Dynamical Systems Approaches to Partial Differential Equations

NSF奖摘要- DMS-0103915数学科学：偏微分方程的动力系统方法摘要DMS-0103915韦恩这个项目使用动力系统理论的工具探索偏微分方程的长期行为。 要研究的方程包括耗散和哈密顿系统。 作为前者的一个例子，韦恩教授将研究纳维尔-斯托克斯方程在涡解附近的长期行为。 他将建设有限维不变流形在相空间中的这些方程，并利用它们来研究长期渐近的解决方案的方程和存在的旋涡解决方案，其中没有明确的公式存在。 在相关的工作，他将推导和严格证明近似方程的运动波表面的流体，特别是研究相互作用的碰撞波。 此外，通过使用思想首先开发研究有限维，近可积，哈密顿系统，他将调查的有效性常用的梁和板模型运动的弹性材料在薄域。 最后，韦恩教授将与布朗大学的数学生物学家合作，研究神经组织模型中脉冲的存在和稳定性。所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，但应用至少需要对它们的解的行为有定性的了解。 例如，描述弹性材料振动的方程是如此复杂，以至于即使使用现代计算工具，它们的解仍然非常耗时。 因此，工程师们已经推导出了许多近似模型的行为，这样的系统，特别是在情况下，一个维度的系统是远远小于其他的，如梁和板的情况。 然而，关于这些模型实际上模拟梁或板的真实行为的程度，我们所知甚少。 本研究的目的是提供准确的估计，在使用这些模型中发生的错误，并开发一种算法，允许系统地改善模型。 同样，韦恩教授也将研究海洋上的波浪模型。 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，从而在海岸造成重大损害。 这些波在尾流中产生的机制还不清楚。 了解模型问题的孤立波和实际水波问题的解之间的关系，可以揭示这种类型的尾流的产生和传播。

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.