportunities for both undergraduate and graduate students. Moreover, to promote study in the STEM fields, this project will include K-12 outreach activities.This research entails the development of novel methodologies to address issues of existence, uniqueness, longtime behavior, and numerical approximation of solutions to multilayered FSI systems. Part of the project research aims to: (I.i) establish novel mathematical methodologies to determine the existence and uniqueness of solutions to those FSI that consist of multilayered elastic equations coupled to incompressible fluid flows; (I.ii) ascertain the qualitative behavior of such solutions, including the possibility of obtaining optimal rates of rational decay, as time evolves. Such results in (I.i) and (I.ii) could provide qualitative insight concerning the incidence and pathology of those aneurysms caused by arterial wall deformations during the mammalian blood transportation process. Moreover, the project research aims to: (II.i) provide intrinsically novel mixed variational formulations to obtain solutions of coupled (compressible and incompressible) Navier Stokes-fully nonlinear Kirchoff plate FSI systems which describe certain phenomena in civil engineering; (II.ii) construct a mathematical control theory relative to the boundary control of said FSI PDE systems, in the physically relevant case that boundary control is active in the plate component. This project anticipates that the mixed variational approaches to wellposedness noted in (II.i) will give rise to implementable numerical approximation schemes for the solutions of multilayered FSI systems, with faster convergence rates than those in the existing literature, and with less computational cost. Moreover, the boundary controllability project work noted in (II.ii) is consonant with certain fluid mechanical applications; namely, the intent of the boundary control law is to induce a mixing of the fluid velocity within its 3D chamber, to ultimately attain a nonchaotic or quiescent flow state.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.