Decay Properties of Axially Symmetric D-Solutions to the Steady Navier–Stokes Equations

Abstract

We investigate the decay properties of smooth axially symmetric D-solutions to the steady Navier-Stokes equations. The achievements of this paper are two folds. One is improved decay rates of u θ and ∇u, especially we show that |u θ (r, z)| ≤ c log r r 1 2 for any smooth axially symmetric D-solutions to the Navier-Stokes equations. These improvement are based on improved weighted estimates of ω θ and A p weight for singular integral operators, which yields good decay estimates for (∇u r , ∇u z ) and (ω r , ω z ), where ω = curl u = ω r e r + ω θ e θ + ω z e z . Another is the first decay rate estimates in the Oz-direction for smooth axially symmetric flows without swirl. We do not need any small assumptions on the forcing term.Mathematics Subject Classifications 2010: Primary 76D05; Secondary 35Q35.

Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations

“…Comparing with (2.20) and (2.21), heuristically, one may guess that the integrability of u θ in the z-direction is enough, the decay rate of u θ in the radial direction maybe a key issue. Unfortunately, the decay rates obtained in [8] seemed not good enough, this issue will be further investigated in [18].…”

Remarks on Liouville type theorems for the 3D steady axially symmetric Navier–Stokes equations

“…Firstly, we introduce a representation formula of u r , u z and u θ with the help of the vorticity. Since b = u r e r + u z e z and ∇ × b = w θ e θ , ∇ × (u θ e θ ) = w r e r + w z e z by Biot-Savart law, we can get the integral representation of the velocity as follows(for example, see Lemma 2.2 for a local version by Choe-Jin [6], also see Lemma 3.10 by Weng [17]).…”

“…The condition ru θ ∈ L q with some q ≥ 1 or b ∈ L 3 is enough, see Chae-Weng in [5]. Specially, for the axially symmetric case, the decay of the velocity or the vorticity can be obtained: Choe-Jin [6], Weng [17] proved that…”

Decay properties of Axially Symmetric D-solutions to the Steady Incompressible Magnetohydrodynamic Equations

“…Recently there are several papers working on decay rate estimates for the axisymmetric D-solutions to the steady Navier-Stokes equations. Our goal here is to extend the results in [3,10,35] to the axisymmetric steady MHD equations. Note that the weighted energy method developed in [10,35] needs to use the special structure of the vorticity equations, such an extension may be quite nontrivial, although the steady MHD equations has the same scaling as the steady Navier-Stokes equations, the scaling technique developed in [3] still works in the MHD setting.…”

“…In [19], they further found that the weak solution with finite Dirichlet integral may not be bounded, but it must grow more slowly than (ln r) 1/2 . By further adapting the ideas in [18,19] to the 3D axisymmetric setting, the authors in [10,35] obtained some decay rates for smooth axially symmetric solutions to steady Navier-Stokes equations. On the other hand, it is well-known that if (u, p) solves stationary Navier-Stokes equations, so does (u λ , p λ ) for all λ > 0, where u λ (x) = λu(λx) and p λ (x) = λ 2 p(λx).…”