Asymptotic Properties of Axis-Symmetric D-Solutions of the Navier–Stokes Equations

Abstract

We consider asymptotic behavior of Leray's solution which expresses axis-symmetric incompressible Navier-Stokes flow past an axis-symmetric body. When the velocity at infinity is prescribed to be nonzero constant, Leray's solution is known to have optimum decay rate, which is in the class of physically reasonable solution. When the velocity at infinity is prescribed to be zero, the decay rate at infinity has been shown under certain restrictions such as smallness on the data. Here we find an explicit decay rate when the flow is axis-symmetric by decoupling the axial velocity and the horizontal velocities. (2000). 35Q30, 76D07. Mathematics Subject Classification

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

“…The vorticity ω = ∂ x 2 u 1 − ∂ x 1 u 2 decayed more rapidly than r − 3 4 (log r) 1 8 and the first derivatives of the velocity decayed more rapidly than r − 3 4 (log r) 9 8 at infinity. Inspired by [16] and [17], Choe and Jin [7] first obtained the following decay rates for smooth axially symmetric solutions to (1.1): Let Ω be an exterior domain, suppose f ∈ H 1 (Ω) be an axially symmetric vector field with…”

“…Remark 3.4. The case δ = 0 was obtained in [7]. We can also extend these arguments to the exterior domain case by choosing large enough r 0 in the definition of the cut-off function η.…”

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…”

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

“…Let's show the detail of the integration by parts. The key point here is I 1 contains a singular integral since 3 and thus is not integrable when y is close to x. So we should be careful when doing IBP (short for integration by parts).…”