Asymptotic Behavior of the Navier-Stokes Equations with Nonzero Far-Field Velocity - Jaiok Roh (Abstract and Applied Analysis)
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).