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 $||\frac{u_r}{r}{\bf 1}_{\{u_r< -\frac {1}{r}\}}||_{L^{3/2}(\mathbb{R}^3)} < C_{\sharp}$ where $C_{\sharp}$ is a universal constant to be specified. In particular, if $u_r(r,z)\geq -\frac1r$ for $\forall (r,z) \in [0,\infty) \times \mathbb{R}$, then ${\bf u}\equiv 0$. Liouville theorems also hold if $\displaystyle\lim_{|x|\to \infty}\Gamma =0$ or $\Gamma\in L^q(\mathbb{R}^3)$ for some $q\in [2,\infty)$ where $\Gamma= r u_{\theta}$. We also established some interesting inequalities for $\Omega := \frac{\partial_z u_r-\partial_r u_z}{r}$, showing that $\nabla\Omega$ can be bounded by $\Omega$ itself. All these results are extended to the axially symmetric MHD and Hall-MHD equations with ${\bf u}=u_r(r,z){\bf e}_r +u_{\theta}(r,z) {\bf e}_{\theta} + u_z(r,z){\bf e}_z, {\bf h}=h_{\theta}(r,z){\bf e}_{\theta}$, 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 $\Phi=\frac {1}{2} (|{\bf u}|^2+|{\bf h}|^2)+p$ for this special solution class.