Smoothness of moments of the solutions of discrete coagulation equations with diffusion

Breden, Maxime; Desvillettes, Laurent; Fellner, Klemens

doi:10.1007/s00605-016-0969-y

Cited by 9 publications

(17 citation statements)

References 29 publications

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“…In particular this implies that strong enough fragmentation can prevent gelation even for superlinear coagulation, a statement which was only known up to now in the homogeneous setting. We also use this control of superlinear moments to extend a recent result from [3], about the regularity of the solutions in the pure coagulation case, to strong fragmentation models.…”

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confidence: 92%

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Applications of improved duality lemmas to the discrete coagulation-fragmentation equations with diffusion

Breden¹

2018

Kinetic &Amp; Related Models

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In this paper, we investigate the use of so called "duality lemmas" to study the system of discrete coagulation-fragmentation equations with diffusion. When the fragmentation is strong enough with respect to the coagulation, we show that we have creation and propagation of superlinear moments. In particular this implies that strong enough fragmentation can prevent gelation even for superlinear coagulation, a statement which was only known up to now in the homogeneous setting. We also use this control of superlinear moments to extend a recent result from [3], about the regularity of the solutions in the pure coagulation case, to strong fragmentation models.

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confidence: 92%

“…In this paper, we investigate other consequences of the L p estimates of [3], this time in presence of fragmentation. Our main theorem states that, when the fragmentation is strong enough compared to the coagulation, we have creation and propagation of superlinear moments.…”

Section: Introductionmentioning

confidence: 99%

Applications of improved duality lemmas to the discrete coagulation-fragmentation equations with diffusion

Breden¹

2018

Kinetic &Amp; Related Models

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“…with Neumann boundary conditions (19) and with initial data (20) in the following sense: identity (29) holds a.e., and for all ϕ, ψ ∈ C 2 c ([0, +∞) ×Ω) such that ∇ x ϕ · n | ∂Ω = 0, ∇ x ψ · n | ∂Ω = 0,…”

Section: Rigorous Results Of Convergencementioning

confidence: 99%

“…A refined version of the same lemma (cf. for example [19] or [20]) yields in fact the better estimate…”

Section: Rigorous Results For the Passages To The Limit In Microscopimentioning

confidence: 99%

“…On one hand, we use the duality lemmas devised for reaction-diffusion systems by M. Pierre and D. Schmitt [18]. More precisely, we use an improved version of those lemmas allowing to recover L p bounds for p > 2 for the solutions of such systems [19,20]. On the other hand, we also use entropy-like functionals which are strongly reminiscent of those used in works in which microscopic models for the Shigesada-Kawasaki-Teramoto system [21] are studied [22].…”

Section: General Presentationmentioning

confidence: 99%

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About reaction–diffusion systems involving the Holling-type II and the Beddington–DeAngelis functional responses for predator–prey models

Conforto

Desvillettes

Soresina

2018

Nonlinear Differ. Equ. Appl.

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We consider in this paper a microscopic model (that is, a system of three reaction-diffusion equations) incorporating the dynamics of handling and searching predators, and show that its solutions converge when a small parameter tends to 0 towards the solutions of a reaction-cross diffusion system of predator-prey type involving a Holling-type II or Beddington-DeAngelis functional response. We also provide a study of the Turing instability domain of the obtained equations and (in the case of the Beddington-DeAngelis functional response) compare it to the same instability domain when the cross diffusion is replaced by a standard diffusion.If J * 11 < 0, we have J * 11− J * ∆22 + + J * 22 − J * ∆11 + − J * 12 − J * ∆21 − , so that condition (90) does not hold and, as in the case of linear diffusion, no Turing instability can appear. On the opposite, if J * 11 > 0, we have J * 11 + J * ∆22 + + J * 22 − J * ∆11 + − J * 12 − J * ∆21 − .Then, for any k = 0 (so that λ k > 0, we can select D 2 large enough and D 3 ∼ D 2 so that det J − (J * 11 J * ∆22 − J * 12 J * ∆21 )λ k < 0. Then, when D 1 > 0 is small enough, det M < 0 and the Turing instability appears.We then compare the Turing instability regions in the cases 2 and 3, that is when the characteristic matrices are

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Uniqueness of strong solutions and weak–strong stability in a system of cross-diffusion equations

et al. 2019

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Proving the uniqueness of solutions to multi-species cross-diffusion systems is a difficult task in the general case, and there exist very few results in this direction. In this work, we study a particular system with zero-flux boundary conditions for which the existence of a weak solution has been proven in [14]. Under additional assumptions on the value of the crossdiffusion coefficients, we are able to show the existence and uniqueness of non-negative strong solutions. The proof of the existence relies on the use of an appropriate linearized problem and a fixed-point argument. In addition, a weak-strong stability result is obtained for this system in dimension one which also implies uniqueness of weak solutions.

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Smoothness of moments of the solutions of discrete coagulation equations with diffusion

Cited by 9 publications

References 29 publications

Applications of improved duality lemmas to the discrete coagulation-fragmentation equations with diffusion

Applications of improved duality lemmas to the discrete coagulation-fragmentation equations with diffusion

About reaction–diffusion systems involving the Holling-type II and the Beddington–DeAngelis functional responses for predator–prey models

Uniqueness of strong solutions and weak–strong stability in a system of cross-diffusion equations

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