Research Intiation Award: Long Time Behavior for Systems of Coupled Partial Differential Equations

研究启动奖为历史悠久的黑人学院和大学的初级和职业生涯中期教师提供支持，这些教师正在建立新的研究计划或重新定向和重建现有的研究计划。预计该奖项有助于进一步提高教师的研究能力和有效性，改善其所在机构的研究和教学，并让本科生参与研究经验。授予弗吉尼亚州立大学的奖项可能会在许多领域产生更广泛的影响。本项目的目的是研究给定适当初始构形的强耦合偏微分方程解的存在唯一性，并研究解的长时间行为。这些模型在航空航天工程、土木工程和环境科学等领域有着广泛的应用。本科生将获得研究经验，并将加强常微分方程和偏微分方程组的课程。该项目的目标是研究以具有物理意义的系统为中心的控制、优化和稳定性分析，这些系统由动态的相互作用的非均匀结构组成，其行为受非线性耦合偏微分方程组(PDE)的控制。这两个PDE组件作用于单独和相邻的介质。所考虑的两个具体模型是：(1)流体-结构相互作用(FSI)，其中模型由具有动态弹性的界面上耦合的Navier-Stokes方程组成；(2)结构声学相互作用(SAI)，在以弹性或热弹性壳体为柔性壁的声腔中。SAI模型由声波方程和壳方程的混合耦合组成，壳方程可能是非线性的。需要研究的控制理论问题是：(A)镇定，特别是FSI中不稳定平衡点的镇定和弱耗散下SAI的镇定；(B)适定性，特别是寻求适当的反馈控制，使具有移动界面的FSI的解是适定的。这两种模型都可以推广到其他结构，所发展的数学技术也可以应用于其他耦合系统。

Research Initiation Awards provide support for junior and mid-career faculty at Historically Black Colleges and Universities who are building new research programs or redirecting and rebuilding existing research programs. It is expected that the award helps to further the faculty member's research capability and effectiveness, improves research and teaching at his home institution, and involves undergraduate students in research experiences. The award to Virginia State University has potential broader impact in a number of areas. The goal of the project is to study the existence and uniqueness of solutions of systems of strongly coupled partial differential equations given an appropriate initial configuration and to study the long time behavior of the solutions. The models have wide applications in areas such as aerospace engineering, civil engineering, and environmental sciences. Undergraduate students will gain research experiences and courses in ordinary and partial differential equations will be enhanced.The goal of the project is to study the control, optimization and stability analysis centered on physically significant systems composed of dynamical "interactive" inhomogeneous structures, whose behavior is governed by nonlinear systems of coupled partial differential equations (PDEs). The two PDE-components act on separate and adjacent media. Two specific models under consideration are: (1) Fluid-structure interaction (FSI), where the model consists of the Navier Stokes equation coupled on the interface with dynamic elasticity; and (2) Structure acoustic interaction (SAI), in an acoustic chamber with an elastic or thermoelastic shell as a flexible wall. The SAI model consists of hybrid coupling between an acoustic wave equation and a shell equation which is possibly nonlinear. Control theoretic issues to be studied are: (a) stabilization, particularly stabilization of unstable equilibria in FSI and stabilization of SAI subject to weak dissipation; and (b) well-posedness, particularly seeking suitable feedback control such that the solution to FSI with moving interface is well-posed. Both models could be generalized to other structures where the developed mathematical technology could be applied to other coupled systems.