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

Abstract

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 ${\int }_{{\mathbb{R}}^3}|\nabla u{|}^{q}dx<\infty$ for $2<q<\infty$. We do not impose any zero or nonzero constant vector asymptotic assumption on the solution at infinity. Our results generalize those in \cite{CJ:2009JMFM,Ws:2018JMFM,CPZ2018} where the authors focused on the case $q=2$ and the velocity field approaches zero at infinity. Meanwhile, when $q\to 2_{+}$ and the velocity field approaches zero at infinity, our results coincide with the results in \cite{CJ:2009JMFM,Ws:2018JMFM,CPZ2018}.

24 pages. Final version, to appear in Commun. Contemp. Math

References in corpus (1)

• Liouville theorem of axially symmetric Navier-Stokes equations with growing velocity at infinity

Cited by in corpus (1)

• Asymptotic properties of steady solutions to the 3D axisymmetric Navier-Stokes equations with no swirl