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

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.

40 pages incorporated (with thanks) referees' helpful suggestions

References in corpus (5)

• Liouville Type Theorem for Stationary Navier-Stokes Equations
• Decay and vanishing of some D-solutions of the Navier-Stokes equations
• Decay properties of smooth axially symmetric D-solutions to the steady Navier-Stokes equations
• On ancient periodic solutions to Axially-Symmetric Navier-Stokes Equations
• A Liouville Theorem for Axi-symmetric Navier-Stokes Equations on ${\mathbb{R}}^2\times {\mathbb{T}}^1$

Cited by in corpus (6)

• On ancient periodic solutions to Axially-Symmetric Navier-Stokes Equations
• Liouville theorem for steady-state solutions of simplified Ericksen-Leslie system
• Liouville theorem of axially symmetric Navier-Stokes equations with growing velocity at infinity
• Remarks on Liouville type theorems for the 3D steady axially symmetric Navier-Stokes equations
• A Liouville type theorem for axially symmetric $D$-solutions to steady Navier-Stokes Equations
• Asymptotic properties of the plane shear thickening fluids with bounded energy integral