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

该项目旨在对某些流体结构相互作用(FSI)偏微分方程(PDE)系统进行定性和定量分析。这种耦合的PDE系统在数学上描述了各种生物现象，例如包膜血管壁结构内的血流相互作用。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液动力，会发生巨大的变形。动脉壁和不可压缩血流之间的这种相互作用是通过一种复合(多层)FSI PDE在数学上实现的，该PDE正是本项目所考虑的类型。此外，该项目还将重点介绍FSI PDE模型，这些模型通常由耦合到可压缩或不可压缩的Stokes或Navier Stokes流体的弹性动力学组成，这些模型已知可以描述土木工程中看到的各种现象：例如，流体或气体流动与置换弹性膜的相互作用，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发展一种连续的数值逼近理论，该理论与流体流动PDE组件表现出Navier-Stokes非线性以及当平板PDE组件发出非线性时相关。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领域的研究，该项目将包括K-12推广活动。这项研究需要开发新的方法来解决多层FSI系统解的存在性、唯一性、长期行为和数值逼近等问题。项目研究的一部分目的是：(I)建立新的数学方法来确定由耦合到不可压缩流体流动的多层弹性方程组成的FSI解的存在和唯一性；(Ii)确定这种解的定性行为，包括随着时间的演变获得最优有理衰减率的可能性。(I.i)和(I.ii)中的这些结果可以为哺乳动物血液运输过程中动脉壁变形引起的动脉瘤的发生率和病理提供定性的见解。此外，项目研究的目的是：(Ii)提供内在新颖的混合变分公式，以获得描述土木工程中某些现象的耦合(可压缩和不可压缩)Navier-Stokes-全非线性Kirchoff板FSI系统的解；(Ii)在物理上相关的情况下，构建与所述FSI PDE系统的边界控制相关的数学控制理论。这个项目期望(II)中提到的关于适定性的混合变分方法将产生可实现的多层FSI系统解的数值逼近格式，比现有文献中的收敛速度更快，并且具有更少的计算成本。此外，(Ii)中提到的边界可控性项目工作符合某些流体机械应用；即，边界控制法的目的是在其3D腔内诱导流体速度的混合，最终达到非混沌或静止的流动状态。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.