This paper is concerned with an initial boundary value problem for the planar magnetohydrodynamic compressible flow with temperature dependent heat conductivity in a half-line. In particular, the transverse magnetic field is assumed to satisfy the Neumann boundary condition, which was first investigated by Kazhikhov in 1987. We establish the global existence of the unique strong solutions to the MHD equations without any smallness conditions on the initial data. More precisely, our result can be regarded as a natural generalization of Kazhikov's result for applying the constant heat-conductivity in bounded domains to the degenerate case in unbounded domains.
Mengdi TONG1
,
Xue Wang2
,
Rong ZHANG3
,
*
. GLOBAL STRONG SOLUTIONS TO THE PLANAR COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS WITH DEGENERATE HEAT-CONDUCTIVITY IN THE HALF-LINE[J]. Acta mathematica scientia, Series B, 2026
, 46(1)
: 189
-208
.
DOI: 10.1007/s10473-026-0112-7
[1] Amosov A A, Zlotnik A A. A difference scheme on a non-uniform mesh for the equations of one-dimensional magnetic gas dynamics. USSR Computational Mathematics and Mathematical Physics, 1989, ${\bf 29}$: 129-139
[2] Antontsev S N, Kazhikhov A V, Monakhov V N.Boundary Value Problems in Mechanics of Nonhomogeneous Fluids. New York: North-Holland, 1990
[3] Cai G C, Chen C P, Peng Y F. Nonlinearly stability of solutions on the outer pressure problem of compressible Navier-Stokes system with temperature-dependent heat conductivity. Journal of Mathematical Fluid Mechanics, 2022, ${\bf 24}$: Article 60
[4] Cao Y B, Peng Y, Sun Y. Global existence of strong solutions to MHD with density-depending viscosity and temperature-depending heat-conductivity in unbounded domains. Journal of Mathematical Physics, 2021, ${\bf 62}$: Article 011508
[5] Chen G Q, Wang D H. Global solutions for nonlinear magnetohydrodynamics with large initial data. Journal of Differential Equations, 2002, ${\bf 182}$: 344-376
[6] Chen G Q, Wang D H. Existence and continuous dependence of large solutions for the magnetohydrodynamics equations. Zeitschrift Fur Angewandte Mathematik Und Physik, 2003, ${\bf 54}$: 608-632
[7] Hu Y, Ju Q. Global large solutions of magnetohydrodynamics with temperature-dependent heat conductivity. Zeitschrift Fur Angewandte Mathematik Und Physik, 2015, ${\bf 66}$: 865-889
[8] Huang B, Shi X D. Nonlinearly exponential stability of compressible Navier Stokes system with degenerate heat-conductivity. Journal of Differential Equations, 2020, ${\bf 268}$: 2464-2490
[9] Huang B, Shi X D, Sun Y. Global strong solutions to magnetohydrodynamics with density-dependent viscosity and degenerate heat-conductivity. Nonlinearity, 2019, ${\bf 32}$: 4395-4412
[10] Huang B, Shi X D, Sun Y. Large-time behavior of magnetohydrodynamics with temperature-dependent heat-conductivity. Journal of Mathematical Fluid Mechanics, 2021, ${\bf 23}$: Article 67
[11] Jenssen H K, Karper T K. One-dimensional compressible flow with temperature dependent transport coefficients. Journal on Mathematical Analysis, 2010, ${\bf 42}$: 904-930
[12] Jiang S. On the asymptotic behavior of the motion of a viscous, heat-conducting, one-dimensional real gas. Mathematische Zeitschrift, 1994, ${\bf 216}$: 317-336
[13] Jiang S. Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain. Communications in Mathematical Physics, 1996, ${\bf 178}$: 339-374
[14] Jiang S. Large-time behavior of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains. Communications in Mathematical Physics, 1999, ${\bf 200}$: 181-193
[15] Jiang S. Remarks on the asymptotic behavior of solutions to the compressible Navier-Stokes equations in the half line. Proceedings of the Royal Society of Edinburgh: Section A mathematics, 2002, ${\bf 132}$: 627-638
[16] Kawashima S, Nishida T. Global solutions to the initial value problem for the equations of one dimensional motion of viscous polytropic gases. Journal of Mathematics of Kyoto University, 1981, ${\bf 21}$: 825-837
[17] Kazhikhov A V. Cauchy problem for viscous gas equations. Siberian Mathematical Journal, 1982, ${\bf 23}$: 44-49
[18] Kazhikhov A V.A priori estimates for the solutions of equations of magnetic gas dynamics//Boundary value problems for equations of mathematical physics. Krasnoyarsk: 1987 (In Russian)
[19] Kazhikhov A V, Shelukhin V V. Unique global solution with respect to time of initial boundary value problems for one-dimensional equations of a viscous gas. Journal of Applied Mathematics and Mechanics, 1977, ${\bf 41}$: 273-282
[20] Li J, Liang Z L. Some uniform estimates and large-time behavior of solutions to one-dimensional compressible Navier-Stokes system in unbounded domains with large data. Archive for Rational Mechanics and Analysis, 2016, ${\bf 220}$: 1195-1208
[21] Li K X, Shu X L, Xu X J. Global existence of strong solutions to compressible Navier-Stokes system with degenerate heat conductivity in unbounded domains. Mathematical Methods in the Applied Sciences, 2020, ${\bf 43}$: 1543-1554
[22] Lv B Q, Shi X D, Xiong C F. Global existence of strong solutions to the planar compressible magnetohydrodynamic equations with large initial data in unbounded domains. Communications in Mathematical Sciences, 2021, ${\bf 19}$: 1655-1671
[23] Nagasawa J. Le problem de Cauchy pour les quations diffrentielles d un fluide gnral. Bulletin des Sciences Mathematiques, 1962, ${\bf 90}$: 487-497
[24] Nagasawa T. On the one-dimensional motion of the polytropic ideal gas non-fixed on the boundary. Journal of Differential Equations.1986, ${\bf 65}$: 49-67
[25] Okada M, Kawashima S. On the equations of one-dimensional motion of compressible viscous fluids. Journal of Mathematics of Kyoto University, 1983, ${\bf 23}$: 55-71
[26] Pan R H, Zhang W Z. Compressible Navier-Stokes equations with temperature dependent heat conductivities. Communications in Mathematical Sciences, 2015, ${\bf 13}$: 401-425
[27] Tani A. On the first initial-boundary value problem of compressible viscous fluid motion. Publications of the Research Institute for Mathematical Sciences, 1977, ${\bf 13}$: 193-253
[28] Wang D H. Large solutions to the initial-boundary value problem for planar magnetohydrodynamics. SIAM Journal on Applied Mathematics, 2003, ${\bf 63}$: 1424-1441
