International Research Fellowship Program: Hadamard Wellposedness and Asymptotic Stability of Finite Energy Solutions for a Structural Acoustic Interaction Modeled by Nonlinear

国际研究奖学金计划使美国科学家和工程师能够在国外进行9到24个月的研究。该计划的奖项提供了联合研究的机会，并利用国外独特或互补的设施、专业知识和实验条件。该奖项将支持Lorena Bociu博士为期24个月的研究奖学金，与索菲亚·安提波利斯Nice非线型研究所国家研究中心的Jean-Paul Zolsio博士以及法国巴黎Pole University Leonard de Vinci的John Cagnol博士合作。拟议的研究重点是具有曲面墙壁的非线性结构声学模型的有限能量解的适定性和稳定性。结构声相互作用由一组耦合方程来描述：波动方程和动态壳方程，前者模拟三维腔室中的声学介质，后者描述腔体的柔性壁。相应地，动力壳的运动可用一组双曲型的耦合非线性偏微分方程组来描述：面内位移为弹性波，标量法向位移为非线性Kirchhoff方程。结构声学模型由于其广泛的工程应用范围，在工程和数学文献中受到了极大的关注。然而，大多数分析都是在平壁的线性模型上执行的。该研究的主要创新之处在于，它将考虑曲壁(即完全非线性壳体)在与非线性声学介质(非线性波动方程)耦合时的非线性位移。因此，非线性(拓扑学)和几何方面都将是拟议研究的重点，欧几里得平面几何将被黎曼几何取代。更具体地说，在文献中被认为是公开问题的下列问题将被讨论：第一，对于具有非线性边界源的非线性壳：局部和整体(或有限时间爆破)，有限能量解的唯一性和正则性。其次，对于具有粘性阻尼的三维结构声学非线性模型，包括介质界面上的壳体：(I)由临界和超临界源驱动的有限能量解的Hadamard适定性，以及在存在边界(几何约束)阻尼的情况下解的稳定性，以及(Ii)足以确保有限能量解是全局的阻尼的非线性程度的量化。因此，阻尼的非线性是问题的核心。这一问题的解决不仅将为偏微分方程及其控制提供新的和重要的贡献，而且将具有将研究转移到基于工程的设计的深远潜力。该项目将使用M.Delfour和J.P.Zolsio开发的基于内在几何的动态壳模型，这为问题的解析表示提供了巨大的优势。主持人J.P.佐利西奥也是互动结构分析和控制方面的专家--这是提案中的一个主要主题。该项目还将受益于与J.Cagnol的密切互动，J.Cagnol在壳分析方面经验丰富，包括使用固有的基于几何的代码进行计算。这项拟议的研究将为一个在应用中至关重要的物理问题(声学环境中的噪声抑制)提供数学解决方案。它还应该刺激工程设计的新方法，最终影响社会。所提出的方法可以应用于其他具有共同性质的偏微分方程组模型：奇点传播、有限传播速度和超临界。此外，良好的适定性理论是控制理论方法应用的基础。

0802187BociuThe International Research Fellowship Program enables U.S. scientists and engineers to conduct nine to twenty-four months of research abroad. The program's awards provide opportunities for joint research, and the use of unique or complementary facilities, expertise and experimental conditions abroad.This award will support a twenty-four month research fellowship by Dr. Lorena Bociu to work with Dr. Jean-Paul Zolesio at Centre National de la Recherche Scientifique, Institut Non-Lineaire de Nice, Sophia Antipolis, and with Dr. John Cagnol, at Pole Universitaire Leonard de Vinci, in Paris, France.The proposed research is focused on wellposedness and stability of finite energy solutions to nonlinear structural acoustic models with curved walls. Structural acoustic interactions are described by a system of coupled equations: the wave equation, which models the acoustic medium in a 3-D chamber, and the dynamic shell equation, describing the flexible wall of the chamber. In turn, the motion of a dynamic shellis described by a set of coupled nonlinear partial differential equations, both of hyperbolic type: an elastic wave for the in-plane displacement, and a nonlinear Kirchhoff equation for the scalar normal displacement. Structural acoustic models, due to their large spectrum of engineering applications, have received a lot of attention in engineering and mathematical literature. However, most of the analysis has been performed on linear models with flat walls. The main novelty of the proposed research is that it will account for nonlinear displacements of the curved wall (i.e. fully nonlinear shell) in a coupling with a nonlinear acoustic medium (nonlinear wave equation). Thus, both nonlinear (topological) and geometric aspects will be at the focus of the proposed research, with Euclidean flat geometry being replaced by Riemannian geometry. More specifically, the following issues, recognized as open problems in the literature, will be addressed: First, for a nonlinear shell with nonlinear boundary sources: local and global existence (or blow-up in finite time), uniqueness and regularity of finite energy solutions. Second, for a 3-D structural acoustic nonlinear model with viscous damping and involving shells on the interface between the media: (i) Hadamard wellposedness of finite energy solutions driven by critical and supercritical sources, along with stability of solutions in the presence of boundary (geometrically restricted) damping, and (ii) quantification of the level of nonlinearity of the damping that is sufficient to ensure that finite energy solutions be global. Thus, nonlinearity of the damping is at the heart of the problem. The solution to this problem will not only provide a novel and important contribution to PDEs and their control, but will also have far-reaching potential for transferable research into engineering-based design. The project will use the dynamic shell model based on intrinsic geometry developed by M.Delfour and J.P.Zolesio, which offers great advantages for an analytic formulation of the problem. Host J.P.Zolesio is also an expert in analysis and control of interactive structures - a dominant theme in the proposal. The project will also benefit from strong interaction with J. Cagnol, who is well experienced in shell analysis, including computations with intrinsic geometry-based codes. The proposed research will provide a mathematical solution to a physical problem that is fundamental in application (noise suppression in an acoustic environment). It should also stimulate new approaches in engineering design, eventually impacting society. The proposed methods could be applied to other PDE models sharing common properties: propagation of singularities, finite speed of propagation and supercriticality. Moreover, good wellposedness theory is fundamental for control theory methods to be applied.