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

Abstract

An old problem since Leray [Le] asks whether homogeneous D solutions of the 3 dimensional Navier-Stokes equation in R 3 or some noncompact domains are 0. In this paper, we give a positive solution to the problem in two cases: (1) full 3 dimensional slab case R 2 × [0, 1] with Dirichlet boundary condition (Theorem 1.1); (2) when the solution is axially symmetric and periodic in the vertical variable (Theorem 1.3).Also, in the slab case, we prove that even if the Dirichlet integral has some growth, axially symmetric solutions with Dirichlet boundary condition must be swirl free, namely u θ = 0, thus reducing the problem to essentially a "2 dimensional" problem. In addition, a general Dsolution (without the axial symmetry assumption) vanishes in R 3 if, in spherical coordinates, the positive radial component of the velocity decays at order -1 of the distance. The paper is self contained comparing with [CPZ] although the general idea is related.Since J ln r,

A Liouville type theorem for axially symmetric D-solutions to steady Navier–Stokes equations

“…As a result, letting R → ∞ in both sides of (2.5) gives (2.2). Now we will derive the decay rate of p by (2.2) under the assumption (1.4) by using the method of Lemma 3.2 in [1]. It is easy to check that ∂ y j ∂ y i Γ(x − y) (i, j = 1, 2, 3) is a Calderon-Zygmud kernel since it satisfies, for each i, j,…”

“…Seregin [11] showed that u ∈ L 6 (R 3 ) ∩ BMO −1 also implies Liouville type theorem. On the other hand, authors in [1] considered D-solutions in the slab R 2 × [−π, π] with suitable boundary conditions and proved u ≡ 0.…”

“…For a positive number a, we denote a − as a number smaller but close to a and a + a number bigger but close to a. Recently, in [1], authors gave a simple proof for the decay rate of u by using Brezis-Gallouet inequality [7] and improved the decay of ω to…”

On the Vanishing of Some D-Solutions to the Stationary Magnetohydrodynamics System

“…Our proof of the theorem and corollaries are based on the oscillation estimate of the pressure in [2]. Because of the partly "θ−dependent" of the pressure p and some magnetic related terms, we need a careful treatment for getting the boundedness of u and h up to their second order derivatives and oscillation estimate of p in a dyadic annulus.…”

“…The study is partly motivated by the related Liouville problem of the stationary Navier-Stokes equations, which has attracted much attention in recent years and is still far from being fully understood. See for example [1,2,3,4,5,7,8,9,12,13] and the reference therein. First, in full 3D case, the Liouville-type theorem holds provided the vanishing of u θ and h θ .…”