    October  2016, 36(10): 5267-5285. doi: 10.3934/dcds.2016031

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

 1 Department of Mathematics, Chung-Ang University, Seoul 156-756, South Korea 2 Pohang Mathematics Institute, Pohang University of Science and Technology, Pohang, Gyungbuk, 790-784, South Korea

Received  October 2015 Revised  March 2016 Published  July 2016

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  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.
Citation: Dongho Chae, Shangkun Weng. Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5267-5285. doi: 10.3934/dcds.2016031
##### References:
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##### References:
  M. Acheritogaray, P. Degond, A. Frouvelle and J.-G. Liu, Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system,, Kinet. Relat. Models, 4 (2011), 901.  doi: 10.3934/krm.2011.4.901.  Google Scholar  D. Chae, Liouville-type theorem for the forced Euler equations and the Navier-Stokes equations,, Commun. Math. Phys, 326 (2014), 37.  doi: 10.1007/s00220-013-1868-x.  Google Scholar  D. Chae, P. Degond and J.-G. Liu, Well-posedness for Hall-magnetohydrodynamics,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 555.  doi: 10.1016/j.anihpc.2013.04.006.  Google Scholar  D. Chae and J. Lee, On the regularity of the axisymmetric solutions of the Navier-Stokes equations,, Math. Z., 239 (2002), 645.  doi: 10.1007/s002090100317.  Google Scholar  D. Chae and S. Weng, Singularity formation for the incompressible Hall-MHD equations without resistivity,, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, (2015).  doi: 10.1016/j.anihpc.2015.03.002. Google Scholar  D. Chae and J. Wolf, On partial regularity for the steady Hall magnetohydrodynamics system,, Commun. Math. Phys, 339 (2015), 1147.  doi: 10.1007/s00220-015-2429-2.  Google Scholar  D. Chae and T. Yoneda, On the Liouville theorem for the stationary Navier-Stokes equations in a critical space,, J. Math. Anal. Appl., 405 (2013), 706.  doi: 10.1016/j.jmaa.2013.04.040.  Google Scholar  H. Choe and B. Jin, Asymptotic properties of axi-symmetric D-solutions of the Navier-Stokes equations,, J. Math. Fluid. Mech., 11 (2009), 208.  doi: 10.1007/s00021-007-0256-8.  Google Scholar  G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. In: Steady State problems,, $2^{nd}$ edition, (2011).  doi: 10.1007/978-0-387-09619-3.  Google Scholar  D. Gilbarg and H. F. Weinberger, Asymptotic properties of steady plane solutions of the Navier-Stokes equations with bounded Dirichlet integral,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5 (1978), 381. Google Scholar  C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations,, J. Differential Equations, 213 (2005), 235.  doi: 10.1016/j.jde.2004.07.002.  Google Scholar  T. Hou, Z. Lei and C. Li, Global regularity of the 3D axi-symmetric Navier-Stokes equations with anisotropic data,, Comm. Partial Differential Equations, 33 (2008), 1622.  doi: 10.1080/03605300802108057.  Google Scholar  G. Koch, N. Nadirashvili, G. Seregin and V. Sverak, Liouville theorems for the Navier-Stokes equations and applications,, Acta. Math., 203 (2009), 83.  doi: 10.1007/s11511-009-0039-6.  Google Scholar  M. Korobkov, K. Pileckas and R. Russo, The Liouville theorem for the steady-state Navier-Stokes problem for axially symmetric 3D solutions in absence of swirl,, J. Math. Fluid Mech., 17 (2015), 287.  doi: 10.1007/s00021-015-0202-0.  Google Scholar  M. Korobkov, K. Pileckas and R. Russo, Solution of Leray's problem for the stationary Navier-Stokes equations in plane and axially symmetric spatial domains,, Annals of Mathematics, 181 (2015), 769.  doi: 10.4007/annals.2015.181.2.7.  Google Scholar  Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions,, J. Differential Equations, 259 (2015), 3202.  doi: 10.1016/j.jde.2015.04.017.  Google Scholar  J. Liu and W. Wang, Characterization and regularity for axisymmetric solenoidal vector fields with application to Navier-Stokes equation,, SIAM J. Math. Anal., 41 (2009), 1825.  doi: 10.1137/080739744.  Google Scholar  S. Weng, Decay properties of smooth axially symmetric D-solutions to the steady Navier-Stokes equations,, preprint, ().   Google Scholar  S. Weng, Existence of axially symmetric weak solutions to steady MHD with non-homogeneous boundary conditions,, preprint, ().   Google Scholar  Y. Zhou, Remarks on regularities for the 3D MHD equations,, Discrete Contin. Dyn. Syst., 12 (2005), 881.  doi: 10.3934/dcds.2005.12.881.  Google Scholar
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