Approximate solutions to a model of two-component reactive flow

Piotr Bogusław Mucha; Milan Pokorný; Ewelina Zatorska

doi:10.3934/dcdss.2014.7.1079

Article Contents

2014, Volume 7, Issue 5: 1079-1099. Doi: 10.3934/dcdss.2014.7.1079

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Approximate solutions to a model of two-component reactive flow

1.
Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa
2.
Charles University in Prague, Faculty of Mathematics and Physics, Mathematical Institute of Charles University, Sokolovská 83, 186 75 Praha

Received: September 30, 2012

Revised: December 31, 2012

Published: September 2014

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Abstract

We consider a model of motion of binary mixture, based on the compressible Navier-Stokes system. The mass balances of chemically reacting species are described by the reaction-diffusion equations with generalized form of multicomponent diffusion flux. Under a special relation between the two density dependent viscosity coefficients and for singular cold pressure we construct the weak solutions passing through several levels of approximation.

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Mathematics Subject Classification: 35Q30, 35K55, 35D30, 76N10.

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References

[1]	D. Bothe, On the Maxwell-Stefan approach to multicomponent diffusion, Progress in Nonlinear Differential Equations and their Applications, Springer, Basel, 80 (2011), 81-93.doi: 10.1007/978-3-0348-0075-4_5.
[2]	D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223.
[3]	D. Bresch and B. Desjardins, On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models, J. Math. Pures Appl. (9), 86 (2006), 362-368.doi: 10.1016/j.matpur.2006.06.005.
[4]	D. Bresch and B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids, J. Math. Pures Appl. (9), 87 (2007), 57-90.doi: 10.1016/j.matpur.2006.11.001.
[5]	D. Bresch, B. Desjardins and C.-K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868.doi: 10.1081/PDE-120020499.
[6]	G.-Q. Chen, D. Hoff and K. Trivisa, Global solutions to a model for exothermically reacting, compressible flows with large discontinuous initial data, Arch. Ration. Mech. Anal., 166 (2003), 321-358.doi: 10.1007/s00205-002-0233-6.
[7]	D. Donatelli and K. Trivisa, A multidimensional model for the combustion of compressible fluids, Arch. Ration. Mech. Anal., 185 (2007), 379-408.doi: 10.1007/s00205-006-0043-3.
[8]	E. Feireisl, On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable, Comment. Math. Univ. Carolin., 42 (2001), 83-98.
[9]	E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Birkhäuser Verlag, Basel, 2009.doi: 10.1007/978-3-7643-8843-0.
[10]	E. Feireisl, H. Petzeltová and K. Trivisa, Multicomponent reactive flows: Global-in-time existence for large data, Commun. Pure Appl. Anal., 7 (2008), 1017-1047.doi: 10.3934/cpaa.2008.7.1017.
[11]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.
[12]	V. Giovangigli, Multicomponent Flow Modeling, Birkhäuser Boston Inc., Boston, MA, 1999.doi: 10.1007/978-1-4612-1580-6.
[13]	R. Klein, N. Botta, T. Schneider, C. D. Munz, S. Roller, A. Meister, L. Hoffmann and T. Sonar, Asymptotic adaptive methods for multi-scale problems in fluid mechanics. Special issue on practical asymptotics, J. Engrg. Math., 39 (2001), 261-343.doi: 10.1023/A:1004844002437.
[14]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Trans. Math. Monographs 23, Providence, 1967.
[15]	P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2, Compressible Models, Oxford Science Publications, Oxford, 1998.
[16]	A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995.doi: 10.1007/978-3-0348-9234-6.
[17]	A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equations, Comm. Partial Differential Equations, 32 (2007), 431-452.doi: 10.1080/03605300600857079.
[18]	P. B. Mucha, M. Pokorný and E. Zatorska, Chemically reacting mixtures in terms of degenerated parabolic setting, J. Math. Phys., 54 (2013), 071501.doi: 10.1063/1.4811564.
[19]	A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, Oxford, 2004.
[20]	E. Zatorska, On the steady flow of a multicomponent, compressible, chemically reacting gas, Nonlinearity, 24 (2011), 3267-3278.doi: 10.1088/0951-7715/24/11/013.
[21]	E. Zatorska, On the flow of chemically reacting gaseous mixture, J. Differential Equations, 253 (2012), 3471-3500.doi: 10.1016/j.jde.2012.08.043.