Asymptotic Behavior of the Navier-Stokes Equations with Nonzero Far-Field Velocity

Abstract EN

Concerning the nonstationary Navier-Stokes flow with a nonzero constant velocity at infinity, the temporal stability has been studied by Heywood (1970, 1972) and Masuda (1975) in L2 space and by Shibata (1999) and Enomoto-Shibata (2005) in Lp spaces for p≥3.

However, their results did not include enough information to find the spatial decay.

So, Bae-Roh (2010) improved Enomoto-Shibata's results in some sense and estimated the spatial decay even though their results are limited.

In this paper, we will prove temporal decay with a weighted function by using Lr-Lp decay estimates obtained by Roh (2011).

Bae-Roh (2010) proved the temporal rate becomes slower by (1+σ)/2 if a weighted function is |x|σ for 0<σ<1/2.

In this paper, we prove that the temporal decay becomes slower by σ, where 0<σ<3/2 if a weighted function is |x|σ.

For the proof, we deduce an integral representation of the solution and then establish the temporal decay estimates of weighted Lp-norm of solutions.

This method was first initiated by He and Xin (2000) and developed by Bae and Jin (2006, 2007, 2008).