On a Class of Continuous Coagulation-Fragmentation Equations

Laurençot, Philippe

doi:10.1006/jdeq.2000.3809

Cited by 69 publications

(59 citation statements)

References 19 publications

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“…As in [45], it is possible to extend Theorem 3.8 to perturbations of product kernels of the form K(x, y) = r(x)r(y) +K(x, y) provided 0 ≤K(x, y) ≤ κ 1 r(x)r(y) for x > 0, y > 0, and some κ 1 > 0.…”

Section: Product Kernelsmentioning

confidence: 99%

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Weak Compactness Techniques and Coagulation Equations

Laurençot

2014

Lecture Notes in Mathematics

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Abstract. Smoluchowski's coagulation equation is a mean-field model describing the growth of clusters by successive mergers. Since its derivation in 1916 it has been studied by several authors, using deterministic and stochastic approaches, with a blossoming of results in the last twenty years. In particular, the use of weak L 1 -compactness techniques led to a mature theory of weak solutions and the purpose of these notes is to describe the results obtained so far in that direction, as well as the mathematical tools used.

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Section: Product Kernelsmentioning

confidence: 99%

“…As for the conjecture for gelling kernels (1.9), it was solved rather recently in [27,40]. An intermediate step is the existence of solutions to (1.1)-(1.2) and (1.5) with non-increasing finite mass, that is, satisfying M 1 (f (t)) ≤ M 1 (f (0)) for t ≥ 0, see [24,45,47,52,69,80] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Weak Compactness Techniques and Coagulation Equations

Laurençot

2014

Lecture Notes in Mathematics

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“…An attempt by da Costa [49] to prove that the same behaviour would occur for all solutions, based in a dynamical systems approach, resulted in the identification of a larger family of gelling solutions but did not solve the problem, although it had some use in the numerical analysis of the gelling phenomenon [8]. For the continuous system Laurençot [125] considered a.x; y/ D r.x/r.y/ C˛.x; y/ with˛.x; y/ Ä Ar.x/r.y/ and r.x/ Rx; and proved that all solution exhibit gelation and obtained some results about the density decay and the gelification time.…”

Section: Sketch Of Proofmentioning

confidence: 99%

Mathematical Aspects of Coagulation-Fragmentation Equations

Costa

2015

CIM Series in Mathematical Sciences

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We give an overview of the mathematical literature on the coagulationlike equations, from an analytic deterministic perspective. In Sect. 1 we present the coagulation type equations more commonly encountered in the scientific and mathematical literature and provide a brief historical overview of relevant works. In Sect. 2 we present results about existence and uniqueness of solutions in some of those systems, namely the discrete Smoluchowski and coagulation-fragmentation: we start by a brief description of the function spaces, and then review the results on existence of solutions with a brief description of the main ideas of the proofs. This part closes with the consideration of uniqueness results. In Sects. 3 and 4 we are concerned with several aspects of the solutions behaviour. We pay special attention to the long time convergence to equilibria, self-similar behaviour, and density conservation or lack thereof.

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“…Equation (1) becomes the coagulation and multiple-fragmentation equation introduced in the work of Blatz and Tobolsky [9] and subsequently analysed in References [5,6,10,11]. Many results on the existence and uniqueness of solutions to the various forms of the coagulation-fragmentation equation have already been established using a number of di erent techniques.…”

Section: Introductionmentioning

confidence: 98%

Existence and uniqueness results for the continuous coagulation and fragmentation equation

Lamb

2004

Math Methods in App Sciences

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A non-linear integro-differential equation modelling coagulation and fragmentation is investigated using the theory of strongly continuous semigroups of operators. Under the assumptions that the coagulation kernel is bounded and the overall rate of fragmentation satisfies a linear growth condition, global existence and uniqueness of mass-conserving solutions are established. This extends similar results obtained in earlier investigations. In the case of pure fragmentation, when no coagulation occurs, a precise characterization of the generator of the associated semigroup is also obtained by using perturbation results for substochastic semigroups due to Banasiak (Taiwanese J. Math. 2001; 5: 169-191) and Voigt (Transport Theory Statist. Phys. 1987; 16: 453-466)

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On a Class of Continuous Coagulation-Fragmentation Equations

Cited by 69 publications

References 19 publications

Weak Compactness Techniques and Coagulation Equations

Weak Compactness Techniques and Coagulation Equations

Mathematical Aspects of Coagulation-Fragmentation Equations

Existence and uniqueness results for the continuous coagulation and fragmentation equation

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