Mathematical Sciences: Approximation of the Global Attractors of Evolution Equations

有了这个奖项，主要研究者将使用 分析和数值工具来研究长期行为 一类耗散偏微分方程的解 方程，如Kuramoto-Sivashinsky方程和 纳维-斯托克斯方程 特别感兴趣的是 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，位于一个无限的， 维相空间 为了描述全球 吸引器主要研究人员计划建造一个 代数近似方法，使用光滑有限- 维流形来近似分形集，然后 在计算机上实现该方法。 从抽象的角度来看，人们可以把解决方案 一个偏微分方程，生成一个 无限维的轨迹或曲线。 这是一 这就是为什么研究这些方程很困难的原因。 幸运的是，在这个无限维的空间里 存在着有限维的集合， 在很长的时间里表现。 因为这些集合是有限的- 三维的，可以近似地用 数值和分析技术。 有了这个奖项， 研究人员将研究有限维的物体，称为全球 吸引子出现在方程的解集中， 流体动力学

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.