Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 163, pp. 1-25.
Title: Sharp asymptotic estimates for vorticity
solutions of the 2D Navier-Stokes equation
Author: Yuncheng You (Univ. of South Florida, Tampa, FL, USA)
Abstract:
The asymptotic dynamics of high-order temporal-spatial derivatives
of the two-dimensional vorticity and velocity of an
incompressible, viscous fluid flow in $\mathbb{R}^2$ are studied,
which is equivalent to the 2D Navier-Stokes equation. It is known
that for any integrable initial vorticity, the 2D vorticity
solution converges to the Oseen vortex. In this paper, sharp
exterior decay estimates of the temporal-spatial derivatives of
the vorticity solution are established. These estimates are then
used and combined with similarity and $L^p$ compactness to show
the asymptotical attraction rates of temporal-spatial derivatives
of generic 2D vorticity and velocity solutions by the Oseen
vortices and velocity solutions respectively. The asymptotic estimates
and the asymptotic attraction rates of all the derivatives obtained
in this paper are independent of low or high Reynolds numbers.
Submitted November 5, 2008. Published December 18, 2008.
Math Subject Classifications: 35B40, 35B65, 35K15, 35Q30, 76D05, 76D17.
Key Words: Navier-Stokes equation; vorticity; regularity; asymptotic dynamics;
Biot-Savart law; similarity solution; Oseen vortex.