Differential Equations Seminar
Dr. Animikh Biswas, UMBC
Location
Mathematics/Psychology : 401
Date & Time
November 9, 2015, 11:00 am – 12:00 pm
Description
Tttle: From Gevrey classes to a near Shangri-La class
Abstract:
Abstract:
A basic tool, in both numerical and theoretical study of an infinite dimensional dynamical system, is to approximate it by a suitable finite dimensional system. This may take the form of Galerkin projections or finite elements in computational fluid dynamics, or other suitable interpolant (such as ``determining modes" or readings from spatially dispersed weather stations) in data assimilation and weather forecasting. It turns out that in certain dissipative systems, these approximates converge exponentially fast. This is related to ``gain of analyticity" of solutions over time and ``uniform space analyticity radius" of the attractor. In fluid dynamics, this radius is related to the Kolmogorov length scale and it demarcates the length scale below which the viscous effect dominates the (nonlinear) inertial effect. Foias and Temam introduced an effective approach to estimate space analyticity radius for the Navier-Stokes equations via a class of (exponential) norms called Gevrey norms. We will discuss how Gevrey functional classes (and their generalizations) can be used to study the dynamics for a wide class of equations with applications ranging from turbulence theory to geophysics. In particular, we will show that Gevrey class techniques can be used to obtain higher order decay rate of solutions. Subsequently, we will discuss an ongoing recent work where a certain generalized Gevrey class almost linearizes the 3D Navier-Stokes equations.
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