zbMATH — the first resource for mathematics

Global existence for the discrete diffusive coagulation-fragmentation equations in \(L^1\). (English) Zbl 1036.35089
This paper shows existence of global weak solutions to the discrete coagulation-fragmentation equations with diffusion: \[ \begin{cases} \partial_t c_i -d_i \triangle_xc_i =Q_i (c), &\text{ in } (0, \, + \infty) \times\Omega, \\ \partial_n c_i = 0, &\text{ on } (0, + \infty )\times \partial \Omega, \\ c(0) = c_i^{i_n}, &\text{ in } \Omega. \end{cases} \] Unlike previous works requiring \(L^{\infty}\)-estimates, an \(L^1\)-approach is developed in this paper. The authors show that under rather general assumptions on the kinetic coefficients, there exists at least a weak solution to the above equations for any initial datum \(c^{i_n}=(c_i^{i_n})_{i \geq 1}\) with finite total mass.

35K57 Reaction-diffusion equations
82D60 Statistical mechanics of polymers
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
Full Text: DOI EuDML
[1] Amann, H.: Dual semigroups and second order linear elliptic boundary value problems. Israel J. Math. 45 (1983), 225-254. · Zbl 0535.35017
[2] Amann, H.: Coagulation-fragmentation processes. Arch. Rational Mech. Anal. 151 (2000), 339-366. · Zbl 0977.35060
[3] Ball, J.M. and Carr, J.: The discrete coagulation-fragmentation equa- tions : existence, uniqueness, and density conservation. J. Statist. Phys. 61 (1990), 203-234. · Zbl 1217.82050
[4] Baras, P.: Compacité de l’opérateur f \rightarrow u solution d’une équation non linéaire (du/dt) + Au f . C. R. Acad. Sci. Paris Sér. I Math. 286 (1978), 1113-1116. · Zbl 0389.47030
[5] Baras, P. Hassan, J.C. and Véron, L.: Compacité de l’opérateur définissant la solution d’une équation d’évolution non homog‘ ene. C. R. Acad. Sci. Paris Sér. I Math. 284 (1977), 799-802. · Zbl 0348.47026
[6] Bénilan, Ph. and Wrzosek, D.: On a infinite system of reaction- diffusion equations. Adv. Math. Sci. Appl. 7 (1997), 349-364. P. Laurenc \?ot and S. Mischler · Zbl 0884.35165
[7] Chae, D. and Dubovski\?ı, P.B.: Existence and uniqueness for spatially inhomogeneous coagulation equation with sources and effluxes. Z. Angew. Math. Phys. 46 (1995), 580-594. · Zbl 0833.35142
[8] Collet, J.F. and Poupaud, F.: Existence of solutions to coagulation- fragmentation systems with diffusion. Transport Theory Statist. Phys. 25 (1996), 503-513. · Zbl 0870.35117
[9] Costa, F.P. da: Existence and uniqueness of density conserving solu- tions to the coagulation-fragmentation equations with strong fragmenta- tion. J. Math. Anal. Appl. 192 (1995), 892-914. · Zbl 0839.34015
[10] Drake, R.L.: A general mathematical survey of the coagulation equation, in Topics in Current Aerosol Research (part 2). International Reviews in Aerosol Physics and Chemistry, Pergamon Press, Oxford, 1972, 203-376.
[11] Guias \?, F.: Convergence properties of a stochastic model for coagulation- fragmentation processes with diffusion. Stochastic Anal. Appl. 19 (2001), 245-278. · Zbl 1015.60094
[12] Herrero, M.A., Velázquez, J.J.L. and Wrzosek, D.: Sol-gel transi- tion in a coagulation-diffusion model. Phys. D 141 (2000), 221-247. · Zbl 1033.82011
[13] Jeon, I.: Existence of gelling solutions for coagulation-fragmentation equa- tions. Comm. Math. Phys. 194 (1998), 541-567. · Zbl 0910.60083
[14] Laurenc \?ot, Ph. and Wrzosek, D.: Fragmentation-diffusion model. Ex- istence of solutions and asymptotic behaviour. Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 759-774. · Zbl 0912.35031
[15] Laurenc \?ot, Ph. and Wrzosek, D.: The Becker-Döring model with diffusion. I. Basic properties of solutions. Colloq. Math. 75 (1998), 245-269. · Zbl 0894.35055
[16] McGrady, E.D. and Ziff, R.M.: “Shattering” transition in fragmenta- tion. Phys. Rev. Lett. 58 (1987), 892-895.
[17] Perelson, A.S. and Samsel, R.W.: Kinetics of red blood cell aggrega- tion : an example of geometric polymerization, in Kinetics of Aggregation and Gelation. F. Family and D.P. Landau (eds.), Elsevier, 1984.
[18] Safronov, V.S.: Evolution of the Protoplanetary Cloud and Formation of the Earth and the Planets, Israel Program for Scientific Translations Ltd., Jerusalem, 1972.
[19] Simons, S. and Simpson, D.R.: The effect of particle coagulation on the diffusive relaxation of a spatially inhomogeneous aerosol. J. Phys. A 21 (1988), 3523-3536.
[20] Slemrod, M.: Coagulation-diffusion systems : derivation and existence of solutions for the diffuse interface structure equations. Phys. D 46 (1990), 351-366. · Zbl 0732.35103
[21] Smoluchowski, M.: Drei Vorträge über Diffusion, Brownsche Moleku- larbewegung und Koagulation von Kolloidteilchen. Physik. Zeitschr. 17 (1916), 557-599. 745
[22] Smoluchowski, M.: Versuch einer mathematischen Theorie der Koagu- lationskinetik kolloider Lösungen. Zeitschrift f. physik. Chemie 92 (1917), 129-168.
[23] Spouge, J.L.: An existence theorem for the discrete coagulation- fragmentation equations. Math. Proc. Cambridge Philos. Soc. 96 (1984), 351-357. · Zbl 0541.92029
[24] Wrzosek, D.: Existence of solutions for the discrete coagulation- fragmentation model with diffusion. Topol. Methods Nonlinear Anal. 9 (1997), 279-296. · Zbl 0892.35077
[25] Wrzosek, D.: Mass-conserving solutions to the discrete coagulation- fragmentation model with diffusion. Nonlinear Anal. 49 (2002), 297-314. · Zbl 1001.35059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.