Asymptotic properties of generalized D-solutions to the stationary axially symmetric Navier–Stokes equations

Abstract

Listen

Translate article

In this paper, we derive asymptotic properties of both the velocity and the vorticity fields to the 3-dimensional axially symmetric Navier–Stokes equations at infinity under the generalized D-solution assumption [Formula: see text] for [Formula: see text]. We do not impose any zero or non-zero constant vector asymptotic assumption to the solution at infinity. Our results generalize those in [H. Choe and B. Jin, Asymptotic properties of axis-symmetric D-solutions of the Navier–Stokes equations, J. Math. Fluid Mech. 11(2) (2009) 208–232; S. Weng, Decay properties of axially symmetric D-solutions to the steady Navier–Stokes equations, J. Math. Fluid Mech. 20(1) (2018) 7–25; B. Carrillo, X. Pan and Q. S. Zhang, Decay and vanishing of some axially symmetric D-solutions of the Navier–Stokes equations, J. Funct. Anal. 279(1) (2020) 108504], where the authors focused on the case [Formula: see text] and the velocity field approaches zero at infinity. Meanwhile, when [Formula: see text] and the velocity field approaches zero at infinity, our results coincide with the results in [H. Choe and B. Jin, Asymptotic properties of axis-symmetric D-solutions of the Navier–Stokes equations, J. Math. Fluid Mech. 11(2) (2009) 208–232; S. Weng, Decay properties of axially symmetric D-solutions to the steady Navier–Stokes equations, J. Math. Fluid Mech. 20(1) (2018) 7–25; B. Carrillo, X. Pan and Q. S. Zhang, Decay and vanishing of some axially symmetric D-solutions of the Navier–Stokes equations, J. Funct. Anal. 279(1) (2020) 108504].