On one-dimensional compressible Navier–Stokes equations for a reacting mixture in unbounded domains

Open Access


In this paper we consider the one-dimensional Navier–Stokes system for a heat-conducting, compressible reacting mixture which describes the dynamic combustion of fluids of mixed kinds on unbounded domains. This model has been discussed on bounded domains by Chen (SIAM J Math Anal 23:609–634, 1992) and Chen–Hoff–Trivisa (Arch Ration Mech Anal 166:321–358, 2003), among others, in which the reaction rate function is a discontinuous function obeying the Arrhenius’ law of thermodynamics. We prove the global existence of weak solutions to this model on one-dimensional unbounded domains with large initial data in \(H^1\). Moreover, the large-time behaviour of the weak solution is identified. In particular, the uniform-in-time bounds for the temperature and specific volume have been established via energy estimates. For this purpose we utilise techniques developed by Kazhikhov–Shelukhin (cf. Kazhikhov in Siber Math J 23:44–49, 1982; Solonnikov and Kazhikhov in Annu Rev Fluid Mech 13:79–95, 1981) and refined by Jiang (Commun Math Phys 200:181–193, 1999, Proc R Soc Edinb Sect A 132:627–638, 2002), as well as a crucial estimate in the recent work by Li–Liang (Arch Ration Mech Anal 220:1195–1208, 2016). Several new estimates are also established, in order to treat the unbounded domain and the reacting terms.


Navier–Stokes equations Compressible Reacting mixture Combustion Global existence Uniform estimates Large-time behaviour 

Mathematics Subject Classification

Primary 35Q30 35Q35 35Q79 Secondary 76N10 76N15 


  1. 1.
    Chen, G.-Q.: Global solutions to the compressible Navier–Stokes equations for a reacting mixture. SIAM J. Math. Anal. 23, 609–634 (1992)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Chen, G.-Q., 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
  3. 3.
    Chen, G.-Q., Kratka, M.: Global solutions to the Navier–Stokes equations for compressible heat-conducting flow with symmetry and free boundary. Commun. PDE 27, 907–943 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chen, G.-Q., Wagner, D.H.: Global entropy solutions to exothermically reacting, compressible Euler equations. J. Differ. Equ. 191, 277–322 (2003)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Donatelli, D., Trivisa, K.: On the motion of a viscous compressible radiative-reacting gas. Commun. Math. Phys 265, 463–491 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Documet, B., Zlotnik, A.: Lyapunov functional method for 1D radiative and reactive viscous gas dynamics. Arch. Ration. Mech. Anal. 177, 185–229 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Huang, X., Li, J., Wang, Y.: Serrin-type blowup criterion for full compressible Navier–Stokes system. Arch. Ration. Mech. Anal. 207, 303–316 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Jiang, S.: Large-time behavior of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains. Commun. Math. Phys. 200, 181–193 (1999)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Jiang, S.: Remarks on the asymptotic behaviour of solutions to the compressible Navier–Stokes equations in the half-line. Proc. R. Soc. Edinb. Sect. A 132, 627–638 (2002)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kanel’, Y.I.: On a model system of equations of one-dimensional gas motion. Differ. Equ. 4, 374–380 (1968)MATHGoogle Scholar
  11. 11.
    Kazhikhov, A.V.: Cauchy problem for viscous gas equations. Siber. Math. J. 23, 44–49 (1982)CrossRefMATHGoogle Scholar
  12. 12.
    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, (PMM vol. 41, n=2, 1977, pp. 282–291). J. Appl. Math. Mech. 41, 273–282 (1977)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Li, J., Liang, Z.: 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. 220, 1195–1208 (2016)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Solonnikov, V.A., Kazhikhov, A.V.: Existence theorems for the equations of motion of a compressible viscous fluid. Ann. Rev. Fluid Mech. 13, 79–95 (1981)CrossRefMATHGoogle Scholar
  15. 15.
    Wang, D.: Global solutions for the mixture of real compressible reacting flows in combustion. Commun. Pure Appl. Anal. 3, 775–790 (2004)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK

Personalised recommendations