# American Institute of Mathematical Sciences

February  2012, 5(1): 49-59. doi: 10.3934/dcdss.2012.5.49

## The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction

 1 Center of Smart Interfaces, Technical University Darmstadt, Petersenstr. 32, 64287 Darmstadt, Germany 2 ENS Cachan Bretagne, IRMAR, EUB, Campus de Ker Lann, 35170 Bruz, France

Received  August 2009 Revised  January 2010 Published  February 2011

We consider reaction-diffusion systems which, in addition to certain slow reactions, contain a fast irreversible reaction in which chemical components A and B form a product P. In this situation and under natural assumptions on the RD-system we prove the convergence of weak solutions, as the reaction speed of the irreversible reaction tends to infinity, to a weak solution of a limiting system. The limiting system is a Stefan-type problem with a moving interface at which the chemical reaction front is localized.
Citation: Dieter Bothe, Michel Pierre. The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 49-59. doi: 10.3934/dcdss.2012.5.49
##### References:
 [1] H. Amann, Global existence for semilinear parabolic problems, J. Reine Angew. Math., 360 (1985), 47-83. doi: 10.1515/crll.1985.360.47.  Google Scholar [2] P. Baras and M. Pierre, Problèmes paraboliques semi-linéaires avec données mesures, (French, English summary), Applicable Analysis, 18 (1984), 111-149. doi: 10.1080/00036818408839514.  Google Scholar [3] Ph. Bénilan, M. G. Crandall and A. Pazy, "Nonlinear Evolution Equations in Banach Spaces,", preprint book., ().   Google Scholar [4] M. Bisi, F. Confort and L. Desvillettes, Quasi-steady-state approximation for reaction-diffusion equations, Bull. Inst. Math., Acad. Sin. (N.S.), 2 (2007), 823-850.  Google Scholar [5] D. Bothe, The instantaneous limit of a reaction-diffusion system, in "Evolution Equations and Their Applications in Physical and Life Sciences" (eds. G. Lumer, L. Weis), Marcel Dekker, Lect. Notes Pure Appl. Math., 215 (2000), 215-224.  Google Scholar [6] D. Bothe, Instantaneous limits of reversible chemical reactions in presence of macroscopic convection, J. Differ. Equations, 193 (2003), 27-48. doi: 10.1016/S0022-0396(03)00148-7.  Google Scholar [7] D. Bothe and D. Hilhorst, A reaction-diffusion system with fast reversible reaction, J. Math. Anal. Appl., 286 (2003), 125-135. doi: 10.1016/S0022-247X(03)00457-8.  Google Scholar [8] D. Bothe, A. Lojewski and H.-J. Warnecke, Computational analysis of an instantaneous chemical reaction in a T-microreactor, AIChE J., 56 (2010), 1406-1415. Google Scholar [9] D. Bothe and M. Pierre, Quasi-steady-state approximation for a reaction-diffusion system with fast intermediate, J. Math. Anal. Appl., 368 (2010), 120-132. doi: 10.1016/j.jmaa.2010.02.044.  Google Scholar [10] H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590. doi: 10.2969/jmsj/02540565.  Google Scholar [11] J. R. Cannon and C. D. Hill, On the movement of a chemical reaction interface, Indiana Univ. Math. J., 20 (1970), 429-453. doi: 10.1512/iumj.1970.20.20037.  Google Scholar [12] J. R. Cannon and A. Fasano, Boundary value multidimensional problems in fast chemical reactions, Archive Rat. Mech. Anal., 53 (1973), 1-13. doi: 10.1007/BF00735697.  Google Scholar [13] E. N. Dancer, D. Hilhorst, M. Mimura and L. A. Peletier, Spatial segregation limit of a competition-diffusion system, European J. Appl. Math., 10 (1999), 97-115. doi: 10.1017/S0956792598003660.  Google Scholar [14] L. Desvillettes, K. Fellner, M. Pierre and J. Vovelle, About global existence for quadratic systems of reaction-diffusion, Advanced Nonlinear Studies, 7 (2007), 491-511.  Google Scholar [15] P. Erdi and J. Toth, "Mathematical Models of Chemical Reactions," Manchester Univ. Press, Manchester 1989.  Google Scholar [16] L. C. Evans, A convergence theorem for a chemical diffusion-reaction system, Houston J. Math., 6 (1980), 259-267.  Google Scholar [17] D. Hilhorst, M. Mimura and H. Ninomiya, Fast reaction limit of competition-diffusion systems, in "Handbook of Differential Equations" (eds. C. M. Dafermos and M. Pokorny), Elsevier, Evolution Equations 5 (2009).  Google Scholar [18] A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,'' Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser Verlag, Basel, 1995.  Google Scholar [19] R. H. Martin and M. Pierre, Nonlinear reaction-diffusion systems, in "Nonlinear Equations in the Applied Sciences" (eds. W.F. Ames and C. Rogers), Acad. Press, New York, Math. Sci. Eng., 185 (1991).  Google Scholar [20] M. Pierre, Weak solutions and supersolutions in $L^1$ for reaction-diffusion systems, J. Evol. Equ., 3 (2003), 153-168. doi: 10.1007/s000280300007.  Google Scholar [21] M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass, SIAM J. Math. Anal., 28 (1997), 259-269. doi: 10.1137/S0036141095295437.  Google Scholar [22] M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass, SIAM Rev., 42 (2000), 93-106 (electronic). doi: 10.1137/S0036144599359735.  Google Scholar [23] Y. Tonegawa, On the regularity of a chemical reaction interface, Comm. PDEs, 23 (1998), 1181-1207.  Google Scholar [24] J. L. Vazquez, "The Porous Medium Equation, Mathematical Theory," Clarendon-Press, Oxford University Press, Oxford, 2007.  Google Scholar

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##### References:
 [1] H. Amann, Global existence for semilinear parabolic problems, J. Reine Angew. Math., 360 (1985), 47-83. doi: 10.1515/crll.1985.360.47.  Google Scholar [2] P. Baras and M. Pierre, Problèmes paraboliques semi-linéaires avec données mesures, (French, English summary), Applicable Analysis, 18 (1984), 111-149. doi: 10.1080/00036818408839514.  Google Scholar [3] Ph. Bénilan, M. G. Crandall and A. Pazy, "Nonlinear Evolution Equations in Banach Spaces,", preprint book., ().   Google Scholar [4] M. Bisi, F. Confort and L. Desvillettes, Quasi-steady-state approximation for reaction-diffusion equations, Bull. Inst. Math., Acad. Sin. (N.S.), 2 (2007), 823-850.  Google Scholar [5] D. Bothe, The instantaneous limit of a reaction-diffusion system, in "Evolution Equations and Their Applications in Physical and Life Sciences" (eds. G. Lumer, L. Weis), Marcel Dekker, Lect. Notes Pure Appl. Math., 215 (2000), 215-224.  Google Scholar [6] D. Bothe, Instantaneous limits of reversible chemical reactions in presence of macroscopic convection, J. Differ. Equations, 193 (2003), 27-48. doi: 10.1016/S0022-0396(03)00148-7.  Google Scholar [7] D. Bothe and D. Hilhorst, A reaction-diffusion system with fast reversible reaction, J. Math. Anal. Appl., 286 (2003), 125-135. doi: 10.1016/S0022-247X(03)00457-8.  Google Scholar [8] D. Bothe, A. Lojewski and H.-J. Warnecke, Computational analysis of an instantaneous chemical reaction in a T-microreactor, AIChE J., 56 (2010), 1406-1415. Google Scholar [9] D. Bothe and M. Pierre, Quasi-steady-state approximation for a reaction-diffusion system with fast intermediate, J. Math. Anal. Appl., 368 (2010), 120-132. doi: 10.1016/j.jmaa.2010.02.044.  Google Scholar [10] H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590. doi: 10.2969/jmsj/02540565.  Google Scholar [11] J. R. Cannon and C. D. Hill, On the movement of a chemical reaction interface, Indiana Univ. Math. J., 20 (1970), 429-453. doi: 10.1512/iumj.1970.20.20037.  Google Scholar [12] J. R. Cannon and A. Fasano, Boundary value multidimensional problems in fast chemical reactions, Archive Rat. Mech. Anal., 53 (1973), 1-13. doi: 10.1007/BF00735697.  Google Scholar [13] E. N. Dancer, D. Hilhorst, M. Mimura and L. A. Peletier, Spatial segregation limit of a competition-diffusion system, European J. Appl. Math., 10 (1999), 97-115. doi: 10.1017/S0956792598003660.  Google Scholar [14] L. Desvillettes, K. Fellner, M. Pierre and J. Vovelle, About global existence for quadratic systems of reaction-diffusion, Advanced Nonlinear Studies, 7 (2007), 491-511.  Google Scholar [15] P. Erdi and J. Toth, "Mathematical Models of Chemical Reactions," Manchester Univ. Press, Manchester 1989.  Google Scholar [16] L. C. Evans, A convergence theorem for a chemical diffusion-reaction system, Houston J. Math., 6 (1980), 259-267.  Google Scholar [17] D. Hilhorst, M. Mimura and H. Ninomiya, Fast reaction limit of competition-diffusion systems, in "Handbook of Differential Equations" (eds. C. M. Dafermos and M. Pokorny), Elsevier, Evolution Equations 5 (2009).  Google Scholar [18] A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,'' Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser Verlag, Basel, 1995.  Google Scholar [19] R. H. Martin and M. Pierre, Nonlinear reaction-diffusion systems, in "Nonlinear Equations in the Applied Sciences" (eds. W.F. Ames and C. Rogers), Acad. Press, New York, Math. Sci. Eng., 185 (1991).  Google Scholar [20] M. Pierre, Weak solutions and supersolutions in $L^1$ for reaction-diffusion systems, J. Evol. Equ., 3 (2003), 153-168. doi: 10.1007/s000280300007.  Google Scholar [21] M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass, SIAM J. Math. Anal., 28 (1997), 259-269. doi: 10.1137/S0036141095295437.  Google Scholar [22] M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass, SIAM Rev., 42 (2000), 93-106 (electronic). doi: 10.1137/S0036144599359735.  Google Scholar [23] Y. Tonegawa, On the regularity of a chemical reaction interface, Comm. PDEs, 23 (1998), 1181-1207.  Google Scholar [24] J. L. Vazquez, "The Porous Medium Equation, Mathematical Theory," Clarendon-Press, Oxford University Press, Oxford, 2007.  Google Scholar
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