Decay Properties of Axially Symmetric D-Solutions to the Steady Navier–Stokes Equations

• Published in: Journal of mathematical fluid mechanics 2018-03, Vol.20 (1), p.7-25
• Main Author: Weng, Shangkun
• Format: Article
• Language: English
• Subjects: Classical and Continuum Physics; Decay rate; Estimates; Fluid dynamics; Fluid flow; Fluid mechanics; Fluid- and Aerodynamics; Mathematical analysis; Mathematical Methods in Physics; Navier-Stokes equations; Operators (mathematics); Physics; Physics and Astronomy; Theoretical mathematics
• ISSN: 1422-6928; 1422-6952
• DOI: 10.1007/s00021-016-0310-5

We investigate the decay properties of smooth axially symmetric D-solutions to the steady Navier–Stokes equations. The achievements of this paper are two folds. One is improved decay rates of u θ and ∇ u , especially we show that | u θ ( r , z ) | ≤ c ( log r r ) 1 2 for any smooth axially symmetric D-solutions to the Navier–Stokes equations. These improvement are based on improved weighted estimates of ω θ and A p weight for singular integral operators, which yields good decay estimates for ( ∇ u r , ∇ u z ) and ( ω r , ω z ) , where ω = curl u = ω r e r + ω θ e θ + ω z e z . Another is the first decay rate estimates in the Oz -direction for smooth axially symmetric flows without swirl. We do not need any small assumptions on the forcing term.