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

TL;DR

This paper investigates the decay and vanishing of certain D-solutions to the 3D Navier-Stokes equations, providing positive results in specific symmetric and boundary condition cases, advancing understanding of Leray's old problem.

Contribution

It offers new positive results on the vanishing of homogeneous D-solutions in special symmetric and boundary cases, extending previous partial results.

Findings

Homogeneous D-solutions vanish in axially symmetric periodic cases.

Solutions also vanish in the 3D slab with Dirichlet boundary conditions.

Partial results contribute to understanding Leray's problem.

Abstract

An old problem since Leray \cite{Le:1} asks whether homogeneous D solutions of the 3 dimensional Navier-Stokes equation in $\mathbb{R}^3$ or some noncompact domains are 0. In this paper, we give a positive solution to the problem in two special cases: (1) when the solution is axially symmetric and periodic in the vertical variable; (2) full 3 dimensional slab case $\mathbb{R}^2 \times [0, 1]$ with Dirichlet boundary condition. Other partial results are also presented. The paper is self contained comparing with the first part \cite{CPZ:1} although the general idea is related.