| Abstract : ver since the work of Foias and Temam in case of the Navier-Stokes equations (NSE), the use of Gevrey norms has become standard fare in estimating the radius of analyticity in various nonlinear evolution equations. Subsequent work has shown that Gevrey estimates can be used among other things to estimate the decay rate of higher order derivatives of solutions to the NSE and establish exponential decay of the power spectrum. Recently, analytycity properties of the solutions has also been employed to study the global attractor. Motivated by the work of Grujic and Kukavica, we extend the work of Foias and Temam to the study of NSE in certain Banach spaces which include Sobolev type spaces of negative order and establish (generalized) Gevrey regularity for the solutions. We will also discuss some recent results on the time evolution of the radius of analyticity for the 3D Navier-Stokes equations. Our investigation reveals a close relation between the analyticity radius and the regularity problem for the 3D NSE. This is a joint work in part with C. Foias and D. Swanson. |