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

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 ||uᵣ/r 1_ᵣ< -1/r\||_L3/2(R³) < C_ where C_ is a universal constant to be specified. In particular, if uᵣ(r,z)≥ -1r for ∀ (r,z) ∈ [0,∞) × R, then u≡ 0. Liouville theorems also hold if lim|x|→ ∞Γ =0 or Γ∈ Lq(R³) for some q∈ [2,∞) where Γ= r uθ. 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 +uθ(r,z) eθ + uz(r,z) ez, h=hθ(r,z) eθ, 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.