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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 |
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. |
show all references
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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