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تسجيل مستخدم جديدLiouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations
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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 $forall (r,z)in[0,oo)timesmbR$, then ${bf u}equiv 0$. Liouville theorems also hold if $displaystylelim_{|x|to oo}Ga =0$ or $Gain L^q(mbR^3)$ for some $qin [2,oo)$ where $Ga= r u_{th}$. We also established some interesting inequalities for $Omco f{p_z u_r-p_r u_z}{r}$, showing that $ aOm$ 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.
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