A note on optimal decay rates for the axisymmetric D-solutions to the steady Navier-Stokes equations

2017/09/11 by Weng, Shangkun, Xie, Chunjing
#76D05 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1709.03212

Abstract

In this paper, we investigate the decay properties of an axisymmetric D-solutions to stationary incompressible Navier-Stokes systems in ℝ³. We obtain the optimal decay rate |\bf u(x)|≤ (C)/(|x|+1) for axisymmetric flows without swirl. Furthermore, we find a dichotomy for the decay rates of the swirl component u_θ, that is, either O((1)/(r+1))≤ |u_θ(r,z)|≤ \fracClog(r+1)(r+1)^1/2 or |u_θ(r,z)|≤ (C r)/((ρ+1)³), where ρ=√(r²+z²). In the latter case, we can further deduce that the other two components of the velocity field also attain the optimal decay rates: |u_r(r,z)|+ |u_z(r,z)|≤ (C)/(ρ+1). We do not require any small assumptions on the forcing term.

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