Fast Reaction Limit of the Discrete Diffusive Coagulation–Fragmentation Equation

Abstract

The local mass of weak solutions to the discrete diffusive coagulation–fragmentation equation is proved to converge, in the fast reaction limit, to the solution of a nonlinear diffusion equation, the coagulation and fragmentation rates enjoying a detailed balance condition.     

The continuous coagulation equation with multiple fragmentation

We present a proof of the existence of solutions to the continuous coagulation equation with multiple fragmentation whenever the kernels satisfy certain growth conditions. The proof relies on weak L¹ compactness methods applied to suitably chosen approximating equations. The question of uniqueness is also considered.     

Global classical solutions of coagulation-fragmentation equations with unbounded coagulation rates

We study discrete fragmentation coagulation equations in spaces X_p, p>1, consisting of distributions having the pth moments finite. We show that for sufficiently regular fragmentation laws the fragmentation semigroup is analytic in X_p, and fully characterize the domain of its generator. This allows for explicit characterization of the domains of the fractional powers of the generator through real interpolation. Finally, we use the linear results to show the existence of global classical solutions to fragmentation coagulation equations for a class of unbounded coagulation kernels.     

Mass-conserving self-similar solutions to coagulation–fragmentation equations

Existence of mass-conserving self-similar solutions with a sufficiently small total mass is proved for a specific class of homogeneous coagulation and fragmentation coefficients. The proof combines a dynamical approach to construct such solutions for a regularized coagulation–fragmentation equation in scaling variables and a compactness method.     

Existence,Uniqueness and Mass Conservation for the Coagulation–Fragmentation Equation

We prove global existence and uniqueness to the initial value problem for the coagulation–fragmentation equation for an unbounded coagulation kernel with possible linear growth at infinity and a fragmentation kernel from a very large class of unbounded functions. We show that the solutions satisfy the mass conservation law.     

Solutions with peaks for a coagulation–fragmentation equation. Part I: stability of the tails

Abstract

The aim of this two-part paper is to investigate the stability properties of a special class of solutions to a coagulation–fragmentation equation. We assume that the coagulation kernel is close to the diagonal kernel, and that the fragmentation kernel is diagonal. We construct a two-parameter family of stationary solutions concentrated in Dirac masses. We carefully study the asymptotic decay of the tails of these solutions, showing that this behavior is stable. In a companion paper, we prove that for initial data which are sufficiently concentrated, the corresponding solutions approach one of these stationary solutions for large times.     

Approach to Equilibrium for the Coagulation–Fragmentation Equation via a Lyapunov Functional

It is shown that the non-linear coagulation–fragmentation equation with constant kernels has a unique equilibrium solution. This equilibrium solution is given explicitly in terms of the initial data and the kernels. Weak L¹ convergence of time-dependent solutions to the unique equilibrium is demonstrated via an invariance principle employing a suitable lower semicontinuous Lyapunov functional.     

Uniqueness for the coagulation-fragmentation equation with strong fragmentation

The existence of at least one mass-conserving solution for continuous coagulation-fragmentation equation has been established by Escobedo et?al. (J Differ Equ 195:143?C174, 2003) for a large class of coagulation kernels under strong binary fragmentation. In this work, uniqueness of mass-conserving solutions is demonstrated with some additional restrictions on the fragmentation kernels.     

Existence and uniqueness results for the non-autonomous coagulation and multiple-fragmentation equation

An initial-value problem modelling coagulation and fragmentation processes is studied. The results of earlier papers are extended to models where either one or both of the rates of coagulation and fragmentation depend on time. An abstract integral equation, involving the solution operator to the linear fragmentation part, is investigated via the contraction mapping principle. A unique global, non-negative, mass-conserving solution to this abstract equation is shown to exist. The latter solution is used to generate a global, non-negative, mass-conserving solution to the original non-autonomous coagulation and multiple-fragmentation equation. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Weak solutions to the continuous coagulation equation with multiple fragmentation

The existence of weak solutions to the continuous coagulation equation with multiple fragmentation is shown for a class of unbounded coagulation and fragmentation kernels, the fragmentation kernel having possibly a singularity at the origin. This result extends previous ones where either boundedness of the coagulation kernel or no singularity at the origin for the fragmentation kernel was assumed.     

Analytic fragmentation semigroups and continuous coagulation–fragmentation equations with unbounded rates

In this paper we show that continuous fragmentation operators are sectorial for a large range of physically relevant fragmentation rates and use this fact to prove classical solvability of the combined coagulation–fragmentation equation with unbounded coagulation kernels.     

Smoluchowski’s equation: Rate of convergence of the Marcus-Lushnikov process

We derive a satisfying rate of convergence of the Marcus-Lushnikov process towards the solution to Smoluchowski’s coagulation equation. Our result applies to a class of homogeneous-like coagulation kernels with homogeneity degree ranging in (−∞,1]. It relies on the use of a Wasserstein-type distance, which has shown to be particularly well-adapted to coalescence phenomena. It was introduced and used in preceding works (Fournier and Laurençot (2006) [7]) and (Fournier and Löcherbach (2009) [8]).     

The steady spectra of particles in dispersible systems with coagulation and fragmentation

The formation of a steady dimensional distribution of particles (particle spectra) is dispersible systems with coagulation and fragmentation is considered. The relation between versions of the kinetic equation that defines these processes is traced. An analytical solution is obtained for the parametric set of coagulation coefficients and the velocities of paired fragmentation. The steady spectrum of particles is investigated in the case when the fragmentation is of the multiple type.     

Well-posedness of Smoluchowski's coagulation equation for a class of homogeneous kernels

The uniqueness and existence of measure-valued solutions to Smoluchowski's coagulation equation are considered for a class of homogeneous kernels. Denoting by λ∈(-∞,2]?{0} the degree of homogeneity of the coagulation kernel a, measure-valued solutions are shown to be unique under the sole assumption that the moment of order λ of the initial datum is finite. A similar result was already available for the kernels a(x,y)=2, x+y and xy, and is extended here to a much wider class of kernels by a different approach. The uniqueness result presented herein also seems to improve previous results for several explicit kernels. Furthermore, a comparison principle and a contraction property are obtained for the constant kernel.     

Existence globale pour l'équation de Smoluchowski continue non homogène et comportement asymptotique des solutions

We prove global existence of solutions to the continuous nonhomogeneous Smoluchowski equation for coagulation rates satisfying a more general structure condition than the Galkin–Tupchiev monotony hypothesis considered in (Ph. Laurençot, S. Mischler, Arch. Rational Mech. Anal. 162 (1) (2002) 45–99). The Smoluchowski coagulation rate fulfils this condition as well as some rates which vanish on the diagonal. Under the condition of positivity of the coagulation rate outside of the diagonal we prove that solutions tend to 0 in the large time asymptotic. These results depend on a new estimate from below for the dissipation rate of the L^p-norm, p>1. To cite this article: S. Mischler, M. Rodriguez Ricard, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Pseudodifferential operators with non-regular operator-valued symbols

In this paper, we consider pseudodifferential operators with operator-valued symbols and their mapping properties, without assumptions on the underlying Banach space E. We show that, under suitable parabolicity assumptions, the \({W_p^k(\mathbb{R}^n, E)}\) -realization of the operator generates an analytic semigroup. Our approach is based on oscillatory integrals and kernel estimates for them. An application to non-autonomous pseudodifferential Cauchy problems gives the existence and uniqueness of a classical solution. As an example, we include a discussion of coagulation–fragmentation processes.     

Convergence of the Marcus–Lushnikov Process

The Smoluchowski coagulation equation describes the concentration c(t,x) of particles of mass x [ 0,] at the instant t 0, in an infinite system of coalescing particles. It is well-known that in some cases, gelation occurs: a particle with infinite mass appears. But this infinite particle is inert, in the sense that it does not interact with finite particles. We consider the so-called Marcus–Lushnikov process, which is a stochastic finite system of coalescing particles. This process is expected to converge, as the number of particles tends to infinity, to a solution of the Smoluchowski coagulation equation. We show that it actually converges, for t [0,], to a modified Smoluchowski equation, which takes into account a possible interaction between finite and infinite particles.     

Radial equivalence and study of self-similarity for two very fast diffusion equations

In this paper we continue the study of the radial equivalence between the porous medium equation and the evolution p-Laplacian equation, begun in a previous work. We treat the cases m<0 and p<1. We perform an exhaustive study of self-similar solutions for both equations, based on a phase-plane analysis and the correspondences we discover. We also obtain special correspondence relations and self-maps for the limit case m=−1, p=0, which is particularly important in applications in image processing. We also find self-similar solutions for the very fast p-Laplacian equation that have finite mass and, in particular, some of them that conserve mass, while this phenomenon is not true for the very fast diffusion equation.