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

Laurençot, Philippe; Mischler, Stéphane

doi:10.1007/s00605-005-0308-1

Cited by 12 publications

(9 citation statements)

References 21 publications

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“…A result similar to the latter, but giving uniform convergence only in compact sets, can be found in [9]; weak convergence was proved in [12] using entropy arguments. Hence, the main improvement of Theorem 1.1 is that we give an explicit rate of convergence, which is to our knowledge a new result, and that the convergence in eq.…”

Section: Introduction 1smoluchowski's Equation With a Constant Kernelmentioning

confidence: 65%

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

Cañizo¹,

Mischler²,

Mouhot³

2010

SIAM J. Math. Anal.

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We show that solutions to Smoluchowski's equation with a constant coagulation kernel and an initial datum with some regularity and exponentially decaying tail converge exponentially fast to a selfsimilar profile. This convergence holds in a weighted Sobolev norm which implies the L 2 convergence of derivatives up to a certain order k depending on the regularity of the initial condition. We prove these results through the study of the linearized coagulation equation in self-similar variables, for which we show a spectral gap in a scale of weighted Sobolev spaces. We also take advantage of the fact that the Laplace or Fourier transforms of this equation can be explicitly solved in this case.

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Section: Introduction 1smoluchowski's Equation With a Constant Kernelmentioning

confidence: 65%

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

Cañizo¹,

Mischler²,

Mouhot³

2010

SIAM J. Math. Anal.

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“…In the case of Equations (6) and (11) respectively, we get formulae (9) and (12), which discrete versions are expressed by (29) and (27).…”

Section: Discussionmentioning

confidence: 99%

“…By making use of flux cytometry technologies for instance, it is possible to determine cell populations with certain properties as protein content on a large scale of tenths of thousands of cells. In other applications, like coagulation fragmentation equation [8,9,10,11,12], or prion aggregation and fragmentation [13,14,15], similar equations arise, and much less is known on aggregate size repartition. The division rate B(x), on the contrary, is not directly measurable.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of an inverse problem in size-structured population dynamics

Doumic¹,

Perthame²,

Zubelli³

2009

Inverse Problems

124

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We consider a size-structured model for cell division and address the question of determining the division (birth) rate from the measured stable size distribution of the population. We propose a new regularization technique based on a filtering approach. We prove convergence of the algorithm and validate the theoretical results by implementing numerical simulations, based on classical techniques. We compare the results for direct and inverse problems, for the filtering method and for the quasi-reversibility method proposed in [1].

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“…C1. This idea was successfully implemented, in the framework of weak convergence in L 1 , in the case of constant coefficients a.x; y/ Á 1; using Lyapunov functions whose construction were strongly dependent of the known form of the limit˚ [134], which turns the potentially promising method useless if the form of is not known. The idea was also applied with success in the prove of existence and stability of self-similar solutions in the Oort-Hulst-Safronov equations with constant [127], with additive [9], and with multiplicative [128] coefficients.…”

Section: Sketch Of Proofmentioning

confidence: 98%

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

Cited by 12 publications

References 21 publications

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

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

Numerical solution of an inverse problem in size-structured population dynamics

Mathematical Aspects of Coagulation-Fragmentation Equations

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