# American Institute of Mathematical Sciences

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 [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.
Citation: Dongho Chae, Shangkun Weng. Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5267-5285. doi: 10.3934/dcds.2016031
##### References:
 [1] 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), 901-918. doi: 10.3934/krm.2011.4.901. [2] D. Chae, Liouville-type theorem for the forced Euler equations and the Navier-Stokes equations, Commun. Math. Phys, 326 (2014), 37-48. doi: 10.1007/s00220-013-1868-x. [3] D. Chae, P. Degond and J.-G. Liu, Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 555-565. doi: 10.1016/j.anihpc.2013.04.006. [4] D. Chae and J. Lee, On the regularity of the axisymmetric solutions of the Navier-Stokes equations, Math. Z., 239 (2002), 645-671. doi: 10.1007/s002090100317. [5] 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), Available from: http://dx.doi.org/10.1016/j.anihpc.2015.03.002. doi: 10.1016/j.anihpc.2015.03.002. [6] D. Chae and J. Wolf, On partial regularity for the steady Hall magnetohydrodynamics system, Commun. Math. Phys, 339 (2015), 1147-1166. doi: 10.1007/s00220-015-2429-2. [7] 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), 706-710. doi: 10.1016/j.jmaa.2013.04.040. [8] H. Choe and B. Jin, Asymptotic properties of axi-symmetric D-solutions of the Navier-Stokes equations, J. Math. Fluid. Mech., 11 (2009), 208-232. doi: 10.1007/s00021-007-0256-8. [9] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. In: Steady State problems, $2^{nd}$ edition, Springer Monographs in Mathematics. Springer, New York, 2011. doi: 10.1007/978-0-387-09619-3. [10] 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), 381-404. [11] C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254. doi: 10.1016/j.jde.2004.07.002. [12] 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), 1622-1637. doi: 10.1080/03605300802108057. [13] G. Koch, N. Nadirashvili, G. Seregin and V. Sverak, Liouville theorems for the Navier-Stokes equations and applications, Acta. Math., 203 (2009), 83-105. doi: 10.1007/s11511-009-0039-6. [14] 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), 287-293. doi: 10.1007/s00021-015-0202-0. [15] 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), 769-807. doi: 10.4007/annals.2015.181.2.7. [16] Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215. doi: 10.1016/j.jde.2015.04.017. [17] 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), 1825-1850. doi: 10.1137/080739744. [18] S. Weng, Decay properties of smooth axially symmetric D-solutions to the steady Navier-Stokes equations,, preprint, (). [19] S. Weng, Existence of axially symmetric weak solutions to steady MHD with non-homogeneous boundary conditions,, preprint, (). [20] Y. Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst., 12 (2005), 881-886. doi: 10.3934/dcds.2005.12.881.

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##### References:
 [1] 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), 901-918. doi: 10.3934/krm.2011.4.901. [2] D. Chae, Liouville-type theorem for the forced Euler equations and the Navier-Stokes equations, Commun. Math. Phys, 326 (2014), 37-48. doi: 10.1007/s00220-013-1868-x. [3] D. Chae, P. Degond and J.-G. Liu, Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 555-565. doi: 10.1016/j.anihpc.2013.04.006. [4] D. Chae and J. Lee, On the regularity of the axisymmetric solutions of the Navier-Stokes equations, Math. Z., 239 (2002), 645-671. doi: 10.1007/s002090100317. [5] 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), Available from: http://dx.doi.org/10.1016/j.anihpc.2015.03.002. doi: 10.1016/j.anihpc.2015.03.002. [6] D. Chae and J. Wolf, On partial regularity for the steady Hall magnetohydrodynamics system, Commun. Math. Phys, 339 (2015), 1147-1166. doi: 10.1007/s00220-015-2429-2. [7] 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), 706-710. doi: 10.1016/j.jmaa.2013.04.040. [8] H. Choe and B. Jin, Asymptotic properties of axi-symmetric D-solutions of the Navier-Stokes equations, J. Math. Fluid. Mech., 11 (2009), 208-232. doi: 10.1007/s00021-007-0256-8. [9] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. In: Steady State problems, $2^{nd}$ edition, Springer Monographs in Mathematics. Springer, New York, 2011. doi: 10.1007/978-0-387-09619-3. [10] 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), 381-404. [11] C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254. doi: 10.1016/j.jde.2004.07.002. [12] 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), 1622-1637. doi: 10.1080/03605300802108057. [13] G. Koch, N. Nadirashvili, G. Seregin and V. Sverak, Liouville theorems for the Navier-Stokes equations and applications, Acta. Math., 203 (2009), 83-105. doi: 10.1007/s11511-009-0039-6. [14] 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), 287-293. doi: 10.1007/s00021-015-0202-0. [15] 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), 769-807. doi: 10.4007/annals.2015.181.2.7. [16] Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215. doi: 10.1016/j.jde.2015.04.017. [17] 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), 1825-1850. doi: 10.1137/080739744. [18] S. Weng, Decay properties of smooth axially symmetric D-solutions to the steady Navier-Stokes equations,, preprint, (). [19] S. Weng, Existence of axially symmetric weak solutions to steady MHD with non-homogeneous boundary conditions,, preprint, (). [20] Y. Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst., 12 (2005), 881-886. doi: 10.3934/dcds.2005.12.881.
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