April  2018, 11(2): 279-301. doi: 10.3934/krm.2018014

Applications of improved duality lemmas to the discrete coagulation-fragmentation equations with diffusion

1. 

CMLA, ENS Cachan, CNRS, Université Paris-Saclay, 94235 Cachan, France

2. 

Technical University of Munich, Faculty of Mathematics, Research Unit "Multiscale and Stochastic Dynamics", 85748 Garching b. München, Germany

Received  February 2017 Published  January 2018

Fund Project: The author was partially supported by french "ANR blanche" project Kibord: ANR-13-BS01-0004.

In this paper, we investigate the use of so called "duality lemmas" to study the system of discrete coagulation-fragmentation equations with diffusion. When the fragmentation is strong enough with respect to the coagulation, we show that we have creation and propagation of superlinear moments. In particular this implies that strong enough fragmentation can prevent gelation even for superlinear coagulation, a statement which was only known up to now in the homogeneous setting. We also use this control of superlinear moments to extend a recent result from [3], about the regularity of the solutions in the pure coagulation case, to strong fragmentation models.

Citation: Maxime Breden. Applications of improved duality lemmas to the discrete coagulation-fragmentation equations with diffusion. Kinetic and Related Models, 2018, 11 (2) : 279-301. doi: 10.3934/krm.2018014
References:
[1]

J. M. Ball and J. Carr, The discrete coagulation-fragmentation equations: Existence, uniqueness, and density conservation, Journal of Statistical Physics, 61 (1990), 203-234.  doi: 10.1007/BF01013961.

[2]

J. M. BallJ. Carr and O. Penrose, The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions, Communications in Mathematical Physics, 104 (1986), 657-692.  doi: 10.1007/BF01211070.

[3]

M. BredenL. Desvillettes and K. Fellner, Smoothness of moments of the solutions of discrete coagulation equations with diffusion, Monatsh. Math., 183 (2017), 437-463.  doi: 10.1007/s00605-016-0969-y.

[4]

J. CanizoL. Desvillettes and K. Fellner, Regularity and mass conservation for discrete coagulation-fragmentation equations with diffusion, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 27 (2010), 639-654.  doi: 10.1016/j.anihpc.2009.10.001.

[5]

J. CanizoL. Desvillettes and K. Fellner, Improved duality estimates and applications to reaction-diffusion equations, Communications in Partial Differential Equations, 39 (2014), 1185-1204.  doi: 10.1080/03605302.2013.829500.

[6]

J. Carr, Asymptotic behaviour of solutions to the coagulation-fragmentation equations. Ⅰ. The strong fragmentation case, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 121 (1992), 231-244.  doi: 10.1017/S0308210500027888.

[7]

F. P. Da Costa, Existence and uniqueness of density conserving solutions to the coagulation-fragmentation equations with strong fragmentation, Journal of Mathematical Analysis and Applications, 192 (1995), 892-914.  doi: 10.1006/jmaa.1995.1210.

[8]

L. Desvillettes and K. Fellner, Duality and entropy methods in coagulation-fragmentation models, Revista di Matematica della Universita di Parma, 4 (2013), 215-263. 

[9]

L. DesvillettesK. FellnerM. Pierre and J. Vovelle, Global existence for quadratic systems of reaction-diffusion, Advanced Nonlinear Studies, 7 (2007), 491-511. 

[10]

R. L. Drake, A general mathematical survey of the coagulation equation, International Reviews in Aerosol Physics and Chemistry, Oxford, (1972), 203-376.

[11]

M. EscobedoP. LaurençotS. Mischler and B. Perthame, Gelation and mass conservation in coagulation and fragmentation models, Journal of Differential Equations, 195 (2003), 143-174.  doi: 10.1016/S0022-0396(03)00134-7.

[12]

M. EscobedoS. Mischler and B. Perthame, Gelation in coagulation and fragmentation models, Communications in Mathematical Physics, 231 (2002), 157-188.  doi: 10.1007/s00220-002-0680-9.

[13]

A. Hammond and F. Rezakhanlou, Moment bounds for the Smoluchowski equation and their consequences, Communications in Mathematical Physics, 276 (2007), 645-670.  doi: 10.1007/s00220-007-0304-5.

[14]

E. M. HendriksM. H. Ernst and R. M. Ziff, Coagulation equations with gelation, Journal of Statistical Physics, 31 (1983), 519-563.  doi: 10.1007/BF01019497.

[15]

P. Laurençot and S. Mischler, Global existence for the discrete diffusive coagulation-fragmentation equations in $L^{1}$, Revista Matemática Iberoamericana, 18 (2002), 731-745. 

[16]

P. Laurençot and S. Mischler, On coalescence equations and related models, In Modeling and computational methods for kinetic equations, Model. Simul. Sci. Eng. Technol. Birkhäuser Boston, Boston, MA, (2004), 321-356.

[17]

M. Pierre, Global existence in reaction-diffusion systems with control of mass: A survey, Milan J. Math., 78 (2010), 417-455.  doi: 10.1007/s00032-010-0133-4.

[18]

F. Rezakhanlou, Moment bounds for the solutions of the Smoluchowski equation with coagulation and fragmentation, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 140 (2010), 1041-1059.  doi: 10.1017/S0308210509000924.

[19]

F. Rezakhanlou, Pointwise bounds for the solutions of the Smoluchowski equation with diffusion, Archive for Rational Mechanics and Analysis, 212 (2014), 1011-1035.  doi: 10.1007/s00205-013-0716-7.

[20]

M. Smoluchowski, Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen, Physik. Zeitschr., 17 (1916), 557-599. 

[21]

M. Smoluchowski, Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen, Zeitschrift f. physik. Chemie, 92 (1917), 129-168.  doi: 10.1515/zpch-1918-9209.

[22]

D. Wrzosek, Existence of solutions for the discrete coagulation-fragmentation model with diffusion, Topological Methods in Nonlinear Analysis, 9 (1997), 279-296.  doi: 10.12775/TMNA.1997.014.

[23]

D. Wrzosek, Weak solutions to the Cauchy problem for the diffusive discrete coagulation-fragmentation system, J. Math. Anal. Appl., 289 (2004), 405-418.  doi: 10.1016/j.jmaa.2003.08.022.

show all references

References:
[1]

J. M. Ball and J. Carr, The discrete coagulation-fragmentation equations: Existence, uniqueness, and density conservation, Journal of Statistical Physics, 61 (1990), 203-234.  doi: 10.1007/BF01013961.

[2]

J. M. BallJ. Carr and O. Penrose, The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions, Communications in Mathematical Physics, 104 (1986), 657-692.  doi: 10.1007/BF01211070.

[3]

M. BredenL. Desvillettes and K. Fellner, Smoothness of moments of the solutions of discrete coagulation equations with diffusion, Monatsh. Math., 183 (2017), 437-463.  doi: 10.1007/s00605-016-0969-y.

[4]

J. CanizoL. Desvillettes and K. Fellner, Regularity and mass conservation for discrete coagulation-fragmentation equations with diffusion, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 27 (2010), 639-654.  doi: 10.1016/j.anihpc.2009.10.001.

[5]

J. CanizoL. Desvillettes and K. Fellner, Improved duality estimates and applications to reaction-diffusion equations, Communications in Partial Differential Equations, 39 (2014), 1185-1204.  doi: 10.1080/03605302.2013.829500.

[6]

J. Carr, Asymptotic behaviour of solutions to the coagulation-fragmentation equations. Ⅰ. The strong fragmentation case, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 121 (1992), 231-244.  doi: 10.1017/S0308210500027888.

[7]

F. P. Da Costa, Existence and uniqueness of density conserving solutions to the coagulation-fragmentation equations with strong fragmentation, Journal of Mathematical Analysis and Applications, 192 (1995), 892-914.  doi: 10.1006/jmaa.1995.1210.

[8]

L. Desvillettes and K. Fellner, Duality and entropy methods in coagulation-fragmentation models, Revista di Matematica della Universita di Parma, 4 (2013), 215-263. 

[9]

L. DesvillettesK. FellnerM. Pierre and J. Vovelle, Global existence for quadratic systems of reaction-diffusion, Advanced Nonlinear Studies, 7 (2007), 491-511. 

[10]

R. L. Drake, A general mathematical survey of the coagulation equation, International Reviews in Aerosol Physics and Chemistry, Oxford, (1972), 203-376.

[11]

M. EscobedoP. LaurençotS. Mischler and B. Perthame, Gelation and mass conservation in coagulation and fragmentation models, Journal of Differential Equations, 195 (2003), 143-174.  doi: 10.1016/S0022-0396(03)00134-7.

[12]

M. EscobedoS. Mischler and B. Perthame, Gelation in coagulation and fragmentation models, Communications in Mathematical Physics, 231 (2002), 157-188.  doi: 10.1007/s00220-002-0680-9.

[13]

A. Hammond and F. Rezakhanlou, Moment bounds for the Smoluchowski equation and their consequences, Communications in Mathematical Physics, 276 (2007), 645-670.  doi: 10.1007/s00220-007-0304-5.

[14]

E. M. HendriksM. H. Ernst and R. M. Ziff, Coagulation equations with gelation, Journal of Statistical Physics, 31 (1983), 519-563.  doi: 10.1007/BF01019497.

[15]

P. Laurençot and S. Mischler, Global existence for the discrete diffusive coagulation-fragmentation equations in $L^{1}$, Revista Matemática Iberoamericana, 18 (2002), 731-745. 

[16]

P. Laurençot and S. Mischler, On coalescence equations and related models, In Modeling and computational methods for kinetic equations, Model. Simul. Sci. Eng. Technol. Birkhäuser Boston, Boston, MA, (2004), 321-356.

[17]

M. Pierre, Global existence in reaction-diffusion systems with control of mass: A survey, Milan J. Math., 78 (2010), 417-455.  doi: 10.1007/s00032-010-0133-4.

[18]

F. Rezakhanlou, Moment bounds for the solutions of the Smoluchowski equation with coagulation and fragmentation, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 140 (2010), 1041-1059.  doi: 10.1017/S0308210509000924.

[19]

F. Rezakhanlou, Pointwise bounds for the solutions of the Smoluchowski equation with diffusion, Archive for Rational Mechanics and Analysis, 212 (2014), 1011-1035.  doi: 10.1007/s00205-013-0716-7.

[20]

M. Smoluchowski, Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen, Physik. Zeitschr., 17 (1916), 557-599. 

[21]

M. Smoluchowski, Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen, Zeitschrift f. physik. Chemie, 92 (1917), 129-168.  doi: 10.1515/zpch-1918-9209.

[22]

D. Wrzosek, Existence of solutions for the discrete coagulation-fragmentation model with diffusion, Topological Methods in Nonlinear Analysis, 9 (1997), 279-296.  doi: 10.12775/TMNA.1997.014.

[23]

D. Wrzosek, Weak solutions to the Cauchy problem for the diffusive discrete coagulation-fragmentation system, J. Math. Anal. Appl., 289 (2004), 405-418.  doi: 10.1016/j.jmaa.2003.08.022.

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