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On selfsimilarity and stationary problem for fragmentation and coagulation
"... We prove the existence of a stationary solution of any given mass to the coagulationfragmentation equation without assuming a detailed balance condition, but assuming instead that aggregation dominates fragmentation for small particles whisle fragmentation predominates for large particles. We also s ..."
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Cited by 47 (9 self)
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We prove the existence of a stationary solution of any given mass to the coagulationfragmentation equation without assuming a detailed balance condition, but assuming instead that aggregation dominates fragmentation for small particles whisle 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 by applying a simple abstract result of functional analysis based on the Tykhonov fixed point theorem, and which slightly generalizes a method introduced in [26]. Moreover, we show that the solutions to the fragmentation equation with initial data of a given mass behaves, as t → +∞, as the self similar solution of the same mass. 1 Introduction and
Exponential decay towards equilibrium for the inhomogeneous AizenmanBak model
 Comm. Math. Phys
"... In this work, we show how the entropy method enables to get in an elementary way (and without linearization) estimates of exponential decay towards equilibrium for solutions of reactiondiusion equations corresponding to a reversible reaction. Explicit rates of convergence combining the dissipative ..."
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Cited by 36 (18 self)
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In this work, we show how the entropy method enables to get in an elementary way (and without linearization) estimates of exponential decay towards equilibrium for solutions of reactiondiusion equations corresponding to a reversible reaction. Explicit rates of convergence combining the dissipative eects of diusion and reaction are given.
Dust and selfsimilarity for the Smoluchowski coagulation equation
, 2004
"... We establish the wellposedness of the Cauchy problem for the Smoluchowski coagulation equation in the homogeneous space ˙ L 1 1 for a class of homogeneous coagulation rates of degree λ ∈ [0, 2). For any initial datum fin ∈ ˙ L 1 1 we build a weak solution which conserves the mass when λ ≤ 1 and lo ..."
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Cited by 26 (4 self)
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We establish the wellposedness of the Cauchy problem for the Smoluchowski coagulation equation in the homogeneous space ˙ L 1 1 for a class of homogeneous coagulation rates of degree λ ∈ [0, 2). For any initial datum fin ∈ ˙ L 1 1 we build a weak solution which conserves the mass when λ ≤ 1 and loses mass in finite time (gelation phenomena) when λ> 1. We then extend the existence result to a measure framework allowing dust source term. In that case we prove that the income dust instantaneously aggregates and the solution does not contain dust phase. On the other hand, we investigate the qualitative properties of selfsimilar solutions to the Smoluchowski’s coagulation equation when λ < 1. We prove regularity results and sharp uniform small and large size behavior for the selfsimilar profiles.
On the selfsimilar asymptotics for generalized nonlinear kinetic Maxwell models
"... Abstract. Maxwell models for nonlinear kinetic equations have many applications in physics, dynamics of granular gases, economy, etc. In the present manuscript we consider such models from a very general point of view, including those with arbitrary polynomial nonlinearities and in any dimension sp ..."
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Cited by 26 (5 self)
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Abstract. Maxwell models for nonlinear kinetic equations have many applications in physics, dynamics of granular gases, economy, etc. In the present manuscript we consider such models from a very general point of view, including those with arbitrary polynomial nonlinearities and in any dimension space. It is shown that the whole class of generalized Maxwell models satisfies properties which one of them can be interpreted as an operator generalization of usual Lipschitz conditions. This property allows to describe in detail a behavior of solutions to the corresponding initial value problem. In particular, we prove in the most general case an existence of self similar solutions and study the convergence, in the sense of probability measures, of dynamically scaled solutions to the Cauchy problem to those selfsimilar solutions, as time goes to infinity. The properties of these selfsimilar solutions, leading to non classical equilibrium stable states, are studied in detail. We apply the results to three different specific problems related to the Boltzmann equation (with elastic and inelastic interactions) and show that all physically relevant properties of solutions follow directly from the general theory developed in this paper. Contents
Dynamical scaling in Smoluchowski’s coagulation equations: uniform convergence
, 2006
"... Smoluchowski’s coagulation equation is a fundamental meanfield model of clustering dynamics. We consider the approach to selfsimilarity (or dynamical scaling) of the cluster size distribution for the “solvable” rate kernels K(x, y) =2,x+ y, and xy. In the case of continuous cluster size distributi ..."
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Cited by 20 (2 self)
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Smoluchowski’s coagulation equation is a fundamental meanfield model of clustering dynamics. We consider the approach to selfsimilarity (or dynamical scaling) of the cluster size distribution for the “solvable” rate kernels K(x, y) =2,x+ y, and xy. In the case of continuous cluster size distributions, we prove uniform convergence of densities to a selfsimilar solution with exponential tail, under the regularity hypothesis that a suitable moment have an integrable Fourier transform. For discrete size distributions, we prove uniform convergence under optimal moment hypotheses. Our results are completely analogous to classical local convergence theorems for the normal law in probability theory. The proofs rely on the Fourier inversion formula and the solution for the Laplace transform by the method of characteristics in the complex plane.
Generalized kinetic Maxwell type models of granular gases
 In Mathematical models of granular matter, volume 1937 of Lecture Notes in Math
, 2008
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Dynamics and selfsimilarity in mindriven clustering
, 2008
"... We study a meanfield model for a clustering process that may be described informally as follows. At each step a random integer k is chosen with probability pk, and the smallest cluster merges with k randomly chosen clusters. We prove that the model determines a continuous dynamical system on the sp ..."
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Cited by 6 (2 self)
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We study a meanfield model for a clustering process that may be described informally as follows. At each step a random integer k is chosen with probability pk, and the smallest cluster merges with k randomly chosen clusters. We prove that the model determines a continuous dynamical system on the space of probability measures supported in (0, ∞), and we establish necessary and sufficient conditions for approach to selfsimilar form. We also characterize eternal solutions for this model via a LévyKhintchine formula. The analysis is based on an explicit solution formula discovered by Gallay and Mielke, extended using a careful choice of time scale.
OPTIMAL BOUNDS FOR SELFSIMILAR SOLUTIONS TO COAGULATION EQUATIONS WITH MULTIPLICATIVE KERNEL
"... Abstract. We consider massconserving selfsimilar solutions of Smoluchowski’s coagulation equation with multiplicative kernel of homogeneity 2λ ∈ (0,1). We establish rigorously that such solutions exhibit a singular behavior of the form x −(1+2λ) as x → 0. This property had been conjectured, but on ..."
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Cited by 5 (1 self)
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Abstract. We consider massconserving selfsimilar solutions of Smoluchowski’s coagulation equation with multiplicative kernel of homogeneity 2λ ∈ (0,1). We establish rigorously that such solutions exhibit a singular behavior of the form x −(1+2λ) as x → 0. This property had been conjectured, but only weaker results had been available up to now. 1.
Similarity solutions of a BeckerDöring system with timedependent monomer input
 J. Phys. A; Math. Gen
"... Abstract. We formulate the BeckerDoring equations for cluster growth in the presence of a timedependent source of monomer input. In the case of sizeindependent aggregation and fragmentation rate coecients we nd similarity solutions which are approached in the large time limit. The form of the sol ..."
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Cited by 5 (2 self)
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Abstract. We formulate the BeckerDoring equations for cluster growth in the presence of a timedependent source of monomer input. In the case of sizeindependent aggregation and fragmentation rate coecients we nd similarity solutions which are approached in the large time limit. The form of the solutions depends on the rate of monomer input and whether fragmentation is present in the model; four distinct types of solution are found. PACS numbers: 64.60.i, 82.60.Nh, 05.70.Fh, aggregation equations, coagulationfragmentation equations, selfsimilar behaviour. 1.
Universality classes in Burgers turbulence
, 2006
"... We establish necessary and sufficient conditions for the shock statistics to approach selfsimilar form in Burgers turbulence with Lévy process initial data. The proof relies upon an elegant closure theorem of Bertoin and Carraro and Duchon that reduces the study of shock statistics to Smoluchowski’ ..."
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Cited by 5 (4 self)
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We establish necessary and sufficient conditions for the shock statistics to approach selfsimilar form in Burgers turbulence with Lévy process initial data. The proof relies upon an elegant closure theorem of Bertoin and Carraro and Duchon that reduces the study of shock statistics to Smoluchowski’s coagulation equation with additive kernel, and upon our previous characterization of the domains of attraction of selfsimilar solutions for this equation. Keywords: Burgers turbulence, Smoluchowski’s coagulation equation, Lévy processes, dynamic scaling, regular variation, agglomeration, coagulation,