\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 76D05; Secondary: 35Q35.

    Citation:

    \begin{equation} \\ \end{equation}
  • [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, arXiv:1511.00752.

    [19]

    S. Weng, Existence of axially symmetric weak solutions to steady MHD with non-homogeneous boundary conditions, preprint, arXiv:1511.02546. Commun. Math. Sci., accepted.

    [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.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(263) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return