We present a new a priori estimate for discrete coagulation–fragmentation systems with size-dependent diffusion within a bounded, regular domain confined by homogeneous Neumann boundary conditions. Following from a duality argument, this a priori estimate provides a global ${L}^{2}$ bound on the mass density and was previously used, for instance, in the context of reaction–diffusion equations.In this paper we demonstrate two lines of applications for such an estimate: On the one hand, it enables to simplify parts of the known existence theory and allows to show existence of solutions for generalised models involving collision-induced, quadratic fragmentation terms for which the previous existence theory seems difficult to apply. On the other hand and most prominently, it proves mass conservation (and thus the absence of gelation) for almost all the coagulation coefficients for which mass conservation is known to hold true in the space homogeneous case.

Classification: 35B45, 35Q72, 82D60

Keywords: Discrete coagulation–fragmentation systems, Mass conservation, Duality arguments

@article{AIHPC_2010__27_2_639_0, author = {Ca\~nizo, J.A. and Desvillettes, L. and Fellner, K.}, title = {Regularity and mass conservation for discrete coagulation--fragmentation equations with diffusion}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, publisher = {Elsevier}, volume = {27}, number = {2}, year = {2010}, pages = {639-654}, doi = {10.1016/j.anihpc.2009.10.001}, zbl = {1193.35091}, mrnumber = {2595194}, language = {en}, url = {http://www.numdam.org/item/AIHPC_2010__27_2_639_0} }

Regularity and mass conservation for discrete coagulation–fragmentation equations with diffusion. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 2, pp. 639-654. doi : 10.1016/j.anihpc.2009.10.001. http://www.numdam.org/item/AIHPC_2010__27_2_639_0/

[1] Coagulation–fragmentation processes, Arch. Ration. Mech. Anal. 151 no. 4 (2000), 339-366 | MR 1756908 | Zbl 0977.35060

,[2] The discrete coagulation–fragmentation equations: Existence, uniqueness, and density conservation, J. Stat. Phys. 61 no. 1 (1990), 203-234 | MR 1084278 | Zbl 1217.82050

, ,[3] On an infinite system of reaction–diffusion equations, Adv. Math. Sci. Appl. 7 no. 1 (1997), 351-366 | MR 1454671 | Zbl 0884.35165

, ,[4] J.A. Cañizo, Some problems related to the study of interaction kernels: Coagulation, fragmentation and diffusion in kinetic and quantum equations, PhD thesis, Universidad de Granada, June 2006

[5] Asymptotic behavior of solutions to the coagulation–fragmentation equations. II. Weak fragmentation, J. Stat. Phys. 77 no. 1 (1994), 89-123 | MR 1300530 | Zbl 0838.60089

, ,[6] Exponential decay towards equilibrium for the inhomogeneous Aizenman–Bak model, Comm. Math. Phys. 278 no. 2 (2008), 433-451 | MR 2372765 | Zbl 1144.82080

, , ,[7] Existence of solutions to coagulation–fragmentation systems with diffusion, Transport Theory Statist. Phys. 25 no. 3 (1996), 503-513 | MR 1407550 | Zbl 0870.35117

, ,[8] Entropy methods for reaction–diffusion equations: Slowly growing a priori bounds, Rev. Mat. Iberoamericana 24 no. 2 (2008), 407-431 | MR 2459198 | Zbl 1171.35330

, ,[9] About global existence for quadratic systems of reaction–diffusion, Adv. Nonlinear Stud. 7 no. 3 (2007), 491-511 | MR 2340282 | Zbl 1330.35211

, , , ,[10] Gelation and mass conservation in coagulation–fragmentation models, J. Differential Equations 195 no. 1 (2003), 143-174 | MR 2019246 | Zbl 1133.82316

, , , ,[11] Gelation in coagulation and fragmentation models, Comm. Math. Phys. 231 (2002), 157-188 | MR 1947695 | Zbl 1016.82027

, , ,[12] T. Goudon, A. Vasseur, Regularity analysis for systems of reaction–diffusion equations, Annales de l'École Norm. Supérieure, in press | Numdam | MR 2583266

[13] Convergence properties of a stochastic model for coagulation–fragmentation processes with diffusion, Stoch. Anal. Appl. 19 no. 2 (2001), 245-278 | MR 1841439 | Zbl 1015.60094

,[14] Moment bounds for the Smoluchowski equation and their consequences, Comm. Math. Phys. 276 no. 3 (2007), 645-670 | MR 2350433 | Zbl 1132.35090

, ,[15] Global existence for the discrete diffusive coagulation–fragmentation equation in ${L}^{1}$, Rev. Mat. Iberoamericana 18 (2002), 731-745 | MR 1954870 | Zbl 1036.35089

, ,[16] Weak solutions and supersolutions in ${L}^{1}$ for reaction–diffusion systems, J. Evol. Equ. 3 no. 1 (2003), 153-168 | MR 1977432 | Zbl 1026.35047

,[17] Blowup in reaction–diffusion systems with dissipation of mass, SIAM Rev. 42 (2000), 93-106 | MR 1738101 | Zbl 0942.35033

, ,[18] Existence of solutions for the discrete coagulation–fragmentation model with diffusion, Topol. Methods Nonlinear Anal. 9 no. 2 (1997), 279-296 | MR 1491848 | Zbl 0892.35077

,[19] Mass-conserving solutions to the discrete coagulation–fragmentation model with diffusion, Nonlinear Anal. 49 no. 3 (2002), 297-314 | MR 1886116 | Zbl 1001.35059

,[20] Weak solutions to the Cauchy problem for the diffusive discrete coagulation–fragmentation system, J. Math. Anal. Appl. 289 no. 2 (2004), 405-418 | MR 2026914 | Zbl 1039.35051

,