Mathematical Control Theory and Analysis of Partial Differential Equations Coupled Across a Boundary Interface

在这个项目的一个组成部分中，研究人员将研究从数学上描述气体在壁腔内流动并与空腔边界的振动部分相互作用的方程。在这个数学框架内，研究人员将尝试确定是否可以通过故意放置作用在空腔壁上的力来有利地改变或“控制”气体流动和空腔壁位移。这一计划目标的实现将有助于理解和控制与高速飞机和天然气管道相关的气体流动。此外，还将研究最近导出的描述结构-流体相互作用的数学模型，在该模型中，弹性体完全浸入给定的流体中。这个新的数学模型与以前的结构-流体模型的不同之处在于，对新模型的解的数值模拟表明，相关的结构位移以及流体流场导致了比之前认为的更静态的行为。研究人员打算对观察到的这种结构-流体动力学的“稳定性”进行数学验证。这项工作的成功结束可能会增强对细胞动力学的理解：这种结构-流体相互作用模型被用来描述细胞内的核体，因为它与周围的粘性细胞质相互作用。此外，还将研究一个描述结构腔体内声波流动的数学模型，其中一个腔体的壁是柔性的。特别是，它的目的是确定这种结构模型中的声波是否会随着时间的推移而消失。这种数学环境下的波浪稳定性结论将对现实世界的控制工程产生影响：为了促进飞机机舱内部的静态声场，可以想象，该项目的稳定性结果可以在飞机机身设计中考虑在内。在实现项目目标的过程中，包括第一代大学生和女性在内的研究生将接受必要的定性分析和数值计算方面的培训。上述气体流动-空腔模型由三维可压缩Stokes或Navier-Stokes流动方程和描述腔壁柔性部分位移的弹性板方程耦合而成。该偏微分方程组受空腔边界控制项的约束。研究人员将确定在规定光滑度的边界控制项的影响下，此类受控(气体)流动结构PDE模型解的最优正则性。随后，他们打算解决与边界控制函数的适当空间相关的零能控性问题。此外，在上述结构-流体相互作用中，结构由“厚”层和“薄”层组成：厚层由三维波动方程描述，而薄层由二维波动方程描述，构成了厚波方程和三维流体组分方程的边界界面。研究人员打算确定这个3D波-2D波-3D热PDE系统的经典解至少表现出多项式的衰减率，这与数值观察到的结果是一致的。此外，还将考虑结构声学相互作用的有理衰减问题，其中有界域内的声波方程与边界界面上的耗散弹性方程相互作用。此外，还将开发一种算法，确认所获得的多项式比率是最优的。这一奖项反映了NSF的法定使命，并已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。

In one component of this project, the investigators will study equations which mathematically describe the flow of gas as it streams within a walled cavity and interacts with a vibrating portion of the cavity's boundary. Within this mathematical framework, the investigators will attempt to determine if the gas flow and cavity wall displacement can be favorably altered or "controlled" by means of the deliberate placement of forces which act on the cavity wall. The accomplishment of this program objective could lend insight into understanding and controlling the gas flows which are associated with high-speed aircraft and natural gas pipelines. Furthermore, a mathematical model which has recently been derived to describe a structure-fluid interaction in which an elastic body is wholly immersed within a given fluid will be studied. This new mathematical model differs from previous structure-fluid models, in that numerical simulations of solutions to the new model suggest that the associated structural displacements, as well as the fluid flow field, elicit a much more quiescent behavior than previously thought. The investigators intend to mathematically verify this observed "stability" for such structure-fluid dynamics. A successful conclusion to this work could potentially enhance the understanding of cellular dynamics: such structure-fluid interaction models have been invoked to describe a nuclear body, within a cell, as it interacts with the surrounding viscous cytoplasm. Moreover, a mathematical model which describes acoustic wave flow within a structural chamber, with one of the chamber walls being flexible, will be studied. In particular, it is intended to determine if the acoustic waves in this structural model will die out as time evolves. Such a conclusion of wave stability in this mathematical setting would have implications in real-world control engineering: With a view of promoting a quiescent acoustic field within the interior of an aircraft cabin, the stability results of the project could conceivably be taken into account in aircraft fuselage design. In the course of fulfilling the project objectives, graduate students, including first generation university students and women, will receive training in the necessary qualitative analysis and numerical computation. The gas flow-cavity model described above consists of the 3D compressible Stokes or Navier-Stokes flow equations coupled to an elastic plate equation which describes the displacements of the flexible portion of the cavity wall. This PDE system is subjected to cavity boundary control terms. The investigators will determine the optimal regularity of solutions to such controlled (gas) flow-structure PDE models under the influence of boundary control terms of prescribed smoothness. Subsequently, they intend to solve the associated null controllability problem with respect to the appropriate space of boundary control functions. Moreover, in the aforementioned structure-fluid interaction, the structure is composed of a "thick" and "thin" layer: the thick layer is described by a 3D wave equation, while the the thin layer is described by a 2D wave equation and constitutes the boundary interface between the thick wave equation and the 3D fluid component equation. The investigators intend to establish that classical solutions of this 3D wave-2D wave-3D heat PDE system manifest at least a polynomial rate of decay, which would be in line with what has been observed numerically. In addition, the rational decay problem for the structural acoustic interaction, in which an acoustic wave equation within a bounded domain interacts with a dissipative elastic equation across a boundary interface, will be considered. Moreover, an algorithm which will confirm that the polynomial rates obtained are optimal will be developed.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.