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Nonlinear Rayleigh-Taylor instability for nonhomogeneous incompressible viscous magnetohydrodynamic flows
1. | College of Mathematics and Computer Science, Fuzhou University, Fuzhou 360108, China, China |
2. | Institute of Applied Physics and Computational Mathematics, P.O.Box 8009-28, Beijing 100088 |
References:
[1] |
R. A. Adams and J. F. Fournier, Sobolev Space, 2nd edition, Academic Press, New York, 2005. |
[2] |
H. Cabannes, Theoretical Magnetofluiddynamics, Academic Press, New York, 1972.
doi: 10.1063/1.3070781. |
[3] |
Y. Choa and H. Kim, Unique solvability for the density-dependent Navier-Stokes equations, Nonlinear Analysis, 59 (2004), 465-489.
doi: 10.1016/S0362-546X(04)00267-6. |
[4] |
T. G. Cowling, Magnetohydrodynamics, Interscience Publishers, New York, 1957. |
[5] |
R. Duan, F. Jiang and S. Jiang, On the Rayleigh Taylor instability for incompressible, inviscid magnetohydrodynamic flows, SIAM J. Appl. Math., 71 (2011), 1990-2013.
doi: 10.1137/110830113. |
[6] |
D. Erban, The equations of motion of a perfect fluid with free boundary are not well posed, Comm. PDE, 12 (1987), 1175-1201.
doi: 10.1080/03605308708820523. |
[7] |
C. L. Feffermana, D. S. McCormick, J. C. Robinsonb and J. L. Rodrigo, Higher order commutator estimates and local existence for the non-resistive MHD equations and related models, J. Funct. Anal., 267 (2014), 1035-1056.
doi: 10.1016/j.jfa.2014.03.021. |
[8] |
S. Friedlander, W. Strauss and M. Vishik, Nonlinear instability in an ideal fluid, Ann. Inst. H. Poincare Anal. Non Lineaire, 14 (1997), 187-209.
doi: 10.1016/S0294-1449(97)80144-8. |
[9] |
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems (Second Edition), Academic Press: Springer, 2011.
doi: 10.1007/978-0-387-09620-9. |
[10] |
L. Grafakos, Classical Fourier Analysis, Second Edition, Springer, 2008. |
[11] |
Y. Guo, C. Hallstrom and D. Spirn, Dynamics near unstable, interfacial fluids, Comm. Math. Phys., 270 (2007), 635-689.
doi: 10.1007/s00220-006-0164-4. |
[12] |
Y. Guo and W. Strauss, Instability of periodic BGK equilibria, Comm. Pure Appl. Math., 48 (1995), 861-894.
doi: 10.1002/cpa.3160480803. |
[13] |
Y. Guo and W. A. Strauss, Nonlinear instability of double-humped equilibria, Ann. Inst. H. Poincaré Anal. Non Linéaire, 12 (1995), 339-352. |
[14] |
Y. Guo and I. Tice, Linear Rayleigh-Taylor instability for viscous compressible fluids, SIAM J. Math. Anal., 42 (2011), 1688-1720.
doi: 10.1137/090777438. |
[15] |
Y. Guo and I. Tice, Compressible, inviscid Rayleigh-Taylor instability, Indiana Univ. Math. J., 60 (2011), 677-711.
doi: 10.1512/iumj.2011.60.4193. |
[16] |
Y. Guo and I. Tice, Local well-posedness of the viscous surface wave problem without surface tension, Analysis and PDE, 6 (2013), 287-369.
doi: 10.2140/apde.2013.6.287. |
[17] |
R. Hide, Waves in a heavy, viscous, incompressible, electrically conducting fluid of variable density, in the presence of a magnetic field, Proc. Roy. Soc. (London) A, 233 (1955), 376-396.
doi: 10.1098/rspa.1955.0273. |
[18] |
H. Hwang, Variational approach to nonlinear gravity-driven instability in a MHD setting, Quart. Appl. Math., 66 (2008), 303-324.
doi: 10.1090/S0033-569X-08-01116-1. |
[19] |
H. Hwang and Y. Guo, On the dynamical Rayleigh-Taylor instability, Arch. Rational Mech. Anal., 167 (2003), 235-253.
doi: 10.1007/s00205-003-0243-z. |
[20] |
X. P. Hu and F. H. Lin, Global existence for two dimentional incompressible magnetohydrodynamic flow with zero magnetic diffusivity, arXiv:1405.0082v1 [math.AP] 1 May 2014. |
[21] |
J. Jang and I. Tice, Instability theory of the Navier-Stokes-Poisson equations, Analysis and PDE, 6 (2013), 1121-1181.
doi: 10.2140/apde.2013.6.1121. |
[22] |
F. Jiang and S. Jiang and G. X. Ni, Nonlinear instability for nonhomogeneous incompressible viscous fluids, Science China Math., 56 (2013), 665-686.
doi: 10.1007/s11425-013-4587-z. |
[23] |
F. Jiang, S. Jiang and W. Y. Wang, On the Rayleigh-Taylor instability for the incompressible viscous magnetohydrodynamic equations, Comm. Partial Differential Equations, 39 (2014), 399-438.
doi: 10.1080/03605302.2013.863913. |
[24] |
F. Jiang, S. Jiang and W. W. Wang, On the Rayleigh-Taylor instability for two uniform viscous incompressible flows, Chinese Ann. Math. Ser. B, 35 (2014), 907-940.
doi: 10.1007/s11401-014-0863-7. |
[25] |
F. Jiang and S. Jiang, On instability and stability of three-dimensional gravity driven viscous flows in a bounded domain, Adv. Math., 264 (2014), 831-863.
doi: 10.1016/j.aim.2014.07.030. |
[26] |
S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Ph. D. Thesis, Kyoto University, 1983. |
[27] |
M. Kruskal and M. Schwarzschild, Some instabilities of a completely ionized plasma, Proc. Roy. Soc. (London) A, 223 (1954), 348-360.
doi: 10.1098/rspa.1954.0120. |
[28] |
A. G. Kulikovskiy and G. A. Lyubimov, Magnetohydrodynamics, Addison-Wesley, Reading, MA, 1965. |
[29] |
L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media, Vol.8, 1984 (Translated from the Russian). |
[30] |
X. Li, N. Su and D. Wang, Local strong solution to the compressible magnetohydrodynmic flow with large data, J. Hyper. Diff. Eqns., 8 (2011), 415-436.
doi: 10.1142/S0219891611002457. |
[31] |
F. Lin and P. Zhang, Global small solutions to an MHD type system: The three-dimensional case, Comm. Pure. Appl. Math., 67 (2014), 531-580.
doi: 10.1002/cpa.21506. |
[32] |
A. Novotnỳ and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, USA, 2004. |
[33] |
J. Prüss and G. Simonett, On the Rayleigh-Taylor instability for the two-phase Navier-Stokes equations, Indiana Univ. Math. J., 59 (2010), 1853-1871.
doi: 10.1512/iumj.2010.59.4145. |
[34] |
L. Rayleigh, Analytic solutions of the Rayleigh equations for linear density profiles, Proc. London. Math. Soc., 14 (1883), 170-177. |
[35] |
Y. J. Wang, Critical magnetic number in the magnetohydrodynamic Rayleigh-Taylor instability, J. Math. Phys., 53 (2012), 073701, 22pp.
doi: 10.1063/1.4731479. |
[36] |
Y. Wang and I. Tice, The viscous surface-internal wave problem: Nonlinear Rayleigh-Taylor instability, Comm. PDE, 37 (2012), 1967-2028.
doi: 10.1080/03605302.2012.699498. |
show all references
References:
[1] |
R. A. Adams and J. F. Fournier, Sobolev Space, 2nd edition, Academic Press, New York, 2005. |
[2] |
H. Cabannes, Theoretical Magnetofluiddynamics, Academic Press, New York, 1972.
doi: 10.1063/1.3070781. |
[3] |
Y. Choa and H. Kim, Unique solvability for the density-dependent Navier-Stokes equations, Nonlinear Analysis, 59 (2004), 465-489.
doi: 10.1016/S0362-546X(04)00267-6. |
[4] |
T. G. Cowling, Magnetohydrodynamics, Interscience Publishers, New York, 1957. |
[5] |
R. Duan, F. Jiang and S. Jiang, On the Rayleigh Taylor instability for incompressible, inviscid magnetohydrodynamic flows, SIAM J. Appl. Math., 71 (2011), 1990-2013.
doi: 10.1137/110830113. |
[6] |
D. Erban, The equations of motion of a perfect fluid with free boundary are not well posed, Comm. PDE, 12 (1987), 1175-1201.
doi: 10.1080/03605308708820523. |
[7] |
C. L. Feffermana, D. S. McCormick, J. C. Robinsonb and J. L. Rodrigo, Higher order commutator estimates and local existence for the non-resistive MHD equations and related models, J. Funct. Anal., 267 (2014), 1035-1056.
doi: 10.1016/j.jfa.2014.03.021. |
[8] |
S. Friedlander, W. Strauss and M. Vishik, Nonlinear instability in an ideal fluid, Ann. Inst. H. Poincare Anal. Non Lineaire, 14 (1997), 187-209.
doi: 10.1016/S0294-1449(97)80144-8. |
[9] |
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems (Second Edition), Academic Press: Springer, 2011.
doi: 10.1007/978-0-387-09620-9. |
[10] |
L. Grafakos, Classical Fourier Analysis, Second Edition, Springer, 2008. |
[11] |
Y. Guo, C. Hallstrom and D. Spirn, Dynamics near unstable, interfacial fluids, Comm. Math. Phys., 270 (2007), 635-689.
doi: 10.1007/s00220-006-0164-4. |
[12] |
Y. Guo and W. Strauss, Instability of periodic BGK equilibria, Comm. Pure Appl. Math., 48 (1995), 861-894.
doi: 10.1002/cpa.3160480803. |
[13] |
Y. Guo and W. A. Strauss, Nonlinear instability of double-humped equilibria, Ann. Inst. H. Poincaré Anal. Non Linéaire, 12 (1995), 339-352. |
[14] |
Y. Guo and I. Tice, Linear Rayleigh-Taylor instability for viscous compressible fluids, SIAM J. Math. Anal., 42 (2011), 1688-1720.
doi: 10.1137/090777438. |
[15] |
Y. Guo and I. Tice, Compressible, inviscid Rayleigh-Taylor instability, Indiana Univ. Math. J., 60 (2011), 677-711.
doi: 10.1512/iumj.2011.60.4193. |
[16] |
Y. Guo and I. Tice, Local well-posedness of the viscous surface wave problem without surface tension, Analysis and PDE, 6 (2013), 287-369.
doi: 10.2140/apde.2013.6.287. |
[17] |
R. Hide, Waves in a heavy, viscous, incompressible, electrically conducting fluid of variable density, in the presence of a magnetic field, Proc. Roy. Soc. (London) A, 233 (1955), 376-396.
doi: 10.1098/rspa.1955.0273. |
[18] |
H. Hwang, Variational approach to nonlinear gravity-driven instability in a MHD setting, Quart. Appl. Math., 66 (2008), 303-324.
doi: 10.1090/S0033-569X-08-01116-1. |
[19] |
H. Hwang and Y. Guo, On the dynamical Rayleigh-Taylor instability, Arch. Rational Mech. Anal., 167 (2003), 235-253.
doi: 10.1007/s00205-003-0243-z. |
[20] |
X. P. Hu and F. H. Lin, Global existence for two dimentional incompressible magnetohydrodynamic flow with zero magnetic diffusivity, arXiv:1405.0082v1 [math.AP] 1 May 2014. |
[21] |
J. Jang and I. Tice, Instability theory of the Navier-Stokes-Poisson equations, Analysis and PDE, 6 (2013), 1121-1181.
doi: 10.2140/apde.2013.6.1121. |
[22] |
F. Jiang and S. Jiang and G. X. Ni, Nonlinear instability for nonhomogeneous incompressible viscous fluids, Science China Math., 56 (2013), 665-686.
doi: 10.1007/s11425-013-4587-z. |
[23] |
F. Jiang, S. Jiang and W. Y. Wang, On the Rayleigh-Taylor instability for the incompressible viscous magnetohydrodynamic equations, Comm. Partial Differential Equations, 39 (2014), 399-438.
doi: 10.1080/03605302.2013.863913. |
[24] |
F. Jiang, S. Jiang and W. W. Wang, On the Rayleigh-Taylor instability for two uniform viscous incompressible flows, Chinese Ann. Math. Ser. B, 35 (2014), 907-940.
doi: 10.1007/s11401-014-0863-7. |
[25] |
F. Jiang and S. Jiang, On instability and stability of three-dimensional gravity driven viscous flows in a bounded domain, Adv. Math., 264 (2014), 831-863.
doi: 10.1016/j.aim.2014.07.030. |
[26] |
S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Ph. D. Thesis, Kyoto University, 1983. |
[27] |
M. Kruskal and M. Schwarzschild, Some instabilities of a completely ionized plasma, Proc. Roy. Soc. (London) A, 223 (1954), 348-360.
doi: 10.1098/rspa.1954.0120. |
[28] |
A. G. Kulikovskiy and G. A. Lyubimov, Magnetohydrodynamics, Addison-Wesley, Reading, MA, 1965. |
[29] |
L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media, Vol.8, 1984 (Translated from the Russian). |
[30] |
X. Li, N. Su and D. Wang, Local strong solution to the compressible magnetohydrodynmic flow with large data, J. Hyper. Diff. Eqns., 8 (2011), 415-436.
doi: 10.1142/S0219891611002457. |
[31] |
F. Lin and P. Zhang, Global small solutions to an MHD type system: The three-dimensional case, Comm. Pure. Appl. Math., 67 (2014), 531-580.
doi: 10.1002/cpa.21506. |
[32] |
A. Novotnỳ and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, USA, 2004. |
[33] |
J. Prüss and G. Simonett, On the Rayleigh-Taylor instability for the two-phase Navier-Stokes equations, Indiana Univ. Math. J., 59 (2010), 1853-1871.
doi: 10.1512/iumj.2010.59.4145. |
[34] |
L. Rayleigh, Analytic solutions of the Rayleigh equations for linear density profiles, Proc. London. Math. Soc., 14 (1883), 170-177. |
[35] |
Y. J. Wang, Critical magnetic number in the magnetohydrodynamic Rayleigh-Taylor instability, J. Math. Phys., 53 (2012), 073701, 22pp.
doi: 10.1063/1.4731479. |
[36] |
Y. Wang and I. Tice, The viscous surface-internal wave problem: Nonlinear Rayleigh-Taylor instability, Comm. PDE, 37 (2012), 1967-2028.
doi: 10.1080/03605302.2012.699498. |
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