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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

 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.
Citation: Piotr Bogusław Mucha, Milan Pokorný, Ewelina Zatorska. Approximate solutions to a model of two-component reactive flow. Discrete and 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. [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.

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. [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.
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