Decay and vanishing of some axially symmetric D-solutions of the Navier-Stokes equations

TL;DR

This paper establishes decay estimates and vanishing results for axially symmetric D-solutions of the 3D Navier-Stokes equations, advancing understanding of their behavior without requiring smallness or decay assumptions.

Contribution

It provides new decay estimates for velocity and vorticity, and proves the first vanishing results for 3D D-solutions under minimal conditions.

Findings

Decay estimates for velocity and vorticity in axially symmetric solutions

Vanishing of solutions in periodic and bounded domains under certain conditions

First known vanishing results for 3D D-solutions without extra assumptions

Abstract

We study axially symmetric D-solutions of the 3 dimensional Navier-Stokes equations. The first result is an a priori decay estimate of the velocity for general domains. The second is an a priori decay estimate of the vorticity in $\bR^3$, which improves the corresponding results in the literature. In addition, we prove a similar decay of full 3d solutions except for a small set of angles. Next we turn to D-solutions which are periodic in the third variable and prove vanishing result under a reasonable condition. As a corollary we prove that axially symmetric D-solutions in the slab $\bR^2 \times I$ with suitable boundary condition is $0$. Here $I$ is any finite interval. To the best of our knowledge, this seems to be the first vanishing result on a 3 dimensional D-solution without extra integral or decay or smallness assumption on the solution. The tools used include Brezis-Gallouet inequality, dimension reduction, scaling, Green's function bound and Liouville theorems for Navier-Stokes equations.