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

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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.