Mathematical Sciences: Approximation of the Global Attractors of Evolution Equations

数学科学：进化方程全局吸引子的近似

• 批准号: 9007802
• 负责人: Michael Jolly
• 金额: $ 9.5万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-06-01 至 1993-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9007802&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Approximation; Global; Attractors

With this award the principal investigator will use analytical and numerical tools to study the long-time behavior of solutions of certain dissipative partial differential equations such as the Kuramoto-Sivashinsky equation and the Navier-Stokes equations. Of particular interest is the approximation of a set called the global attractor, which is a finite-dimensional, fractal set that lies inside of an infinite- dimensional phase space. In order to describe the global attractor the principal investigators plan to construct an algebraic approximation method that uses smooth finite- dimensional manifolds to approximate the fractal set and then to implement this method on a computer. From an abstract point of view one can regard the solutions of a partial differential equation as generating a space of trajectories or curves that is infinite-dimensional. This is one reason why the study of such equations is difficult. Fortunately, though, within this infinite-dimensional space reside sets of finite dimension that characterize how solutions behave for very large times. And since these sets are finite- dimensional, it is possible to describe them approximately using numerical and analytical techniques. With this award the researchers will study finite-dimensional objects called global attractors that arise in the solution sets of equations from fluid dynamics.

这一奖项的主要研究者将使用分析和数值工具来研究某些耗散偏微分方程解的长期行为，例如Kuramoto-Sivashinsky方程和Navier-Stokes方程。特别令人感兴趣的是一种称为全局吸引子的集合的近似，它是位于无限维相空间内的有限维的分形集。为了描述全局吸引子，主要研究人员计划构造一种代数逼近方法，该方法使用光滑的有限维流形来逼近分形集，然后在计算机上实现该方法。从抽象的观点来看，人们可以将偏微分方程解视为生成无限维的轨迹或曲线的空间。这就是研究这类方程困难的原因之一。然而，幸运的是，在这个无限维空间中，存在着有限维的集合，这些集合表征了解在非常大的时间内的行为。由于这些集合是有限维的，因此可以使用数值和分析技术对其进行近似描述。有了这个奖项，研究人员将研究在流体动力学方程解集中出现的称为全局吸引子的有限维对象。