Archive for Rational Mechanics and Analysis

, Volume 218, Issue 1, pp 553–587 | Cite as

Global Existence of Renormalized Solutions to Entropy-Dissipating Reaction–Diffusion Systems

  • J. Fischer


In the present work we introduce the notion of a renormalized solution for reaction–diffusion systems with entropy-dissipating reactions. We establish the global existence of renormalized solutions. In the case of integrable reaction terms our notion of a renormalized solution reduces to the usual notion of a weak solution. Our existence result in particular covers all reaction–diffusion systems involving a single reversible reaction with mass-action kinetics and (possibly species-dependent) Fick-law diffusion; more generally, it covers the case of systems of reversible reactions with mass-action kinetics which satisfy the detailed balance condition. For such equations the existence of any kind of solution in general was an open problem, thereby motivating the study of renormalized solutions.


Entropy Weak Solution Diffusion Equation Global Existence Diffusion System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alexandre R.: A definition of renormalized solutions for Boltzmann equation without cutoff. C. R. Acad. Sci. Paris Sér. I Math., 328(11), 987–991 (1999)Google Scholar
  2. 2.
    Alexandre R., Villani C.: On the Landau approximation in plasma physics. Ann. Inst. H. Poincaré Anal. Non Linéaire, 21(1), 61–95 (2004)Google Scholar
  3. 3.
    Alt H.W., Luckhaus S.: Quasilinear Elliptic-Parabolic Differential Equations. Math Z. 183, 311–341 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bothe D., Pierre M.: Quasi-steady-state approximation for a reaction–diffusion system with fast intermediate. J. Math. Anal. Appl. 368(1), 120–132 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Canizo J.A., Desvillettes L., Fellner K.: Improved duality estimates and applications to reaction–diffusion equations. Comm. Partial Differ. Equ. 39, 1185–1204 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Cristina Caputo M., Vasseur A.: Global regularity of solutions to systems of reaction–diffusion with sub-quadratic growth in any dimension. Comm. Partial Differ. Equ., 34, 1228–1250 (2009)Google Scholar
  7. 7.
    Dal Maso G., Murat F., Orsina L., Prignet A.: Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28, 741–808 (1999)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Desvillettes L., Fellner K.: Exponential decay toward equilibrium via entropy methods for reaction–diffusion equations. J. Math. Anal. Appl. 319, 157–176 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Desvillettes, L., Fellner, K.: Entropy methods for reaction–diffusion equations with degenerate diffusion arising in reversible chemistry. accepted for the Proceedings of the Equadiff (2007)Google Scholar
  10. 10.
    Desvillettes L., Fellner K.: Entropy methods for reaction–diffusion systems. Discrete Contin. Dyn. Syst. Suppl. 24, 304–312 (2007)MathSciNetGoogle Scholar
  11. 11.
    Desvillettes L., Fellner K.: Entropy methods for reaction–diffusion equations: slowly growing a-priori bounds. Rev. Mat. Iberoamericana 24, 407–431 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Desvillettes L., Fellner K., Pierre M., Vovelle J.: About global existence for quadratic systems of reaction–diffusion. Adv. Nonlinear Stud. 7, 491–511 (2007)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Diaz J.I., Veron L.: Local vanishing properties of solutions of elliptic and parabolic quasilinear equations. Trans. Amer. Math. Soc. 290(2), 787–814 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    DiPerna R.J., Lions P.-L.: On the Fokker-Planck-Boltzmann equation. Commun. Math. Phys. 120, 1–23 (1988)zbMATHMathSciNetADSCrossRefGoogle Scholar
  15. 15.
    DiPerna R.J., Lions P.-L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math. 130(2), 321–366 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    DiPerna R.J., Lions P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–517 (1989)zbMATHMathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Feinberg M., Horn F.J.M.: Chemical mechanism structure and the coincidence of the stoichiometric and kinetic subspaces. Arch. Ration. Mech. Anal. 66, 83–97 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Fiebach A., Glitzky A., Linke A.: Uniform global bounds for solutions of an implicit voronoi finite volume method for reaction–diffusion problems. Numer. Math. 128(1), 31–72 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Gajewski H., Gröger K.: Reaction–diffusion processes of electrically charged species. Math. Nachr. 177(1), 109–130 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Gajewski, H., Skrypnik, I.V.: Existence and uniqueness results for reaction–diffusion processes of electrically charged species. Nonlinear Elliptic and Parabolic Problems, Vol. 64 Prog. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 151–188, 2005Google Scholar
  21. 21.
    Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)zbMATHGoogle Scholar
  22. 22.
    Glitzky, A., Hünlich, R.: Global estimates and asymptotics for electro-reaction–diffusion systems in heterostructures. Appl. Anal. 66(3–4), 205–225 (1997)Google Scholar
  23. 23.
    Glitzky A., Mielke A.: A gradient structure for systems coupling reaction–diffusion effects in bulk and interfaces. Z. Angew. Math. Phys. 64(1), 29–52 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Goudon T., Vasseur A.: Regularity analysis for systems of reaction–diffusion equations. Ann. Sci. Éc. Norm. Supér. 368, 120–132 (2010)Google Scholar
  25. 25.
    Jordan R., Kinderlehrer D., Otto F.: The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29, 1–17 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Kräutle S.: Existence of global solutions of multicomponent reactive transport problems with mass action kinetics in porous media. J. Appl. Anal. Comput. 1(4), 497–515 (2011)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Liero M., Mielke A.: Gradient structures and geodesic convexity for reaction–diffusion systems. Phil. Trans. R. Soc. A 371, 20120346 (2013)MathSciNetADSCrossRefGoogle Scholar
  28. 28.
    Mielke A.: A gradient structure for reaction–diffusion systems and for energy-drift-diffusion systems. Nonlinearity 24, 1329–1346 (2011)zbMATHMathSciNetADSCrossRefGoogle Scholar
  29. 29.
    Mielke, A., Haskovec, J., Markowich, P.: On uniform decay of the entropy for reaction–diffusion systems. J. Dynam. Differ. Equ. doi: 10.1007/s10884-014-9394-x (2014, in press)
  30. 30.
    Mincheva M., Siegel D.: Stability of mass action reaction–diffusion systems. Nonlinear Anal. 56(8), 1105–1131 (2004)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Murat, F.: Solutiones renormalizadas de EDP elipticas non lineares. Technical report R93023 (1993)Google Scholar
  32. 32.
    Otto F.: Dynamics of labyrinthine pattern formation in magnetic fluids: A mean-field theory. Arch. Ration. Mech. Anal. 141(1), 63–103 (1998)zbMATHCrossRefGoogle Scholar
  33. 33.
    Pierre M.: Weak solutions and supersolutions in L 1 for reaction–diffusion systems. J. Evol. Equ. 3, 153–168 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Pierre M.: Global existence in reaction–diffusion systems with control of mass: a survey. Milan J. Math. 78(2), 417–455 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Pierre M., Schmitt D.: Blow-up in reaction–diffusion systems with dissipation of mass. SIAM J. Math. Anal. 28, 259–269 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Pierre M., Schmitt D.: Blow-up in reaction–diffusion systems with dissipation of mass. SIAM Rev. 42, 93–106 (2000)zbMATHMathSciNetADSCrossRefGoogle Scholar
  37. 37.
    Schuster S., Schuster R.: A generalization of Wegscheiders condition, Implications for properties of steady states and for quasi-steady-state approximation. J. Math. Chem. 3, 25–42 (1989)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Simon J.: Compact Sets in the Space L p(0, T ; B). Ann. Mat. Pura Appl. (4) 146(1), 65–96 (1986)ADSCrossRefGoogle Scholar
  39. 39.
    Villani C.: On the Cauchy problem for the Landau equation: sequential stability, global existence. Adv. Differ. Equ. 1(5), 793–816 (1996)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

Personalised recommendations