# American Institute of Mathematical Sciences

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

## 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

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.
Citation: Fei Jiang, Song Jiang, Weiwei Wang. Nonlinear Rayleigh-Taylor instability for nonhomogeneous incompressible viscous magnetohydrodynamic flows. Discrete and 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. [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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##### 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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