On the Motion of Chemically Reacting Fluids Through Porous Medium

  • Eduard FeireislEmail author
  • Jiří Mikyška
  • Hana Petzeltová
  • Peter Takáč
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 209)


We consider a parabolic-hyperbolic system of nonlinear partial differential equations modeling the motion of a chemically reacting mixture through porous medium. The existence of classical as well as weak solutions is established under several physically relevant choices of the constitutive equations and relevant boundary conditions.


Chemically reacting fluid Porous medium DiPerna Lions theory 



The research of E.F. leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ ERC Grant Agreement 320078. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840. J.M. was supported by the project Computational methods in thermodynamics of multicomponent mixtures, KONTAKT LH12064, 2012–2015 of the Czech Ministry of Education, Youth, and Sports. H.P. was supported by the Institute of Mathematics of the Academy of Sciences of the Czech Republic, RVO:67985840. P.T. was partially supported by the Deutsche Forschungsgemeinschft (D.F.G., Germany) under Grant # TA 213 / 16-1.


  1. 1.
    Amann, H.: Maximal regularity and quasilinear parabolic boundary value problems. In: Recent advances in elliptic and parabolic problems, pp. 1–17. World Scientific Publishing, Hackensack, NJ, (2005)Google Scholar
  2. 2.
    Ambrosio, L.: Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158, 227–260 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Crippa, G., De Lellis, C.: Estimates and regularity results for the DiPerna-Lions flow. J. Reine Angew. Math. 616, 15–46 (2008)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Denk, R., Hieber, M., Prüss, J.: R-boundedness, Fourier multipliers, and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166(788) (2003)Google Scholar
  5. 5.
    Ashyralyev, A., Sobolevskii, P.E.: Well-posedness of Parabolic Difference Equations, in Operator Theory: Advances and Applications, vol. 69. Birkhäuser Verlag, BaselBostonBerlin (1994)Google Scholar
  6. 6.
    Di Benedetto, E.: Continuity of weak solutions to a general porous media equations. Indiana Univ. Math. J. 32, 83–118 (1983)MathSciNetCrossRefGoogle Scholar
  7. 7.
    DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    E. Feireisl, A. Novotný, and T. Takahashi.: Homogenization and singular limits for the complete Navier-Stokes-Fourier system. J. Math. Pures Appl. (9), 94(1):33–57, (2010)Google Scholar
  9. 9.
    Feireisl, E., Petzeltová, H., Trivisa, K.: Multicomponent reactive flows: global-in-time existence for large data. Commun. Pure Appl. Anal. 7, 1017–1047 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Giovangigli, V.: Multicomponent Flow Modeling. Birkhäuser, Basel (1999)CrossRefzbMATHGoogle Scholar
  11. 11.
    Masmoudi, N.: Homogenization of the compressible Navier-Stokes equations in a porous medium. ESAIM: Control. Optim. Calc. Var. 8, 885–906 (2002)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Eduard Feireisl
    • 1
    Email author
  • Jiří Mikyška
    • 2
  • Hana Petzeltová
    • 1
  • Peter Takáč
    • 3
  1. 1.Institute of Mathematics of the Academy of Sciences of the Czech RepublicPrague 1Czech Republic
  2. 2.Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical EngineeringDepartment of MathematicsPrague 2Czech Republic
  3. 3.Institut Für MathematikUniversität RostockRostockGermany

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