Abstract
The global existence and large time behavior of solutions are established for a model which describes the dynamic combustion of a compressible, exothermically reacting fluid. Necessary and sufficient conditions for complete combustion are presented in certain cases. The adiabatic “constant” and specific heat depend on the mass fraction of the reactant and therefore vary in time and space. This model is formulated by the Navier-Stokes equations expressing the conservation of mass, the balance of momentum and energy, and two-species chemical kinetics. The results are obtained for large and discontinuous initial data (cf. Chen, Hoff and Trivisa [7].
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References
A. A. Amosov and A. A. Ziotnick, A semidiscrete method for solving equations of the one-dimensional motion of a non-homogeneous viscous heat-conducting gas with nonsmooth data, Izv. Vyssh. Uchebn. Zaved. Mat. 1997, 3–19 (Russian); transl. in Russian Math. (Iz. VUZ) 41(1997), 1-17.
A. Bourlioux, A. J. Majda, and V. Roytburd, Theoretical and numerical structure for unstable one-dimensional detonations, SIAM J. Appl. Math. bd51 (1991), 303–343.
T. Chang and L. Hsiao, The Riemann problem and interaction of waves in gas dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics, 41, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc.: New York, 1989.
G.-Q. Chen, Global solutions to the compressible Navier-Stokes equations for a reacting mixture, SIAM J. Math. Anal. bd23 (1992), 609–634.
G.-Q. Chen, D. Hoff, and K. Trivisa, Global solutions of the compressible Navier-Stokes equations with large discontinuous initial data, Comm. Partial Duff. Eqs. bd25 (2000), 2233–2257.
G.-Q. Chen, D. Hoff, and K. Trivisa, On the Navier-Stokes equations for exothermically reacting compressible fluids, Acta Math. Appi. Sinica, bd18 (2002), 15–36.
G. Q. Chen, D. Hoff and K. Trivisa. Global Solutions to a Model for Exothermically Reacting, Compressible Flows with Large Discontinuous Initial Data. Submitted to Arch. Ration. Mech. Anal.
G.-Q. Chen and D. Wagner, Global entropy solutions to exothermically reacting, compressible Euler equations, J. Duff. Eqs. (to appear).
P. Collela, A. Majda and V. Roytburd, Theoretical and numerical structure for reacting shock waves, SIAM J. Sci. Stat. Comput. bd7 (1986), 1059–1080.
R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Inter-science: New York, 1948. 11. C. M. Dafermos, Global smooth solutions to the initial-boundary value problem for the equations of one-dimensional nonlinear thermoviscoelasticity, SIAM J. Math. Anal. bd13 (1982), 397–408.
J. Glimm, The continuous structure of discontinuities, Lecture Notes in Physics bd344 (1989), 177–186.
E. Godlewski and P. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Appl. Math. Sc. bd118, Springer-Verlag: New York, 1996.
D. Hoff, Global well-posedness of the Cauchy problem for nonisentropic gas dynamics with discontinuous initial data, J. Duff. Eqs. bd95 (1992), 33–74.
D. Hoff, Discontinuous solutions of the Navier-Stokes equations for compressible flow, Arch. Rational Mech. Anal. bd114 (1991), 15–46.
D. Hoff. Discontinuous solutions of the Navier-Stokes equations for multidimensional heat-conducting fluids, Arch. Rational Mech. Anal. bd139 (1997), 303–354.
D. Hoff and D. Serre. The failure of continuous dependence on the initial data for the Navier Stokes equations of compressible flow, SIAM J. Appl. Math. bd51 (1991), 887–898.
D. Hoff and M. Ziane, The global attraction and finite determining nodes for the Navier-Stokes equations of one dimensional compressible flow with discontinuous initial data, Indiana Univ. Math. J. 49(3) (2000), 843–889.
Y. Kanel, On a model system of equations of one-dimensional gas motion, J. Duff. Eq. bd4 (1968), 374–380.
A. V. Kazhikhov. On the theory of initial-boundary-value problems for the equations of one-dimensional nonstationary motion of a viscous heat-conductive gas, rDin. Sploshnoi Sredy bd50 (1981), 37–62 (Russian).
A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary-value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech. bd41 (1977), 273–282.
A. Majda. High Mach number combustion, Lecture Notes in Appl. Math. bd24 (1986), 109–184.
G. S. S. Ludford. Low Mach number combustion, Lecture Notes in Appl. Math. bd24 (1986), 3–74.
A. Matsumura and S. Yanagi, Uniform boundedness of the solutions for a one-dimensional isentropic model system of compressible viscous gas, Comm. Math. Phys. bd175 (1996), 259–274.
K. Trivisa. Global Existence and Asymptotic Analysis on a Model for the Dynamic Combustion of a Compressible, Reacting Fluid. In preparation.
F. A. Williams, Combustion Theory, Addison-Wesley, Reading, MA, 1965.
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Chen, GQ., Hoff, D., Trivisa, K. (2003). Analysis on a Model for the Dynamic Combustion of a Compressible, Reacting Fluid. In: Hou, T.Y., Tadmor, E. (eds) Hyperbolic Problems: Theory, Numerics, Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55711-8_39
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DOI: https://doi.org/10.1007/978-3-642-55711-8_39
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