On self-similarity and stationary problem for fragmentation and coagulation models

Abstract

We prove the existence of a stationary solution of any given mass to the coagulation-fragmentation equation without assuming a detailed balance condition, but assuming instead that aggregation dominates fragmentation for small particles while fragmentation predominates for large particles. We also show the existence of a self-similar solution of any given mass to the coagulation equation and to the fragmentation equation for kernels satisfying a scaling property. These results are obtained, following the theory of Poincaré-Bendixson on dynamical systems, by applying the Tykonov fixed point theorem on the semigroup generated by the equation or by the associated equation written in "self-similar variables". Moreover, we show that the solutions to the fragmentation equation with initial data of a given mass behaves, as t → +∞, as the unique self similar solution of the same mass.  2004 Elsevier SAS. All rights reserved. RésuméPour toute masse donnée, nous démontrons l'existence d'au moins une solution stationnaire pour l'équation de coagulationfragmentation. Nous ne faisons pas d'hypothèse d'équilibre en détails sur les coefficients mais nous supposons que la coagulation domine la fragmentation pour les particules de petite taille et que la fragmentation est prépondérante pour les particules de grande taille. Nous démontrons également l'existence de solutions auto-similaires pour l'équation de coagulation et pour l'équation de fragmentation sous une hypothèse d'homogéneité sur les noyaux. Ces résultats sont obtenus, s'inspirant de la preuve

Rate of Convergence to Self-Similarity for Smoluchowski's Coagulation Equation with Constant Coefficients

“…(6). When k = −1 or 0, the condition g 0 k,ν < ∞ is redundant, as it is already implied by g 0 e νy < ∞ and g ′ 0 ∈ L 1 .…”

“…It is expected that for a general class of homogeneous coagulation kernels, there is a unique self-similar profile with given mass ρ > 0, and that for a very wide class of initial conditions solutions to Smoluchowski's equation approximate, for large times, the self-similar solution with the same mass, in a sense to be precised. For general homogeneous coagulation rates there are no rigorous proofs of this behavior, except for recent results that have shown the existence of self-similar profiles [7,6] and given some of their properties [8,4]. Nevertheless, for the specific coagulation rates given by a(x, y) = x + y and a(x, y) = 1 (our case), it is known that the self-similar profile is unique for each given finite mass, and the convergence to it has been proved in [9,14,15,12].…”

Weak Compactness Techniques and Coagulation Equations

“…In both cases the expected behaviour is of self-similar form (except for some particular kernels with homogeneity 1) but the time and mass scales are only well identified for non-gelling kernels and for the multiplicative kernel K 2 (x, y) = xy, see the survey articles [18,51] and the references therein. Existence of mass-conserving self-similar solutions for a large class of nongelling kernels have been constructed recently [29,31] and their properties studied in [11,26,32,60,65]. Still for non-gelling kernels, the existence of other self-similar solutions (with a different scaling and possibly infinite mass) is uncovered in [8] for the additive kernel K 1 (x, y) = x + y and in [62] for the constant kernel K 0 (x, y) = 2, both results relying on the use of the Laplace transform which maps (1.5) either to Burgers' equation or to an ordinary differential equation.…”

Liapunov Functionals for Smoluchowski’s Coagulation Equation and Convergence to Self-Similarity

“…In fact, for other homogeneous coagulation kernels, the first difficulty encountered is the existence of the scaling profile g S which satisfies a nonlinear and nonlocal integro-differential equation. For a wide class of homogeneous coagulation kernels, such an existence result has been recently obtained in [9,11]. Still, the uniqueness of the profile (in a suitable class) and the convergence to self-similar solutions are open problems.…”