Mathematical Sciences: Approximate Inertial Manifolds for a Benard Convection System: Analysis and Computation

这项研究将评估近似惯性流形在二维和三维几何中通过一个更简单的Benard对流系统的计算效用。众所周知，Benard系统的解随着系统能量的增加而变得越来越复杂，因此研究人员发展的解析和数值方法将对研究由Navier-Stokes方程支配的更复杂的流体动力学现象大有裨益。许多本质上是非线性的物理系统的解随着一些外部参数的变化而表现出越来越复杂的行为。这项建议涉及从解析和数值两个方面研究装满液体的二维和三维容器内的热传导。研究这个正则系统所获得的洞察力将有助于我们理解与湍流和混沌有关的更复杂的现象。

This research will assess the computational utility of approximate inertial manifolds by means of a simpler Benard convection system in two-and three-dimensional geometries. Solutions of the Benard system are known to become more and more complicated as the energy input to the system is increased, and so the analytical and numerical methods developed by the investigators will be of great use in studying more complicated fluid dynamical phenomena governed by the Navier-Stokes equations. Many physical systems that are nonlinear in nature have solutions that diaplay increasingly complicated behavior as some external parameter is varied. This proposal concerns the study of heat conduction inside two-and three-dimensional containers filled with liquid both analytically and numerically. The insight gained from the study of this canonical system will aid in our understanding of much more complicated phenomena related to turbulence and chaos.