Dynamical Systems Approaches to Partial Differential Equations

NSF奖摘要-DMS-0103915数学科学：动力系统解偏微分方程式的方法摘要DMS-0103915这个项目使用动力系统理论的工具探索偏微分方程式的长期行为。所研究的方程既包括耗散系统，也包括哈密顿系统。作为前者的一个例子，韦恩教授将研究涡旋解附近的Navier-Stokes方程的长期行为。他将在这些方程的相空间中构造有限维不变流形，并用它们来研究方程解的长时间渐近性和涡旋解的存在性。在相关工作中，他将推导并严格证明波在流体表面上运动的近似方程，特别是研究碰撞波的相互作用。此外，他还将利用最初研究有限维、几乎可积的哈密顿系统的思想，研究用于弹性材料在薄域中运动的常用梁和板模型的有效性。最后，韦恩教授将与布朗大学的一位数学生物学家合作，研究神经组织模型中脉冲的存在和稳定性。正在研究的系统出现在许多应用中，包括材料科学、流体力学和生物学。尽管不可能明确地求解控制它们运动的方程，但应用程序至少需要对其解的行为有一个定性的了解。例如，描述弹性材料振动的方程是如此复杂，即使使用现代计算工具，它们的解也仍然非常耗时。因此，工程师们已经为这类系统的行为导出了许多近似模型，特别是在系统的一个维度比其他维度小得多的情况下，就像梁和板的情况一样。然而，关于这些模型实际模拟梁或板的真实行为的程度，人们知之甚少。这项研究的目的是对使用这些模型时出现的误差提供准确的估计，并开发一种算法，使人们能够系统地改进模型。同样，韦恩教授还将研究海洋上的海浪模型。最近人们认识到，高速渡轮的尾流可以产生足够大的孤立波，对海岸造成重大破坏。这些波在尾流中产生的机制尚不清楚。了解模型问题的孤立波与实际水波问题的解之间的关系可能有助于了解这类尾迹的产生和传播。

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.