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- W2963669794 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 $displaystylelim_{|x|to infty}Gamma =0$ or $Gammain L^q(mathbb{R}^3)$ for some $qin [2,infty)$ 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 $nablaOmega$ 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." @default.
- W2963669794 created "2019-07-30" @default.
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- W2963669794 date "2016-07-01" @default.
- W2963669794 modified "2023-10-15" @default.
- W2963669794 title "Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations" @default.
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- W2963669794 doi "https://doi.org/10.3934/dcds.2016031" @default.
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