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October  2014, 7(5): 1079-1099. doi: 10.3934/dcdss.2014.7.1079

Approximate solutions to a model of two-component reactive flow

Piotr Bogusław Mucha 1, , Milan Pokorný 2, and  Ewelina Zatorska 1,

1. 

Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland, Poland
2. 

Charles University in Prague, Faculty of Mathematics and Physics, Mathematical Institute of Charles University, Sokolovská 83, 186 75 Praha, Czech Republic

Received  October 2012 Revised  January 2013 Published  May 2014

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.
Keywords: weak solutions., density-dependent viscosity, chemically reacting gas, compressible Navier--Stokes system, Multicomponent flow.
Mathematics Subject Classification: 35Q30, 35K55, 35D30, 76N1.
Citation: Piotr Bogusław Mucha, Milan Pokorný, Ewelina Zatorska. Approximate solutions to a model of two-component reactive flow. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1079-1099. doi: 10.3934/dcdss.2014.7.1079
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar

[12]

V. Giovangigli, Multicomponent Flow Modeling, Birkhäuser Boston Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1580-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2, Compressible Models, Oxford Science Publications, Oxford, 1998.  Google Scholar

[16]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995. doi: 10.1007/978-3-0348-9234-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, Oxford, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar

[12]

V. Giovangigli, Multicomponent Flow Modeling, Birkhäuser Boston Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1580-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2, Compressible Models, Oxford Science Publications, Oxford, 1998.  Google Scholar

[16]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995. doi: 10.1007/978-3-0348-9234-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, Oxford, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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