×

zbMATH — the first resource for mathematics

On the large-time behavior of 1D radiative and reactive viscous flows for higher-order kinetics. (English) Zbl 1083.35109
Summary: We consider the system of quasilinear equations describing 1D radiative and reactive viscous flows with arbitrarily large data. The large-time behavior of solutions in the case of first-order kinetics has been recently studied. In this paper, we present new results concerning the case of higher-order kinetics for fairly general kinetics law (unbounded with respect to density and temperature, and dealing with the ignition phenomenon), including \(L^2\) and \(H^1\)-stabilization rate bounds of power type. The power exponents of bounds improve essentially those known for related problems and are partially proved to be sharp. An effect of ”faster equaling” of values in space for the concentration of unburned gas is also found. Finally we show how our results are modified in the case of reaction without diffusion.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
80A25 Combustion
35L80 Degenerate hyperbolic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Antontsev, S.N.; Kazhikhov, A.V.; Monakhov, V.N., Boundary value problems in mechanics of nonhomogeneous fluids, (1990), North-Holland Publishing Co. Amsterdam · Zbl 0696.76001
[2] Bebernes, J.; Bressan, A., Global a priori estimates for a viscous reactive gas, Proc. roy. soc. Edinburgh, 101A, 321-333, (1985) · Zbl 0614.76076
[3] Bebernes, J.; Eberly, D., Mathematical problems from combustion theory, (1989), Springer New York, Berlin, Heidelberg · Zbl 0692.35001
[4] Chen, G.-Q., Global solution to the compressible navier – stokes equations for a reacting mixture, SIAM J. math. anal., 23, 609-634, (1992) · Zbl 0771.35044
[5] Chen, G.-Q.; Hoff, D.; Trivisa, K., On the navier – stokes equations for exothermically reacting compressible fluids, Acta math. appl. sinica, 18, 15-36, (2002) · Zbl 1032.76056
[6] Chen, G.-Q.; Hoff, D.; Trivisa, K., Global solution to a model for exothermically reacting compressible flows with large discontinuous data, Arch. rational mech. anal., 166, 321-358, (2003) · Zbl 1022.76056
[7] Ducomet, B., A model of thermal dissipation for a one-dimensional viscous reactive and radiative gas, Math. methods appl. sci., 22, 1323-1349, (1999) · Zbl 1027.85005
[8] Ducomet, B., Some asymptotics for a reactive navier – stokes – poisson system, Math. models methods appl. sci., 9, 1039-1076, (1999) · Zbl 1035.76057
[9] Ducomet, B.; Zlotnik, A., Stabilization for 1D radiative and reactive viscous gas flows, C. R. acad. sci. Paris ser. I, 338, 127-132, (2004) · Zbl 1042.35045
[10] B. Ducomet, A. Zlotnik, Lyapunov functional method for 1D radiative and reactive viscous gas dynamics, Arch. Rational Mech. Anal. 177 (2005) 185-229. · Zbl 1070.76044
[11] Guo, B.; Zhu, P., Asymptotic behaviour of the solution to the system for a viscous reactive gas, J. differential equations, 155, 177-202, (1999) · Zbl 0927.76098
[12] Ladyženskaja, O.A.; Solonnikov, V.A.; Ural’ceva, N.N., Linear and quasilinear equations of parabolic type, (1968), American Mathematical Society Providence, RI
[13] Lewicka, M.; Mucha, P.B., On temporal asymptotics for the \(p\)th power viscous reactive gas, Nonlinear anal. TMA, 57, 951-969, (2004) · Zbl 1094.76052
[14] Mihalas, D.M.; Weibel-Mihalas, B., Foundations of radiation hydrodynamics, (1999), Dover Publications Mineola, NY · Zbl 0651.76005
[15] Williams, F., Combustion theory, (1985), Addison-Wesley Publishing Company Reading, MA
[16] Yanagi, S., Asymptotic stability of the solutions to a full one-dimensional system of heat-conducting reactive compressible viscous gas, Jpn. J. ind. appl. math., 15, 423-442, (1998) · Zbl 0912.76077
[17] Zel’dovich, Ya.B.; Raizer, Yu.P., Physics of shock waves and high-temperature hydrodynamic phenomena, (2002), Dover Publications, Inc. Mineola, NY
[18] Zlotnik, A.A., Weak solutions to the equations of motion of viscous compressible reacting binary mixtures: uniqueness and Lipschitz-continuous dependence on data, Math. notes, 75, 278-283, (2004) · Zbl 1122.35118
[19] A.A. Zlotnik, B. Ducomet, Stabilization of one-dimensional flows of radiative and reactive viscous gas for general rate of reaction, Dokl. Math. 72 (2005) 595-600. · Zbl 1140.35554
[20] Zlotnik, A.A.; Puzanov, S.N., Correctness of the problem of viscous gas burning in the case of nonsmooth data and a semidiscrete method of its solution, Math. notes, 65, 793-797, (1999) · Zbl 0948.35119
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.