×

Nonlinear stability and instability in the Rayleigh-Taylor problem of stratified compressible MHD fluids. (English) Zbl 1414.76023

The authors develop the stability criteria for the stratified compressible magnetic Rayleigh-Taylor problem in Lagrangian coordinates. It is shown that a unique solution with an algebraic decay exists, when a vertical component of the magnetic field is imposed. Instability occurs when the magnetic field is small and only horizontal. Compressibility plays a destabilizing effect countered by the pressure, which is stabilizing. When the authors consider the magnetic compressible viscoelastic Rayleigh-Taylor problem, they find that elasticity has a stabilizing effect stronger than the magnetic fields.

MSC:

76E25 Stability and instability of magnetohydrodynamic and electrohydrodynamic flows
76E30 Nonlinear effects in hydrodynamic stability
76W05 Magnetohydrodynamics and electrohydrodynamics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abidi, H.; Zhang, P., On the global solution of a 3-D MHD system with initial data near equilibrium, Commun. Pure Appl. Math., 70, 1509-1561, (2017) · Zbl 1372.35229 · doi:10.1002/cpa.21645
[2] Adams, R.A.: Sobolev Space. Academic Press, New York (1975)
[3] Adams, R.A., John, J.F.F.: Sobolev Space. Academic Press, New York (2005)
[4] Barrett, JW; Lu, Y.; Süli, E., Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Commun. Math. Sci., 15, 1265-1323, (2017) · Zbl 1390.35007 · doi:10.4310/CMS.2017.v15.n5.a5
[5] Boffetta, G.; Mazzino, A.; Musacchio, S.; Vozella, L., Rayleigh-Taylor instability in a viscoelastic binary fluid, J. Fluid Mech., 643, 127-136, (2010) · Zbl 1189.76226 · doi:10.1017/S0022112009992497
[6] Bollada, PC; Phillips, TN, On the mathematical modelling of a compressible viscoelastic fluid, Arch. Ration. Mech. Anal., 205, 1-26, (2012) · Zbl 1314.76008 · doi:10.1007/s00205-012-0496-5
[7] Bucciantini, N.; Amato, E.; Bandiera, R.; Blondin, JM; Zanna, LD, Magnetic Rayleigh-Taylor instability for pulsar wind nebulae in expanding supernova remnants, Astron. Astrophys., 423, 253-265, (2004) · doi:10.1051/0004-6361:20040360
[8] Chandrasekhar, S.: Hydrodynamic and hydromagnetic stability. In: The International Series of Monographson Physics. Clarendon Press, Oxford (1961)
[9] Fan, DY; Zi, RZ, Strong solutions of 3D compressible Oldroyd-B fluids, Math. Methods Appl. Sci., 36, 1423-1439, (2013) · Zbl 1271.76020 · doi:10.1002/mma.2695
[10] Friedlander, S.; Strauss, W.; Vishik, M., Nonlinear instability in an ideal fluid, Ann. I. H. Poincare-An., 14, 187-209, (1997) · Zbl 0874.76026 · doi:10.1016/S0294-1449(97)80144-8
[11] Giaquinta, M., Martinazzi, L.: An introduction to the regularity theory for elliptic systems. In: Harmonic Maps and Minimal Graphs. Scuola Normale Superiore Pisa, Pisa (2012) · Zbl 1262.35001
[12] Guo, Y.; Hallstrom, C.; Spirn, D., Dynamics near unstable, interfacial fluids, Commun. Math. Phys., 270, 635-689, (2007) · Zbl 1112.76033 · doi:10.1007/s00220-006-0164-4
[13] Guo, Y.; Tice, I., Compressible, inviscid Rayleigh-Taylor instability, Indiana Univ. Math. J., 60, 677-712, (2011) · Zbl 1248.35153 · doi:10.1512/iumj.2011.60.4193
[14] Guo, Y.; Tice, I., Linear Rayleigh-Taylor instability for viscous, compressible fluids, SIAM J. Math. Anal., 42, 1688-1720, (2011) · Zbl 1429.76054 · doi:10.1137/090777438
[15] Guo, Y.; Tice, I., Almost exponential decay of periodic viscous surface waves without surface tension, Arch. Ration. Mech. Anal., 207, 459-531, (2013) · Zbl 1320.35259 · doi:10.1007/s00205-012-0570-z
[16] Guo, Y.; Tice, I., Decay of viscous surface waves without surface tension in horizontally infinite domains, Anal. PDE, 6, 1429-1533, (2013) · Zbl 1292.35206 · doi:10.2140/apde.2013.6.1429
[17] Guo, Y.; Strauss, WA, Instability of periodic BGK equilibria, Commun. Pure Appl. Math., 48, 861-894, (1995) · Zbl 0840.45012 · doi:10.1002/cpa.3160480803
[18] Guo, Y.; WA, S., Nonlinear instability of double-humped equilibria, Ann. I. H. Poincare-An., 12, 339-352, (1995) · Zbl 0836.35130 · doi:10.1016/S0294-1449(16)30160-3
[19] Hester, JJ; Stone, JM; Scowen, Paul A., WFPC2 studies of the crab nebula. III. Magnetic Rayleigh-Taylor instabilities and the origin of the filaments, Astrophys. J., 456, 225-233, (1996) · doi:10.1086/176643
[20] Hide, R., Waves in a heavy, viscous, incompressible, electrically conducting fluid of variable density, in the presence of a magnetic field, Proc. R. Soc. Lond. Ser. A., 233, 376-396, (1955) · Zbl 0067.20203
[21] Hillier, A.; Isobe, H.; Shibata, K.; Berger, T., Numerical simulations of the magnetic Rayleigh-Taylor instability in the Kippenhahn-Schlüter prominence model. II. Reconnection-triggered downflows, Astrophys. J., 756, 110, (2012) · doi:10.1088/0004-637X/756/2/110
[22] Hillier, AS, On the nature of the magnetic Rayleigh-Taylor instability in astrophysical plasma: the case of uniform magnetic field strength, Mon. Notices Roy. Astron. Soc., 462, 2256-2265, (2016) · doi:10.1093/mnras/stw1805
[23] Hu, X.P.: Global existence for two dimensional compressible magnetohydrodynamic flows with zero magnetic diffusivity. arXiv:1405.0274v1 [math.AP] 1 May 2014 (2014)
[24] Hu, XP; Wang, DH, Local strong solution to the compressible viscoelastic flow with large data, J. Differ. Equ., 249, 1179-1198, (2010) · Zbl 1197.35205 · doi:10.1016/j.jde.2010.03.027
[25] Hu, XP; Wang, DH, Global existence for the multi-dimensional compressible viscoelastic flows, J. Differ. Equ., 250, 1200-1231, (2011) · Zbl 1208.35111 · doi:10.1016/j.jde.2010.10.017
[26] Hu, XP; Wang, DH, Strong solutions to the three-dimensional compressible viscoelastic fluids, J. Differ. Equ., 252, 4027-4067, (2012) · Zbl 1275.35072 · doi:10.1016/j.jde.2011.11.021
[27] Hu, XP; Wang, DH, The initial-boundary value problem for the compressible viscoelastic flows, Discrete Contin. Dyn. Syst., 35, 917-934, (2015) · Zbl 1304.35548 · doi:10.3934/dcds.2015.35.917
[28] Hu, XP; Wu, GC, Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows, SIAM J. Math. Anal., 45, 2815-2833, (2013) · Zbl 1295.35091 · doi:10.1137/120892350
[29] Hwang, HJ, Variational approach to nonlinear gravity-driven instability in a MHD setting, Quart. Appl. Math., 66, 303-324, (2008) · Zbl 1139.76020 · doi:10.1090/S0033-569X-08-01116-1
[30] Isobe, H.; Miyagoshi, T.; Shibata, K.; Yokoyam, T., Three-dimensional simulation of solar emerging flux using the earth simulator I. Magnetic Rayleigh-Taylor instability at the top of the emerging flux as the origin of filamentary structure, Publ. Astron. Soc. Jpn., 58, 423-438, (2006) · doi:10.1093/pasj/58.2.423
[31] Isobe, H.; Miyagoshi, T.; Shibata, K.; Yokoyam, T., Filamentary structure on the Sun from the magnetic Rayleigh-Taylor instability, Nature, 434, 478-481, (2005) · doi:10.1038/nature03399
[32] Jang, J.; Tice, I.; Wang, YJ, The compressible viscous surface-internal wave problem: local well-posedness, SIAM J. Math. Anal., 48, 2602-2673, (2016) · Zbl 1349.35273 · doi:10.1137/15M1036026
[33] Jang, J.; Tice, I.; Wang, YJ, The compressible viscous surface-internal wave problem: stability and vanishing surface tension limit, Commun. Math. Phys., 343, 1039-1113, (2016) · Zbl 1383.35163 · doi:10.1007/s00220-016-2603-1
[34] Jiang, F.; Jiang, S., On instability and stability of three-dimensional gravity flows in a bounded domain, Adv. Math., 264, 831-863, (2014) · Zbl 1425.76085 · doi:10.1016/j.aim.2014.07.030
[35] Jiang, F.; Jiang, S., On linear instability and stability of the Rayleigh-Taylor problem in magnetohydrodynamics, J. Math. Fluid Mech., 17, 639-668, (2015) · Zbl 1327.76074 · doi:10.1007/s00021-015-0221-x
[36] Jiang, F.; Jiang, S., On the stabilizing effect of the magnetic field in the magnetic Rayleigh-Taylor problem, SIAM J. Math. Anal., 50, 491-540, (2018) · Zbl 1387.76034 · doi:10.1137/16M1069584
[37] Jiang, F., Jiang, S.: On the dynamical stability and instability of Parker problem. Physica D (2019). https://doi.org/10.1016/j.physd.2018.11.004
[38] Jiang, F.; Jiang, S.; Wang, WW, Nonlinear Rayleigh-Taylor instability in nonhomogeneous incompressible viscous magnetohydrodynamic fluids, Discrete Contin. Dyn. Syst., 9, 1853-1898, (2016) · Zbl 1401.76067 · doi:10.3934/dcdss.2016076
[39] Jiang, F.; Jiang, S.; Wang, YJ, On the Rayleigh-Taylor instability for the incompressible viscous magnetohydrodynamic equations, Commun. Partial Differ. Equ., 39, 399-438, (2014) · Zbl 1302.76217 · doi:10.1080/03605302.2013.863913
[40] Jiang, F.; Jiang, S.; Wu, GC, On stabilizing effect of elasticity in the Rayleigh-Taylor problem of stratified viscoelastic fluids, J. Funct. Anal., 272, 3763-3824, (2017) · Zbl 1446.76063 · doi:10.1016/j.jfa.2017.01.007
[41] Jiang, F.; Wu, GC; Zhong, X., On exponential stability of gravity driven viscoelastic flows, J. Differ. Equ., 260, 7498-7534, (2016) · Zbl 1342.35254 · doi:10.1016/j.jde.2016.01.030
[42] Jun, BI; Norman, ML; Stone, JM, A numerical study of Rayleigh-Taylor instability in magnetic fluids, Astrophys. J., 453, 332-349, (1966) · doi:10.1086/176393
[43] Kruskal, M.; Schwarzchild, M., Some instabilities of a completely ionized plasma, Proc. R. Soc. Lond. Ser. A., 223, 348-360, (1954) · Zbl 0057.24006
[44] Lee, J.M.: Introduction to Smooth Manifolds. Springer, Berlin (2013) · Zbl 1258.53002
[45] Lei, Z.; Liu, C.; Zhou, Y., Global solutions for incompressible viscoelastic fluids, Arch. Ration. Mech. Anal., 188, 371-398, (2008) · Zbl 1138.76017 · doi:10.1007/s00205-007-0089-x
[46] Lin, FH, Some analytical issues for elastic complex fluids, Commun. Pure Appl. Math., 65, 893-919, (2012) · Zbl 1426.76041 · doi:10.1002/cpa.21402
[47] Lin, FH; Liu, C.; Zhang, P., On hydrodynamics of viscoelastic fluids, Commun. Pure Appl. Math., 58, 1437-1471, (2005) · Zbl 1076.76006 · doi:10.1002/cpa.20074
[48] Lin, FH; Zhang, P., On the initial-boundary value problem of the incompressible viscoelastic fluid system, Commun. Pure Appl. Math., 61, 539-558, (2008) · Zbl 1137.35414 · doi:10.1002/cpa.20219
[49] Lin, FH; Zhang, P., Global small solutions to an mhd type system: the three-dimensional, Commun. Pure. Appl. Math., 67, 531-580, (2014) · Zbl 1298.35153 · doi:10.1002/cpa.21506
[50] Pacitto, G.; Flament, C.; Bacri, JC; Widom, M., Rayleigh-Taylor instability with magnetic fluids: experiment and theory, Phys. Rev. E., 62, 7941, (2000) · doi:10.1103/PhysRevE.62.7941
[51] Qian, JZ; Zhang, ZF, Global well-posedness for compressible viscoelastic fluids near equilibrium, Arch. Ration. Mech. Anal., 198, 835-868, (2010) · Zbl 1231.35176 · doi:10.1007/s00205-010-0351-5
[52] Rayleigh, L., Investigation of the character of the equilibrium of an in compressible heavy fluid of variable density, Proc. Lond. Math. Soc., 14, 170-177, (1883) · JFM 15.0848.02
[53] Ren, XX; Wu, JH; Xiang, ZY; Zhang, ZF, Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion, J. Funct. Anal., 267, 503-541, (2014) · Zbl 1295.35104 · doi:10.1016/j.jfa.2014.04.020
[54] Sharma, RC; Sharma, KC, Rayleigh-Taylor instability of two viscoelastic superposed fluids, Acta Phys. Acad. Sci. Hung., 45, 213-220, (1978) · doi:10.1007/BF03157252
[55] Stone, MJ; Gardiner, T., Nonlinear evolution of the magnetohydrodynamic Rayleigh-Taylor instability, Phys. Fluids, 19, 306-327, (2007) · Zbl 1182.76723 · doi:10.1063/1.2767666
[56] Stone, MJ; Gardiner, T., The magnetic Rayleigh-Tayolor instability in three dimensions, Astrophys. J., 671, 1726-1735, (2007) · doi:10.1086/523099
[57] Tan, Z.; Wang, YJ, Global well-posedness of an initial-boundary value problem for viscous non-resistive MHD systems, SIAM J. Math. Anal., 50, 1432-1470, (2018) · Zbl 1387.35470 · doi:10.1137/16M1088156
[58] Taylor, GI, The stability of liquid surface when accelerated in a direction perpendicular to their planes, Proc. R. Soc. Lond. Ser. A., 201, 192-196, (1950) · Zbl 0038.12201
[59] Wang, J.H.: Two-Dimensional Nonsteady Flows and Shock Waves. Science Press, Beijing (1994). (in Chinese)
[60] Wang, Y.J.: Sharp nonlinear stability criterion of viscous non-resistive MHD internal waves in 3D. Arch. Ration. Mech. Anal. (2018). https://doi.org/10.1007/s00205-018-1307-4
[61] Wang, YJ; Tice, I.; Kim, C., The viscous surface-internal wave problem: global well-posedness and decay, Arch. Ration. Mech. Anal., 212, 1-92, (2014) · Zbl 1293.35259 · doi:10.1007/s00205-013-0700-2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.