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Large-time behavior of magnetohydrodynamics with temperature-dependent heat-conductivity. (English) Zbl 1472.35299

Summary: For the strong solutions to the equations of a planar magnetohydrodynamic compressible flow with the heat conductivity proportional to a nonnegative power of the temperature, we first prove that both the specific volume and the temperature are proved to be bounded from below and above independently of time. Then, we also show that the global strong solution is nonlinearly exponentially stable as time tends to infinity. This is the first result obtaining the exponential stability behavior of strong solutions to the equations of a planar magnetohydrodynamic compressible flow without any smallness conditions on the data. Our result can be regarded as a natural generalization of the previous ones for the compressible Navier-Stokes system to MHD system with either constant heat-conductivity or nonlinear and temperature-depending heat-conductivity. As a direct consequence, it is shown that the global strong solution to the constant heat-conductivity MHD system whose existence is obtained by A. V. Kazhikhov [“A priori estimates for the solutions of equations of magnetic gas dynamics” (Russian), in: Boundary-value problems for equations of mathematical physics. Krasnoyarsk. 84–94 (1987)] is nonlinearly exponentially stable.

MSC:

35Q35 PDEs in connection with fluid mechanics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
76W05 Magnetohydrodynamics and electrohydrodynamics
35D35 Strong solutions to PDEs
35B35 Stability in context of PDEs
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References:

[1] Amosov, AA; Zlotnik, AA, Global generalized solutions of the equations of the one-dimensional motion of a viscous heat-conducting gas, Soviet Math. Dokl., 38, 1-5 (1989) · Zbl 0850.76595
[2] Amosov, AA; Zlotnik, AA, A difference scheme on a non-uniform mesh for the equations of one-dimensional magnetic gas dynamics, U.S.S.R. Compu. Maths. Math. Phys., 29, 129-139 (1989) · Zbl 0702.76126 · doi:10.1016/0041-5553(89)90018-9
[3] Amosov, AA; Zlotnik, AA, Solvability “in the large” of a system of equations of the one-dimensional motion of an inhomogeneous viscous heat-conducting gas, Math. Notes, 52, 753-763 (1992) · Zbl 0779.76079 · doi:10.1007/BF01236769
[4] Amosov, AA; Zlotnik, AA, On the stability of generalizedsolutions of equations of one-dimensional motion of a viscous heat-conducting gas, Sib. Math. J., 38, 663-684 (1997) · Zbl 0880.35026 · doi:10.1007/BF02674573
[5] Antontsev, SN; Kazhikhov, AV; Monakhov, VN, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids (1990), Amsterdam, New York: North-Holland, Amsterdam, New York · Zbl 0696.76001
[6] Cabannes, H., Theoretical Magnetofluiddynamics (1970), New York: Academic Press, New York
[7] Chapman, S.; Colwing, TG, The mathematical theory of nonuniform gases Cambridge Mathematical Library (1994), New York: Springer, New York
[8] Chen, GQ; Wang, DH, Global solutions for nonlinear magnetohydrodynamics with large initial data, J. Differ. Equ., 182, 344-376 (2002) · Zbl 1001.76118 · doi:10.1006/jdeq.2001.4111
[9] Chen, GQ; Wang, DH, Existence and continuous dependence of large solutions for the magnetohydrodynamics equations, Z. Angew. Math. Phys., 54, 608-632 (2003) · Zbl 1036.35126 · doi:10.1007/s00033-003-1017-z
[10] Duan, R.; Jiang, F.; Jiang, S., On the Rayleigh Taylor instability for incompressible, inviscid magnetohydrodynamic flows, SIAM J. Appl. Math., 71, 1990-2013 (2011) · Zbl 1384.76021 · doi:10.1137/110830113
[11] Fan, JS; Huang, SX; Li, FC, Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vaccum, Kinetic Related Model., 10, 1035-1053 (2017) · Zbl 1357.35244 · doi:10.3934/krm.2017041
[12] Fan, JS; Jiang, S.; Nakamura, G., Vanishing shear viscosity limit in the magnetohydrodynamics equations, Commun. Math. Phys., 270, 691-708 (2007) · Zbl 1190.76172 · doi:10.1007/s00220-006-0167-1
[13] Hoff, D.; Tsyganov, E., Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys., 56, 791-804 (2005) · Zbl 1077.35071 · doi:10.1007/s00033-005-4057-8
[14] Hu, Y.; Ju, Q., Global large solutions of magnetohydrodynamics with temperature-dependent heat conductivity, Z. Angew. Math. Phys., 66, 865-889 (2015) · Zbl 1320.35279 · doi:10.1007/s00033-014-0446-1
[15] Huang, B.; Shi, XD, Nonlinearly exponential stability of compressible Navier-Stokes system with degenerate heat-conductivity, J. Differ. Equ. (2020) · Zbl 1433.35227 · doi:10.1016/j.jde.2019.09.006
[16] Huang, B.; Shi, XD; Sun, Y., Global strong solutions to magnetohydrodynamics with density-dependent viscosity and degenerate heat-conductivity, Nonlinearity, 32, 4395-4412 (2019) · Zbl 1430.35197 · doi:10.1088/1361-6544/ab3059
[17] Jeffrey, A.; Taniuti, T., Non-Linear Wave Propagation (1964), New York: With Applications to Physics and Magnetohydrodynamics. Academic Press, New York · Zbl 0117.21103
[18] Jenssen, HK; Karper, TK, One-dimensional compressible flow with temperature de- pendent transport coefficients, SIAM J. Math. Anal., 42, 904-930 (2010) · Zbl 1429.76084 · doi:10.1137/090763135
[19] Jiang, F.; Jiang, S.; Wang, YJ, On the Rayleigh-Taylor instability for incompressible viscous magnetohydrodynamic equations, Commun. Partial Differ. Equ., 39, 399-438 (2014) · Zbl 1302.76217 · doi:10.1080/03605302.2013.863913
[20] Kazhikhov, A. V.: To a theory of boundary value problems for equations of one-dimensional nonstationary motion of viscous heat-conduction gases, in: Boundary Value Problems for Hydrodynamical Equations, No. 50, Institute of Hydrodynamics, Siberian Branch Acad. USSR, 1981, pp. 37-62, in Russian · Zbl 0515.76076
[21] Kazhikhov, AV, A priori estimates for the solutions of equations of magnetic gas dynamics (1987), Krasnoyarsk: Boundary value problems for equations of mathematical physics, Krasnoyarsk
[22] Kazhikhov, AV; Shelukhin, VV, Unique global solution with respect to time of initial boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41, 273-282 (1977) · Zbl 0393.76043 · doi:10.1016/0021-8928(77)90011-9
[23] Kulikovskiy, A. G., Lyubimov, G. A.: Magnetohydrodynamics. Addison-Wesley, Reading (1965)
[24] Laudau, LD; Lifshitz, EM, Electrodynamics of Continuous Media (1984), New York: Pergamon, New York
[25] Li, J.; Liang, ZL, Some uniform estimates and large-time behavior of solutions to one-dimensional compressible Navier-Stokes system in unbounded domains with large data, Arch. Rational Mech. Anal., 220, 1195-1208 (2016) · Zbl 1334.35242 · doi:10.1007/s00205-015-0952-0
[26] Nagasawa, T., On the one-dimensional motion of the polytropic ideal gas non-fixed on the boundary, J. Diff. Eqs., 65, 49-67 (1986) · Zbl 0598.34021 · doi:10.1016/0022-0396(86)90041-0
[27] Nagasawa, T., On the asymptotic behavior of the one-dimensional motion of the polytropic ideal gas with stress-free condition, Quart. Appl. Math., 46, 665-679 (1988) · Zbl 0693.76075 · doi:10.1090/qam/973382
[28] Nagasawa, T.: On the one-dimensional free boundary problem for the heat-conductive compressible viscous gas. In: Mimura, M., Nishida, T. (eds.) Recent Topics in Nonlinear PDE IV, Lecture Notes in Num. Appl. Anal. 10, Amsterdam, Tokyo: Kinokuniya/North-Holland, 1989, pp. 83-99 · Zbl 0712.35113
[29] Nishida, T.; Nishida, T.; Mimura, M.; Fujii, H., Equations of motion of compressible viscous fluids, Pattern and Waves, 97-128 (1986), Amsterdam, Tokyo: Kinokuniya/North-Holland, Amsterdam, Tokyo · Zbl 0632.76081
[30] Okada, M.; Kawashima, S., On the equations of one-dimensional motion of compressible viscous fluids, J. Math. Kyoto Univ., 23, 55-71 (1983) · Zbl 0529.76070
[31] Polovin, RV; Demutskii, VP, Fundamentals of Magnetohydrodynamics (1990), New York: Consultants Bureau, New York
[32] Qin, Y.: Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, Operator Theory, Advances and Applications, vol. 184. Birkhäuser, Basel, Boston, Berlin (2008) · Zbl 1173.35004
[33] Wang, DH, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math., 63, 1424-1441 (2003) · Zbl 1028.35100 · doi:10.1137/S0036139902409284
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