REMARKS ON THE CAUCHY PROBLEM OF THE ONE-DIMENSIONAL VISCOUS RADIATIVE AND REACTIVE GAS

doi:10.1007/s10473-020-0410-4

Abstract

Abstract: This paper is concerned with the large-time behavior of solutions to the Cauchy problem of a one-dimensional viscous radiative and reactive gas. Based on the elaborate energy estimates, we develop a new approach to derive the upper bound of the absolute temperature by avoiding the use of auxiliary functions Z(t) and W(t) introduced by Liao and Zhao [J. Differential Equations, 2018, 265(5): 2076-2120]. Our results also improve upon the results obtained in Liao and Zhao [J. Differential Equations, 2018, 265(5): 2076-2120].

Key words: large-time behavior, viscous radiative and reactive gas, Cauchy problem, large initial data

CLC Number:

35D05

Yongkai LIAO. REMARKS ON THE CAUCHY PROBLEM OF THE ONE-DIMENSIONAL VISCOUS RADIATIVE AND REACTIVE GAS[J].Acta mathematica scientia,Series B, 2020, 40(4): 1020-1034.

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References

[1] Chen G Q. Global solutions to the compressible Navier-Stokes equations for a reacting mixture. SIAM J Math Anal, 1992, 23(3):609-634
[2] Donatelli D, Trivisa K. On the motion of a viscous compressible radiative-reacting gas. Comm Math Phys, 2006, 265(2):463-491
[3] 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
[4] 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
[5] 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
[6] Feireisl E, Novotný A. Singular limits in thermodynamics of viscous fluids//Advances in Mathematical Fluid Mechanics. Basel:Birkhäuser Verlag, 2009
[7] He L, Liao Y K, Wang T, Zhao H J. One-dimensional viscous radiative gas with temperature dependent viscosity. Acta Math Sci, 2018, 38B(5):1515-1548
[8] Jiang J, Zheng S M. Global solvability and asymptotic behavior of a free boundary problem for the onedimensional viscous radiative and reactive gas. J Math Phys, 2012, 53:1-33
[9] 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
[10] Jiang S. Large-time behavior of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains. Comm Math Phys, 1999, 200:181-193
[11] Kawohl B. Global existence of large solutions to initial boundary value problems for a viscous, heatconducting, one-dimensional real gas. J Differential Equations, 1985, 58(1):76-103
[12] Liao Y K, Wang T, Zhao H J. Global spherically symmetric flows for a viscous radiative and reactive gas in an exterior domain. J Differential Equations, 2019, 266(10):6459-6506
[13] 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) Scientia Sinica Mathematica, 2019, 49:1-20
[14] 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
[15] 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
[16] 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
[17] Mihalas D, Mihalas B W. Foundations of Radiation Hydrodynamics. New York:Oxford Univ Press, 1984
[18] Qin Y M, Hu G L. Global smooth solutions for 1D thermally radiative magnetohydrodynamics. J Math Phys, 2011, 52(2):023102
[19] Qin Y M, Hu G L, Wang T G. Global smooth solutions for the compressible viscous and heat-conductive gas. Quart Appl Math, 2011, 69(3):509-528
[20] Qin Y M, Hu G L, Wang T G, Huang L, Ma Z Y. Remarks on global smooth solutions to a 1D self-gravitating viscous radiative and reactive gas. J Math Anal Appl, 2013, 408(1):19-26
[21] Qin Y M, Liu X, Yang X G. Global existence and exponential stability for a 1D compressible and radiative MHD flow. J Differential Equations, 2012, 253(5):1439-1488
[22] 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
[23] Song C M, Li H, Zhang J L. Global existence and asymptotic behavior of solutions to a free boundary problem for the 1D viscous radiative and reactive gas. Acta Math Sci, 2018, 34B(5):827-842
[24] 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
[25] Umehara M, Tani A. Global solvability of the free-boundary problem for one-dimensinal motion of a selfgravitating viscous radiative and reactive gas. Proc Japan Acad Ser A Math Sci, 2008, 84(7):123-128
[26] Zhang J L. Remarks on global existence and exponential stability of solutions for the viscous radiative and reactive gas with large initial data. Nonlinearity, 2017, 30(4):1221-1261
[27] 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