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

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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

Dieter Bothe ^1, and Michel Pierre ^2,

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

Abstract
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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.

Keywords: instantaneous chemical reaction, Stefan-type problem., Chemical reaction interface, spatial segregation.

Mathematics Subject Classification: Primary: 35K57; Secondary: 35B25, 92E2.

Citation: Dieter Bothe, Michel Pierre. The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction. Discrete and 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.
[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.
[3]	Ph. Bénilan, M. G. Crandall and A. Pazy, "Nonlinear Evolution Equations in Banach Spaces," preprint book.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	P. Erdi and J. Toth, "Mathematical Models of Chemical Reactions," Manchester Univ. Press, Manchester 1989.
[16]	L. C. Evans, A convergence theorem for a chemical diffusion-reaction system, Houston J. Math., 6 (1980), 259-267.
[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).
[18]	A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,'' Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser Verlag, Basel, 1995.
[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).
[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.
[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.
[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.
[23]	Y. Tonegawa, On the regularity of a chemical reaction interface, Comm. PDEs, 23 (1998), 1181-1207.
[24]	J. L. Vazquez, "The Porous Medium Equation, Mathematical Theory," Clarendon-Press, Oxford University Press, Oxford, 2007.

show all references

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.
[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.
[3]	Ph. Bénilan, M. G. Crandall and A. Pazy, "Nonlinear Evolution Equations in Banach Spaces," preprint book.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	P. Erdi and J. Toth, "Mathematical Models of Chemical Reactions," Manchester Univ. Press, Manchester 1989.
[16]	L. C. Evans, A convergence theorem for a chemical diffusion-reaction system, Houston J. Math., 6 (1980), 259-267.
[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).
[18]	A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,'' Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser Verlag, Basel, 1995.
[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).
[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.
[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.
[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.
[23]	Y. Tonegawa, On the regularity of a chemical reaction interface, Comm. PDEs, 23 (1998), 1181-1207.
[24]	J. L. Vazquez, "The Porous Medium Equation, Mathematical Theory," Clarendon-Press, Oxford University Press, Oxford, 2007.

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