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

TL;DR

This paper proves Liouville theorems for smooth steady axially symmetric solutions of Navier-Stokes and MHD equations, showing under certain decay and boundedness conditions that solutions must be trivial, extending previous results and establishing new inequalities.

Contribution

It introduces new Liouville theorems under mild decay conditions, extends results to MHD equations, and establishes maximum principles for the total head pressure.

Findings

Liouville theorems imply triviality of solutions under specific conditions.

Extended Liouville results to magnetohydrodynamic equations.

Established maximum principle for the total head pressure.

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 \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 $\Ga= r u_{\th}$. 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 ${\bf u}=u_r(r,z){\bf e}_r +u_{\th}(r,z) {\bf e}_{\th} + u_z(r,z){\bf e}_z, {\bf h}=h_{\th}(r,z){\bf e}_{\th}$, 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 $\Phi=\f {1}{2} (|{\bf u}|^2+|{\bf h}|^2)+p$ for this special solution class.