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

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.

revised after referee's suggestions

References in corpus (1)

• Liouville Type Theorem for Stationary Navier-Stokes Equations

Cited by in corpus (7)

• Remarks on Liouville type theorems for the steady MHD and Hall-MHD equations
• Decay and vanishing of some axially symmetric D-solutions of the Navier-Stokes equations
• Asymptotic behavior and Liouville-type theorems for axisymmetric stationary Navier-Stokes equations outside of an infinite cylinder with a periodic boundary condition
• Asymptotic properties of generalized D-solutions to the stationary axially symmetric Navier-Stokes equations
• Liouville theorem of axially symmetric Navier-Stokes equations with growing velocity at infinity
• Asymptotic properties of steady solutions to the 3D axisymmetric Navier-Stokes equations with no swirl
• A Review of results on axially symmetric Navier-Stokes equations, with addendum by X. Pan and Q. S. Zhang