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December  2016, 9(6): 1853-1898. doi: 10.3934/dcdss.2016076

Nonlinear Rayleigh-Taylor instability for nonhomogeneous incompressible viscous magnetohydrodynamic flows

Fei Jiang 1, , Song Jiang 2, and  Weiwei Wang 1,

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

Received  March 2015 Revised  August 2016 Published  November 2016

We investigate the nonlinear instability of a smooth Rayleigh-Taylor steady-state solution (including the case of heavier density with increasing height) to the three-dimensional incompressible nonhomogeneous magnetohydrodynamic (MHD) equations of zero resistivity in the presence of a uniform gravitational field. We first analyze the linearized equations around the steady-state solution. Then we construct solutions of the linearized problem that grow in time in the Sobolev space $H^k$, thus leading to the linear instability. With the help of the constructed unstable solutions of the linearized problem and a local well-posedness result of smooth solutions to the original nonlinear problem, we establish the instability of the density, the horizontal and vertical velocities in the nonlinear problem. Moreover, when the steady magnetic field is vertical and small, we prove the instability of the magnetic field. This verifies the physical phenomenon: instability of the velocity leads to the instability of the magnetic field through the induction equation.
Keywords: Incompressible MHD flows, Navier-Stokes equations., Rayleigh-Taylor instability, steady solutions.
Mathematics Subject Classification: Primary: 76E25; Secondary: 35E1.
Citation: Fei Jiang, Song Jiang, Weiwei Wang. Nonlinear Rayleigh-Taylor instability for nonhomogeneous incompressible viscous magnetohydrodynamic flows. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1853-1898. doi: 10.3934/dcdss.2016076
References:
[1]

R. A. Adams and J. F. Fournier, Sobolev Space, 2nd edition, Academic Press, New York, 2005. Google Scholar

[2]

H. Cabannes, Theoretical Magnetofluiddynamics, Academic Press, New York, 1972. doi: 10.1063/1.3070781.  Google Scholar

[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.  Google Scholar

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T. G. Cowling, Magnetohydrodynamics, Interscience Publishers, New York, 1957.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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L. Grafakos, Classical Fourier Analysis, Second Edition, Springer, 2008.  Google Scholar

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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.  Google Scholar

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Y. Guo and W. Strauss, Instability of periodic BGK equilibria, Comm. Pure Appl. Math., 48 (1995), 861-894. doi: 10.1002/cpa.3160480803.  Google Scholar

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Y. Guo and W. A. Strauss, Nonlinear instability of double-humped equilibria, Ann. Inst. H. Poincaré Anal. Non Linéaire, 12 (1995), 339-352.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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. Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Ph. D. Thesis, Kyoto University, 1983. Google Scholar

[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.  Google Scholar

[28]

A. G. Kulikovskiy and G. A. Lyubimov, Magnetohydrodynamics, Addison-Wesley, Reading, MA, 1965.  Google Scholar

[29]

L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media, Vol.8, 1984 (Translated from the Russian). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

A. Novotnỳ and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, USA, 2004. Google Scholar

[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.  Google Scholar

[34]

L. Rayleigh, Analytic solutions of the Rayleigh equations for linear density profiles, Proc. London. Math. Soc., 14 (1883), 170-177. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. F. Fournier, Sobolev Space, 2nd edition, Academic Press, New York, 2005. Google Scholar

[2]

H. Cabannes, Theoretical Magnetofluiddynamics, Academic Press, New York, 1972. doi: 10.1063/1.3070781.  Google Scholar

[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.  Google Scholar

[4]

T. G. Cowling, Magnetohydrodynamics, Interscience Publishers, New York, 1957.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

L. Grafakos, Classical Fourier Analysis, Second Edition, Springer, 2008.  Google Scholar

[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.  Google Scholar

[12]

Y. Guo and W. Strauss, Instability of periodic BGK equilibria, Comm. Pure Appl. Math., 48 (1995), 861-894. doi: 10.1002/cpa.3160480803.  Google Scholar

[13]

Y. Guo and W. A. Strauss, Nonlinear instability of double-humped equilibria, Ann. Inst. H. Poincaré Anal. Non Linéaire, 12 (1995), 339-352.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Ph. D. Thesis, Kyoto University, 1983. Google Scholar

[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.  Google Scholar

[28]

A. G. Kulikovskiy and G. A. Lyubimov, Magnetohydrodynamics, Addison-Wesley, Reading, MA, 1965.  Google Scholar

[29]

L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media, Vol.8, 1984 (Translated from the Russian). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

A. Novotnỳ and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, USA, 2004. Google Scholar

[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.  Google Scholar

[34]

L. Rayleigh, Analytic solutions of the Rayleigh equations for linear density profiles, Proc. London. Math. Soc., 14 (1883), 170-177. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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