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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 \cite{kpr15} 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 $\|\f{u_r}{r}{\bf 1}_{\{u_r< -\f 1r\}}\|_{L^{3/2}(\mbR^3)}< C_{\sharp}$ where $C_{\sharp}$ is a universal constant to be specified. In particular, if $u_r(r,z)\geq -\f1r$ for $\forall (r,z)\in[0,\oo)\times\mbR$, then ${\bf u}\equiv 0$. Liouville theorems also hold if $\displaystyle\lim_{|x|\to \oo}\Ga =0$ or $\Ga\in L^q(\mbR^3)$ for some $q\in [2,\oo)$ where $\Ga= r u_þ$. We also established some interesting inequalities for $\Om\co \f{\p_z u_r-\p_r u_z}{r}$, showing that $\na\Om$ can be bounded by $\Om$ itself. All these results are extended to the axially symmetric MHD and Hall-MHD equations with ${\bf u}=u_r(r,z){\bf e}_r +u_þ(r,z) {\bf e}_þ + u_z(r,z){\bf e}_z, {\bf h}=h_þ(r,z){\bf e}_þ$, 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 $Φ=\f {1}{2} (|{\bf u}|^2+|{\bf h}|^2)+p$ for this special solution class.