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

Abstract

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).