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

Abstract

In this paper we study Liouville properties of smooth steady axially symmetric solutions of the Navier-Stokes equations. First, we provide another version of the Liouville theorem of [14] in the case of zero swirl, where we replaced the Dirichlet integrability condition by mild decay conditions. Then we prove some Liouville theorems under the assumption ura universal constant to be specified. In particular, if ur(r, z) ≥ − 1 r for ∀(r, z) ∈ [0, ∞) × R, then u ≡ 0. Liouville theorems also hold if lim |x|→∞ Γ = 0 or Γ ∈ L q (R 3 ) for some q ∈ [2, ∞) where Γ = ru θ . We also established some interesting inequalities for Ω := ∂z ur −∂r uz r, showing that ∇Ω can be bounded by Ω itself. All these results are extended to the axially symmetric MHD and Hall-MHD equations with u = ur(r, z)er + u θ (r, z)e θ + uz(r, z)ez, h = h θ (r, z)e θ , indicating that the swirl component of the magnetic field does not affect the triviality. Especially, we establish the maximum principle for the total head pressure Φ = 1 2 (|u| 2 + |h| 2 ) + p for this special solution class.

On Liouville-type theorems for the 2D stationary MHD equations

“…In recent years, many mathematicians attempted to bring a complete understanding to the Liouville problem for the MHD system; we cite the works of [1,3,14,22] (and the references contained therein) for interesting results. We also point out the recent contribution of Wang and Wang in [20], where they proved a Liouville theorem for the system in question under some smallness condition on the L 1 -norm of b.…”

On Liouville type results for the stationary MHD in R 3

“…For (1.1), Chae and Weng [4] obtained that if a smooth solution to (1.1) with a finite Dirichlet integral…”

“…Remark 1. The above integrability condition can be regarded as an interpolation for the integrability conditions of the cases (p 1 , p 2 ) = ( 92 , 9 2 ) (see [9]) and (p 1 , p 2 ) = (3, 6) (see [4]). The next one is regarding mixed interpolation type sufficient conditions of L p (R 3 ) and M −1 s,q (R 3 ) Theorem 1.2.…”

Liouville-type theorems for the stationary MHD equations in 2D

“…One may wonder whether the velocity fields also play the leading role in a Liouville theory for MHD equations. Partial progress has been made for the three-dimensional case, where Chae-Weng [4] recently proved the axially symmetric Dirichlet solution (u, b)…”