Special Session 28: Fluids and Turbulence

Ever 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. Recent investigations revealed that these are also closely connected to asymptotic behavior of solutions and the study of the global attractor. In most of the studies to date, by considering an adequate Gevrey norm, one shows that the space analyticity radius grows at least as a power of time upto a certain time point. This time point is usually expressed in terms of a Sobolev norm of the initial data. On the other hand, the solution to the 3D NSE is known to be analytic as long as its Sobolev H norm is nite. We consider an auxilliary equation that couples the the space analyticity radius to the 3D NSE. This allows us to study its time evolution more precisely and thus obtain improved estimates. As a corollary, we show that a close relation exists between the analyticity radius and the regularity of the solution to the 3D NSE.