Acta Mathematica Sinica, English Series

, Volume 34, Issue 5, pp 827–842 | Cite as

Global Existence and Asymptotic Behavior of Solutions to a Free Boundary Problem for the 1D Viscous Radiative and Reactive Gas

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Abstract

In this paper, we study a free boundary problem for the 1D viscous radiative and reactive gas. We prove that for any large initial data, the problem admits a unique global generalized solution. Meanwhile, we obtain the time-asymptotic behavior of the global solutions. Our results improve and generalize the previous work.

Keywords

Radiative and reactive gases global solution free boundary problem asymptotic behavior 

MSMR(2010) Subject Classification

35Q30 35R35 35D35 76N10 

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Notes

Acknowledgements

We thank the referees for their time and valuable comments.

References

  1. [1]
    Chen, G.: Global solution to the compressible Navier–Stokes equations for a reacting mixture. SIAM J. Math. Anal., 23, 609–634 (1992)MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    Chen, G., Hoff, D., Trivisa, K.: Global solutions of the compressible Navier–Stokes equations with large discontinuous initial data. Commun. Partial Differential Equations, 25, 2233–2257 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    Chen, G., Hoff, D., Trivisa, K.: Global solutions to a model for exothermically reacting, compressible flows with large discontinuous initial data. Arch. Ration. Mech. Anal., 166, 321–358 (2003)MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    Chen, G., Trivisa, K.: Analysis on models for exothermically reacting, compressible flows with large discontinuous initial data. Contemp. Math., 371, 73–91 (2005)CrossRefMATHGoogle Scholar
  5. [5]
    Donatelli, D., Trivisa, K.: On the motion of a viscous compressible radiative-reacting gas. Commun. Math. Phys., 265(2), 463–491 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Donatelli, D., Trivisa, K.: A multi-dimensional model for the combustion of compressible fluids. Arch. Ration. Mech. Anal., 185, 379–408 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Ducomet, B.: On the stability of a stellar structure in one dimension II: The reactive case. Math. Model. Numer. Anal., 31, 381–407 (1997)MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    Ducomet, B.: A model of thermal dissipation for a one-dimensional viscous reactive and radiative gas. Math. Methods Appl. Sci., 22, 1323–1349 (1999)MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    Ducomet, B.: Some stability results for reactive Navier–Stokes–Poisson systems. In: Evolution Equations: Existence, Regularity and Singularities (Warsaw, 1998), 83–118, Banach Center Publications, 52, Institute of Mathematics, Polish Academy of Sciences, Warsaw, 2000Google Scholar
  10. [10]
    Ducomet, B., Feireisl, E.: On the dynamics of gaseous stars. Arch. Ration. Mech. Anal., 174, 221–266 (2004)MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    Ducomet, B., Feireisl, E.: The equations of magnetohydrodynamics: on the interaction between matter and radiation in the evolution of gaseous stars. Commun. Math. Phys., 266(3), 595–629 (2006)MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    Ducomet, B., Zlotnik, A.: Stabilization for viscous compressible heat-conducting media equations with nonmonotone state functions. C. R. Acad. Sci. Paris Ser. I, 334, 119–124 (2002)MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    Ducomet, B., Zlotnik, A.: Lyapunov functional method for 1D radiative and reactive viscous gas dynamics. Arch. Ration. Mech. Anal., 177(2), 185–229 (2005)MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    Ducomet, B., Zlotnik, A.: On the large-time behavior of 1D radiative and reactive viscous flows for higherorder kinetics. Nonlinear Anal., 63(8), 1011–1033 (2005)MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    Feireisl, E.: Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004MATHGoogle Scholar
  16. [16]
    Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids, Birkhäuser Verlag, Basel–Boston–Berlin, 2009CrossRefMATHGoogle Scholar
  17. [17]
    Feireisl, E., Petzeltov´a, H., Trivisa, K.: Multicomponent reactive flows: global-in-time existence for large data. Commun. Pure Appl. Anal., 7(5), 1017–1047 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    Guo, B., Zhu, P.: Asymptotic behavior of the solution to the system for a viscous reactive gas. J. Differential Equations, 155, 177–202 (1999)MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    Guo, Z., Li, H., Xin, Z.: Lagrange structure and dynamics for solutions to the spherically symmetric compressible Navier–Stokes equations. Commun. Math. Phys., 309(2), 371–412 (2012)MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    Hoff, D.: Spherically symmetric solutions of the Navier–Stokes equations for compressible, isothermal flow with large, discontinuous initial data. Indiana Univ. Math. J., 41(4), 1225–1302 (1992)MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    Hsiao, L., Luo, T.: Large-time behaviour of solutions for the outer pressure problem of a viscous heatconductive onedimensional real gas. Proc. R. Soc. Edinb. Sect. A Math., 126(6), 1277–1296 (1996)CrossRefMATHGoogle Scholar
  22. [22]
    Jiang, S.: On initial boundary value problems for a viscous heat-conducting one-dimensional real gas. J. Differential Equations, 110, 157–181 (1994)MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    Jiang, S.: On the asymptotic behavior of the motion of a viscous, heat-conducting, one-dimensional real gas. Math. Z., 216(2), 317–336 (1994)MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    Jiang, S.: Global spherically symmetric solutions of the equations of a viscous polytropic ideal gas in an exterior domain. Commun. Math. Phys., 178, 339–374 (1996)MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    Jiang, S., Zhang, P.: Global spherically symmetric solutions of the compressible isentropic Navier–Stokes equations. Commun. Math. Phys., 215, 559–581 (2001)CrossRefMATHGoogle Scholar
  26. [26]
    Jiang, J., Zheng, S.: Global solvability and asymptotic behavior of a free boundary problem for the onedimensional viscous radiative and reactive gas. J. Math. Phys., 53, 1–33 (2012)CrossRefGoogle Scholar
  27. [27]
    Jiang, J., Zheng, S.: Global well-posedness and exponential stability of solutions for the viscous radiative and reactive gas. Z. Angew. Math. Phys., 65, 645–686 (2014)MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    Kawohl, B.: Global existence of large solutions to initial boundary value problems for the equations of one-dimensional motion of viscous polytropic gases. J. Differential Equations, 58, 76–103 (1985)MathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    Kazhikhov, A. V., Shelukhin, V. V.: Unique global solution with respect to time of the initial-boundary value problems for one-dimensional equations of a viscous gas. J. Appl. Math. Mech., 41, 273–282 (1977)MathSciNetCrossRefGoogle Scholar
  30. [30]
    Nagasawa, T.: On the outer pressure problem of the one-dimensional polytropic ideal gas. Japan J. Appl. Math., 5, 53–85 (1988)MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    Nagasawa, T.: On the asymptotic behavior of the one-dimensional motion of the polytropic ideal gas with stress-free condition. Q. Appl. Math., 46(4), 665–679 (1988)MathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    Qin, Y.: Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, Volume 184, Advances in Partial Differential Equations, Birkhäuser Verlag AG, Basel–Boston–Berlin, 2008Google Scholar
  33. [33]
    Qin, Y.: Exponential stability for a nonlinear one-dimensional heat-conductive viscous real gas. J. Math. Anal. Appl., 272, 507–535 (2002)MathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    Qin, Y.: Universal attractor in H4 for the nonlinear one-dimensional compressible Navier–Stokes equations. J. Differential Equations, 207, 21–72 (2004)MathSciNetCrossRefMATHGoogle Scholar
  35. [35]
    Qin, Y., Hu, G.: Global smooth solutions for 1D thermally radiative magnetohydrodynamics. J. Math. Phys., 52(2), 1853–1882 (2011)MathSciNetCrossRefMATHGoogle Scholar
  36. [36]
    Qin, Y., Hu, G., Wang, T.: Global smooth solutions for the compressible viscous and heat-conductive gas. Quar. Appl. Math., 69(3), 509–528 (2011)MathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    Qin, Y., Hu, G., Wang, T., et al.: Remarks on global smooth solutions to a 1D self-gravitating viscous radiative and reactive gas. J. Math. Anal. Appl., 408(1), 19–26 (2013)MathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    Qin, Y., Huang, L.: Global Well-Posedness of Nonlinear Parabolic-Hyperbolic Coupled Systems, Frontiers in Mathematics, Springer Basel AG, 2012CrossRefGoogle Scholar
  39. [39]
    Qin, Y., Zhang, J., Su, X., et al.: Global existence and exponential stability of spherically symmetric solutions to the compressible combustion radiative and reactive gas. J. Math. Fluid Mech., 18, 415–461 (2016)MathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    Shen, W., Zheng, S.: On the coupled Cahn–Hilliard equations. Comm. Partial Differential Equations, 18, 701–727 (1993)MathSciNetCrossRefMATHGoogle Scholar
  41. [41]
    Tani, A.: On the first initial-boundary value problem of compressible viscous fluid motion. Publ. Res. Inst. Math. Sci., 13, 193–253 (1977)CrossRefMATHGoogle Scholar
  42. [42]
    Tani, A.: On the free boundary value problem for the compressible viscous fluid motion. J. Math. Kyoto Univ., 21, 839–859 (1981)MathSciNetCrossRefMATHGoogle Scholar
  43. [43]
    Umehara, M., Tani, A.: Global solution to the one-dimensional equations for a self-gravitating viscous radiative and reactive gas. J. Differential Equations, 234(2), 439–463 (2007)MathSciNetCrossRefMATHGoogle Scholar
  44. [44]
    Umehara, M., Tani, A.: Temporally global solution to the equations for a spherically symmetric viscous radiative and reactive gas over the rigid core. Anal. Appl., 6, 183–211 (2008)MathSciNetCrossRefMATHGoogle Scholar
  45. [45]
    Wang, D.: Global solution for the mixture of real compressible reacting flows in combustion. Commun. Pure Appl. Anal., 3(4), 775–790 (2004)MathSciNetCrossRefMATHGoogle Scholar
  46. [46]
    Xin, Z., Yuan, H.: Vacuum state for spherically symmetric solutions of the compressible Navier–Stokes equations. J. Hyperbolic Differential Equations, 3, 403–442 (2006)MathSciNetCrossRefMATHGoogle Scholar
  47. [47]
    Zhang, J.: Remarks on global existence and exponential stability of solutions for the viscous radiative and reactive gas with large initial data. Nonlinearity, 30, 1221–1261 (2017)MathSciNetCrossRefMATHGoogle Scholar
  48. [48]
    Zhang, J., Xie, F.: Global solution for a one-dimensional model problem in thermally radiative magnetohydrodynamics. J. Differential Equations, 245(7), 1853–1882 (2008)MathSciNetCrossRefMATHGoogle Scholar
  49. [49]
    Zheng, S., Qin, Y.: Universal attractors for the Navier–Stokes equations of compressible and heatconductive fluid in bounded annular domains in R n. Arch. Ration. Mech. Anal., 160(2), 153–179 (2001)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Applied Mathematics, College of ScienceZhongyuan University of TechnologyZhengzhouP. R. China
  2. 2.Department of Applied Physics, College of ScienceZhongyuan University of TechnologyZhengzhouP. R. China

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