ONE-DIMENSIONAL VISCOUS RADIATIVE GAS WITH TEMPERATURE DEPENDENT VISCOSITY

数学物理学报(英文版) ›› 2018, Vol. 38 ›› Issue (5): 1515-1548.

ONE-DIMENSIONAL VISCOUS RADIATIVE GAS WITH TEMPERATURE DEPENDENT VISCOSITY

何躏¹, 廖勇凯², 王涛², 赵会江²

1. Institute of Applied Mathematics, Academy of Mathematics and System Science The Chinese Academy of Sciences, Beijing 100190, China;
2. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China;Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China

收稿日期:2018-02-23 出版日期:2018-11-09 发布日期:2018-11-09
通讯作者: Yongkai LIAO,E-mail:yongkai.liao@whu.edu.cn E-mail:yongkai.liao@whu.edu.cn
作者简介:Lin HE,E-mail:helin19891021@163.com;Tao WANG,E-mail:tao.wang@whu.edu.cn;Huijiang ZHAO,E-mail:hhjjzhao@hotmail.com
基金资助:
Research was supported by National Natural Science Foundation of China (11601398, 11671309, 11731008).

ONE-DIMENSIONAL VISCOUS RADIATIVE GAS WITH TEMPERATURE DEPENDENT VISCOSITY

Lin HE¹, Yongkai LIAO², Tao WANG², Huijiang ZHAO²

1. Institute of Applied Mathematics, Academy of Mathematics and System Science The Chinese Academy of Sciences, Beijing 100190, China;
2. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China;Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China

Received:2018-02-23 Online:2018-11-09 Published:2018-11-09
Contact: Yongkai LIAO,E-mail:yongkai.liao@whu.edu.cn E-mail:yongkai.liao@whu.edu.cn
Supported by:
Research was supported by National Natural Science Foundation of China (11601398, 11671309, 11731008).

摘要/Abstract

摘要： This paper is concerned with the construction of global, large amplitude solutions to the Cauchy problem of the one-dimensional compressible Navier-Stokes system for a viscous radiative gas when the viscosity and heat conductivity coefficients depend on both specific volume and absolute temperature. The data are assumed to be without vacuum, mass concentrations, or vanishing temperatures, and the same is shown to be hold for the global solution constructed. The proof is based on some detailed analysis on uniform positive lower and upper bounds of the specific volume and absolute temperature.

关键词: compressible Navier-Stokes system, temperature-dependent viscosity, viscous radiative gas, global solution, asymptotic behavior

Abstract: This paper is concerned with the construction of global, large amplitude solutions to the Cauchy problem of the one-dimensional compressible Navier-Stokes system for a viscous radiative gas when the viscosity and heat conductivity coefficients depend on both specific volume and absolute temperature. The data are assumed to be without vacuum, mass concentrations, or vanishing temperatures, and the same is shown to be hold for the global solution constructed. The proof is based on some detailed analysis on uniform positive lower and upper bounds of the specific volume and absolute temperature.

Key words: compressible Navier-Stokes system, temperature-dependent viscosity, viscous radiative gas, global solution, asymptotic behavior

何躏, 廖勇凯, 王涛, 赵会江. ONE-DIMENSIONAL VISCOUS RADIATIVE GAS WITH TEMPERATURE DEPENDENT VISCOSITY[J]. 数学物理学报(英文版), 2018, 38(5): 1515-1548.

Lin HE, Yongkai LIAO, Tao WANG, Huijiang ZHAO. ONE-DIMENSIONAL VISCOUS RADIATIVE GAS WITH TEMPERATURE DEPENDENT VISCOSITY[J]. Acta Mathematica Scientia, 2018, 38(5): 1515-1548.

参考文献

[1] Antontsev S N, Kazhikhov A V, Monakhov V N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids. Amsterdam:North-Holland, 1990
[2] Chen G-Q, Hoff D, Trivisa K. Global solutions of the compressible Navier-Stokes equations with large discontinous intial data. Comm Partial Differential Equations, 2000, 25:2233-2257
[3] Chen Q, Zhao H-J, Zou Q-Y. Initial-boundary-value problems for one-dimensional compressible Navier-Stokes equations with degenerate transport coefficients. Proc Roy Soc Edinburgh Sect A, 2017, 147(5):935-969
[4] Dafermos C M, Hsiao L. Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity. Nonlinear Anal, 1982, 6:435-454
[5] Donatelli D, Trivisa K. On the motion of a viscous compressible radiative-reacting gas. Comm Math Phys, 2006, 265(2):463-491
[6] Duan R, Liu H-X, Zhao H-J. Nonlinear stability of rarefaction waves for the compressible Navier-Stokes equations with large initial perturbation. Trans Amer Math Soc, 2009, 361:453-493
[7] Ducomet B. A model of thermal dissipation for a one-dimensional viscous reactive and radiative gas. Math Methods Appl Sci, 1999, 22(15):1323-1349
[8] Ducomet B, Feireisl E. On the dynamics of gaseous stars. Arch Ration Mech Anal, 2004, 174(2):221-266
[9] Ducomet B, Feireisl E. The equations of magnetothydrodynamics:on the interaction between matter and radiation in the evolution of gaseous stars. Comm Math Phys, 2006, 266(3):595-626
[10] Ducomet B, Zlotnik A. Lyapunov functional method for 1D radiative and reactive viscous gas dynamics. Arch Ration Mech Anal, 2005, 177(2):185-229
[11] Ducomet B, Zlotnik A. On the large-time behavior of 1D radiative and reactive viscous flows for higher-order kinetics. Nonlinear Anal, 2005, 63(8):1011-1033
[12] He L, Tang S-J, Wang T. Stability of viscous shock waves for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. Acta Math Sci, 2016, 36B(1):34-48
[13] Huang B-K, Liao Y-K. Global stability of combination of viscous contact wave with rarefaction wave for compressible Navier-Stokes equations with temperature-dependent viscosity. Math Models Methods Appl Sci, 2017, 27(12):2321-2379
[14] Huang B-K, Wang L-S, Xiao Q-H. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. Kinet Relat Models, 2016, 9(3):469-514
[15] Huang F-M, Wang T. Stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system. Indiana Univ Math J, 2016, 65(6):1833-1875
[16] Jenssen H K, Karper T K. One-dimensional compressible flow with temperature dependent transport coefficients. SIAM J Math Anal, 2010, 42:904-930
[17] Jiang J, Zheng S-M. Global solvability and asymptotic behavior of a free boundary problem for the one-dimensional viscous radiative and reactive gas. J Math Phys, 2012, 53:1-33
[18] Jiang J, Zheng S-M. Global well-posedness and exponential stability of solutions for the viscous radiative and reactive gas. Z Angew Math Phys, 2014, 65:645-686
[19] Jiang S. Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain. Comm Math Phys, 1996, 178(2):339-374
[20] Jiang S. Large-time behavior of solutions to the equations of a viscous polytropic ideal gas. Ann Mat Pura Appl, 1998, 175:253-275
[21] Jiang S. Large-time behavior of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains. Commun Math Phys, 1999, 200(1):181-193
[22] Jiang S, Zhang P. Global weak solutions to the Navier-Stokes equations for a 1D viscous polytropic ideal gas. Quart Appl Math, 2003, 61:435-449
[23] Kagei Y, Kawashima S. Local solvability of initial boundary value problem for a quasilinear hyperbolic-parabolic system. J Hyperbolic Diff Equatiopns, 2006, 3(2):195-232
[24] Kanel' Y. On a model system of equations of one-dimensional gas motion. Differ Uravn, 1968, 4:374-380
[25] Kawhohl B. Global existence of large solutions to initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas. J Differential Equations, 1985, 58(1):76-103
[26] Kazhikhov A V, Shelukhin V V. Unique global solution with respect to time of initial boundary value problems for one-dimensinal equations of a viscous gas. J Appl Math Mech, 1977, 41(2):273-282
[27] 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. Arch Ration Mech Anal, 2016, 220:1195-1208
[28] Liang Z-L. Large-time behavior for spherically symmetric flow of viscous polytropic gas in exterior unbounded domain with large initial data. arXiv:1405.0569
[29] Liao Y-K, Wang T, Zhao H-J. Global spherically symmetric flows for a viscous radiative and reactive gas in an exterior domain with large initial data. Preprint at arXiv:1801.07169
[30] Liao Y-K, Xu Z-D, Zhao H-J. Cauchy problem of the one-dimensional compressible visocus radiative and reactive gas with degenerate density dependent viscosity (in Chinese). Sci Sin Math, 2019, 49:1-XX, doi:10.1360/012011-XXX
[31] Liao Y-K, Zhang S-X. Global solutions to the one-dimensional compressible Navier-Stokes equation with radiation. J Math Anal Appl, 2018, 461(2):1009-1052
[32] Liao Y-K, Zhao H-J. Global solutions to one-dimensional equations for a self-gravitating viscous radiative and reactive gas with density-dependent viscosity. Commun Math Sci, 2017, 15(5):1423-1456
[33] Liao Y-K, Zhao H-J. Global existence and large-time behavior of solutions to the Cauchy problem of one-dimensional viscous radiative and reactive gas. J Differential Equations, 2018, 265(5):2076-2120
[34] Liu H-X, Yang T, Zhao H-J, Zou Q-Y. One-dimensional compressible Navier-Stokes equations with temperature dependent transport coefficients and large data. SIAM J Math Anal, 2014, 46(3):2185-2228
[35] Luo T. On the outer pressure problem of a viscous heat-conductive one-dimensional real gas. Acta Math Appl, Sin Engl Ser, 1997, 13(3):251-264
[36] Mihalas D, Mihalas B W. Foundations of Radiation Hydrodynamics. New York:Oxford Univ Press, 1984
[37] Nakamura T, Nishibata S. Large-time behavior of spherically symmetric flow of heat-conductive gas in a field of potential forces. Indiana Univ Math J, 2008, 57(2):1019-1054
[38] Nishihara K, Yang T, Zhao H-J. Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations. SIAM J Math Anal, 2004, 35(6):1561-1597
[39] Okada M, Kawashima S. On the equations of one-dimensional motion of compressible viscous fluids. J Math Kyoto Univ, 1983, 23:55-71
[40] Pan R, Zhang W-Z. Compressible Navier-Stokes equations with temperature dependent heat conductivity. Commun Math Sci, 2015, 13:401-425
[41] Qin X-H. Large-time behaviour of solutions to the outflow problem of full compressible Navier-Stokes equations. Nonlinearity, 2011, 24(5):1369-1394
[42] Qin Y M, Hu G L, Wang T G. Global smooth solutions for the compressible viscous and heatconductive gas. Quart Appl Math, 2011, 69(3):509-528
[43] Qin Y-M, Zhang J-L, Su X, Cao J. Global existence and exponential stability of spherically symmetric solutions to a compressible combustion radiative and reactive gas. J Math Fluid Mech, 2016, 18(3):415-461
[44] Tan Z, Yang T, Zhao H-J, Zou Q-Y. Global solutions to the one-dimensional compressible Navier-Stokes-Poisson equations with large data. SIAM J Math Anal, 2013, 45(2):547-571
[45] Tang S-J, Zhang L. Nonlinear stability of viscous shock profiles for compressible Navier-Stokes equations with large initial perturbation. Acta Math Sci, 2018, 38B(3):973-1000
[46] Umehara M, Tani A. Global solution to one-dimensional equations for a self-gravitating viscous radiative and reactive gas. J Differential Equations, 2007, 234(2):439-463
[47] Umehara M, Tani A. Global solvability of the free-boundary problem for one-dimensinal motion of a self-gravitating viscous radiative and reactive gas. Proc Japan Acad Ser A Math Sci, 2008, 84(7):123-128
[48] Wan L, Wang T. Asymptotic behavior for cylindrically symmetric nonbarotropic flows in exterior domains with large data. Nonlinear Anal Real World Appl, 2018, 39:93-119
[49] Wan L, Wang T. Symmetric flows for compressible heat-conducting fluids with temperature dependent viscosity coefficients. J Differential Equations, 2017, 262(12):5939-5977
[50] Wan L, Wang T, Zhao H-J. Asymptotic stability of wave patterns to compressible viscous and heat-conducting gases in the half-space. J Differential Equations, 2016, 261(11):5949-5991
[51] Wan L, Wang T, Zou Q-Y. Stability of stationary solutions to the outflow problem for full compressible Navier-Stokes equations with large initial perturbation. Nonlinearity, 2016, 29(4):1329-1354
[52] Wang T, Zhao H-J. One-dimensional compressible heat-conducting gas with temperaturedependent viscosity. Math Models Methods Appl Sci, 2016, 26(12):2237-2275
[53] Wang T, Zhao H-J, Zou Q-Y. One-dimensional compressible Navier-Stokes equation with large density oscillation. Kinetic Related Models, 2013, 6(3):649-670
[54] Zel'dovich Y B, Raizer Y P. Physics of Shock Waves and High-temperature Hydrodynamic Phenomena. New York:Dover, 2002
[55] Zhang J-W, Xie F. Global solution for a one-dimensional model problem in thermally radiative magnetohydrodynamics. J Differential Equations, 2008, 245(7):1853-1882

ONE-DIMENSIONAL VISCOUS RADIATIVE GAS WITH TEMPERATURE DEPENDENT VISCOSITY

ONE-DIMENSIONAL VISCOUS RADIATIVE GAS WITH TEMPERATURE DEPENDENT VISCOSITY

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