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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

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 & 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

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.  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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