Dynamical Systems Approaches to Partial Differential Equations

偏微分方程的动力系统方法

• 批准号: 0103915
• 负责人: Clarence Wayne
• 金额: --
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0103915&HistoricalAwards=false
• 关键词: Dynamical; Systems; Approaches; Partial; Differential

NSF Award Abstract - DMS-0103915Mathematical Sciences: Dynamical Systems Approaches to Partial Differential Equations AbstractDMS-0103915WayneThis project explores the long-time behavior of partial differential equations using tools from the theory of dynamical systems. The equations to be studied include both dissipative and Hamiltonian systems. As an example of the former, Professor Wayne will study the long-time behavior of the Navier-Stokes equations in the neighborhood of vortex solutions. He will construct finite dimensional invariant manifolds in the phase space of these equations and use them to study both the long-time asymptotics of solutions of the equations and the existence of vortex solutions for which no explicit formulas exist. In related work he will derive and rigorously justify approximate equations for the motion of waves on the surface of a fluid, studying in particular the interaction of colliding waves. In addition, by using ideas first developed to study finite dimensional, nearly integrable, Hamiltonian systems, he will investigate the validity of commonly used beam and plate models for motion of elastic materials in thin domains. Finally, in collaboration with a mathematical biologist at Brown University, Professor Wayne will study the existence and stability of pulses in models of neural tissue. The systems under study arise in many applications, including materials science, fluid mechanics, and biology. Even though it is impossible to solve the equations that govern their motion explicitly, applications require at least a qualitative understanding of the behavior of their solutions. For instance, the equations that describe vibrations of elastic materials are so complicated that their solution remains very time consuming, even with modern computational tools. Consequently, engineers have derived many approximate models for the behavior of such systems, particularly in situations where one dimension of the system is much smaller than others, as is the case for beams and plates. Very little is known rigorously, however, about how well these models actually mimic the true behavior of the beam or plate. This research aims both to provide accurate estimates of the errors that occur in using such models and to develop an algorithm that permits one to systematically improve the models. In a similar vein, Professor Wayne will also investigate models for waves on the ocean. Recently it has been realized that the wake of high-speed ferries can produce solitary waves of sufficient magnitude to cause significant damage at the shore. The mechanism by which these waves are created in the wake is not yet understood. Understanding the relationship between the solitary waves of the model problem and the solutions of the actual water wave problem may shed light on the creation and propagation of this type of wake.

NSF奖励摘要-DMS-0103915数学科学：偏微分方程的动力学系统方法AppperionDMS-0103915WayneThis项目探索了使用动态系统理论中工具的部分微分方程的长时间行为。 要研究的方程式包括耗散和哈密顿系统。 作为前者的一个例子，韦恩教授将研究Vortex Solutions附近Navier-Stokes方程的长期行为。 他将在这些方程式的相空间中构建有限的维数流形，并使用它们来研究方程解的长期渐近级，以及不存在明确公式的涡旋溶液的存在。 在相关工作中，他将得出并严格地证明在流体表面运动的运动方面的近似方程，特别是研究碰撞波的相互作用。 此外，通过使用最初开发的思想来研究有限的尺寸，几乎可以集成的哈密顿系统，他将研究常用的梁和板模型的有效性，用于薄域中弹性材料的运动。 最后，与布朗大学的数学生物学家合作，韦恩教授将研究神经组织模型中脉冲的存在和稳定性。所研究的系统在许多应用中都出现，包括材料科学，流体力学和生物学。 即使不可能明确地求解其运动的方程式，但应用程序至少需要对解决方案行为的定性理解。 例如，描述弹性材料振动的方程式非常复杂，即使使用现代计算工具，它们的解决方案也非常耗时。 因此，工程师为此类系统的行为提供了许多近似模型，尤其是在系统的一个维度比其他方面小得多的情况下，横梁和板的情况也是如此。 然而，关于这些模型实际上模仿了梁或板的真实行为，鲜为人知的知识很少。 这项研究旨在提供对使用此类模型中发生的错误的准确估计，并开发允许系统改进模型的算法。 同样，韦恩教授还将调查海洋波浪的模型。 最近，人们已经意识到，高速渡轮的唤醒可以产生足够幅度的孤立波，以在岸上造成重大损害。 尚不理解这些波的创建机制。 了解模型问题的孤独波与实际水波问题的解决方案之间的关系可能会揭示这种唤醒的创造和传播。