Mathematical Sciences: Approximation of the Global Attractors of Evolution Equations

有了这笔奖金，首席研究员将使用分析和数值工具研究某些耗散偏微分方程（如Kuramoto-Sivashinsky方程和Navier-Stokes方程）解的长期行为。特别有趣的是一个叫做全局吸引子的集合的近似，它是一个有限维的分形集合，位于无限维的相空间内。为了描述全局吸引子，主要研究人员计划构造一种使用光滑有限维流形近似分形集的代数逼近方法，然后在计算机上实现该方法。从抽象的角度来看，人们可以把偏微分方程的解看作是生成一个无限维的轨迹或曲线空间。这就是研究这类方程困难的原因之一。幸运的是，在这个无限维空间中存在着有限维的集合，它们表征了解在很长时间内的行为。由于这些集合是有限维的，所以可以用数值和解析技术来近似地描述它们。有了这个奖，研究人员将研究被称为全局吸引子的有限维物体，它出现在流体动力学方程的解集中。

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.