Electron. J. Diff. Eqns., Vol. 2008(2008), No. 163, pp. 1-25.

Sharp asymptotic estimates for vorticity solutions of the 2D Navier-Stokes equation

Yuncheng You

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.

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Yuncheng You
Department of Mathematics and Statistics
University of South Florida
Tampa, FL 33620, USA
email: you@math.usf.edu

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