# American Institute of Mathematical Sciences

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 & 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. Google Scholar [2] J. M. Ball, J. 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. Google Scholar [3] M. Breden, L. 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. Google Scholar [4] J. Canizo, L. 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. Google Scholar [5] J. Canizo, L. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [9] L. Desvillettes, K. Fellner, M. Pierre and J. Vovelle, Global existence for quadratic systems of reaction-diffusion, Advanced Nonlinear Studies, 7 (2007), 491-511. Google Scholar [10] R. L. Drake, A general mathematical survey of the coagulation equation, International Reviews in Aerosol Physics and Chemistry, Oxford, (1972), 203-376.Google Scholar [11] M. Escobedo, P. Laurençot, S. 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. Google Scholar [12] M. Escobedo, S. 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. Google Scholar [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. Google Scholar [14] E. M. Hendriks, M. H. Ernst and R. M. Ziff, Coagulation equations with gelation, Journal of Statistical Physics, 31 (1983), 519-563. doi: 10.1007/BF01019497. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] M. Smoluchowski, Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen, Physik. Zeitschr., 17 (1916), 557-599. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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. Google Scholar [2] J. M. Ball, J. 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. Google Scholar [3] M. Breden, L. 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. Google Scholar [4] J. Canizo, L. 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. Google Scholar [5] J. Canizo, L. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [9] L. Desvillettes, K. Fellner, M. Pierre and J. Vovelle, Global existence for quadratic systems of reaction-diffusion, Advanced Nonlinear Studies, 7 (2007), 491-511. Google Scholar [10] R. L. Drake, A general mathematical survey of the coagulation equation, International Reviews in Aerosol Physics and Chemistry, Oxford, (1972), 203-376.Google Scholar [11] M. Escobedo, P. Laurençot, S. 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. Google Scholar [12] M. Escobedo, S. 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. Google Scholar [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. Google Scholar [14] E. M. Hendriks, M. H. Ernst and R. M. Ziff, Coagulation equations with gelation, Journal of Statistical Physics, 31 (1983), 519-563. doi: 10.1007/BF01019497. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] M. Smoluchowski, Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen, Physik. Zeitschr., 17 (1916), 557-599. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
 [1] Jacek Banasiak, Luke O. Joel, Sergey Shindin. The discrete unbounded coagulation-fragmentation equation with growth, decay and sedimentation. Kinetic & Related Models, 2019, 12 (5) : 1069-1092. doi: 10.3934/krm.2019040 [2] Jacek Banasiak. Transport processes with coagulation and strong fragmentation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 445-472. doi: 10.3934/dcdsb.2012.17.445 [3] Pierre Degond, Maximilian Engel. Numerical approximation of a coagulation-fragmentation model for animal group size statistics. Networks & Heterogeneous Media, 2017, 12 (2) : 217-243. doi: 10.3934/nhm.2017009 [4] Miguel A. Herrero, Marianito R. Rodrigo. Remarks on accessible steady states for some coagulation-fragmentation systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 541-552. doi: 10.3934/dcds.2007.17.541 [5] Jacek Banasiak. Blow-up of solutions to some coagulation and fragmentation equations with growth. Conference Publications, 2011, 2011 (Special) : 126-134. doi: 10.3934/proc.2011.2011.126 [6] Prasanta Kumar Barik, Ankik Kumar Giri. A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation. Kinetic & Related Models, 2018, 11 (5) : 1125-1138. doi: 10.3934/krm.2018043 [7] Ankik Kumar Giri. On the uniqueness for coagulation and multiple fragmentation equation. Kinetic & Related Models, 2013, 6 (3) : 589-599. doi: 10.3934/krm.2013.6.589 [8] Marie Doumic, Miguel Escobedo. Time asymptotics for a critical case in fragmentation and growth-fragmentation equations. Kinetic & Related Models, 2016, 9 (2) : 251-297. doi: 10.3934/krm.2016.9.251 [9] Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002 [10] Jacek Banasiak, Wilson Lamb. Coagulation, fragmentation and growth processes in a size structured population. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 563-585. doi: 10.3934/dcdsb.2009.11.563 [11] Wilson Lamb, Adam McBride, Louise Smith. Coagulation and fragmentation processes with evolving size and shape profiles: A semigroup approach. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5177-5187. doi: 10.3934/dcds.2013.33.5177 [12] Daniel Balagué, José A. Cañizo, Pierre Gabriel. Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates. Kinetic & Related Models, 2013, 6 (2) : 219-243. doi: 10.3934/krm.2013.6.219 [13] Jacek Banasiak, Wilson Lamb. The discrete fragmentation equation: Semigroups, compactness and asynchronous exponential growth. Kinetic & Related Models, 2012, 5 (2) : 223-236. doi: 10.3934/krm.2012.5.223 [14] Yingshan Chen, Shijin Ding, Wenjun Wang. Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5287-5307. doi: 10.3934/dcds.2016032 [15] Mouhamed Moustapha Fall. Regularity estimates for nonlocal Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1405-1456. doi: 10.3934/dcds.2019061 [16] Mohammad El Smaily, François Hamel, Lionel Roques. Homogenization and influence of fragmentation in a biological invasion model. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 321-342. doi: 10.3934/dcds.2009.25.321 [17] Xianyi Li, Deming Zhu. Comparison theorems of oscillation and nonoscillation for neutral difference equations with continuous arguments. Communications on Pure & Applied Analysis, 2003, 2 (4) : 579-589. doi: 10.3934/cpaa.2003.2.579 [18] Bingyuan Huang, Shijin Ding, Huanyao Wen. Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1717-1752. doi: 10.3934/dcdss.2016072 [19] Alina Gleska, Małgorzata Migda. Qualitative properties of solutions of higher order difference equations with deviating arguments. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 239-252. doi: 10.3934/dcdsb.2018016 [20] Philippe Laurençot, Barbara Niethammer, Juan J.L. Velázquez. Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel. Kinetic & Related Models, 2018, 11 (4) : 933-952. doi: 10.3934/krm.2018037

2018 Impact Factor: 1.38