Stationary Non-equilibrium Solutions for Coagulation Systems

Ferreira, Marina A.; Lukkarinen, Jani; Nota, Alessia; Velázquez, Juan J. L.

doi:10.1007/s00205-021-01623-w

Cited by 13 publications

(72 citation statements)

References 39 publications

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“…Next we state the existence of stationary injection solutions to (2.2) and (2.1) for kernels K with γ + 2p < 1. This result has been proved in [5] and it is a natural extension from one-to multi-component systems of the result contained in [4].…”

Section: Stationary Injection Solutionsmentioning

confidence: 84%

“…Remark 2.6. The flux (2.10) is obtained by considering in (2.8) the test function ψ(r, θ) = r χ δ (r ), with χ δ (r ) ∈ C ∞ c (R * ) such that χ δ (r ) ∈ [0, 1], χ δ (r ) = 1 for r ∈ [1, z] and χ δ (r ) = 0 for r ≥ z+δ and computing the limit when δ → 0 following similar arguments as in the proof of Lemma 2.7 in [4]. We refer to [5] for the details of the computations.…”

Section: Constant Flux Solutionsmentioning

confidence: 99%

“…Following a common approach in such analysis of asymptotic large-scale properties, the first step is to understand what happens to stationary, i.e., time-independent, solutions to the equations. In a related work [5], we have already extended the analysis from the one-component case in [4], and we have shown that a natural class of coagulation kernels can be easily partitioned into two subclasses depending on whether a stationary solution exists or not. Here we continue the analysis of stationary solutions in the class of kernels which can have a stationary non-equilibrium solution.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we will ignore the fragmentation terms in (1.1), and set α,β = 0. The rationale for this, as discussed in [4], is that in the situations we are interested, the formation of larger particles is energetically favourable and therefore only the coagulation of clusters must be taken into account. This yields the following evolution problem…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, there are only a few papers addressing the problem of the coagulation equations with injection terms like |β|=1 s β δ α,β or η. This issue has been discussed in [4] and we refer to that paper for additional references on earlier related works.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Localization in Stationary Non-equilibrium Solutions for Multicomponent Coagulation Systems

Ferreira

Lukkarinen

Nota

et al. 2021

Commun. Math. Phys.

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We consider the multicomponent Smoluchowski coagulation equation under non-equilibrium conditions induced either by a source term or via a constant flux constraint. We prove that the corresponding stationary non-equilibrium solutions have a universal localization property. More precisely, we show that these solutions asymptotically localize into a direction determined by the source or by a flux constraint: the ratio between monomers of a given type to the total number of monomers in the cluster becomes ever closer to a predetermined ratio as the cluster size is increased. The assumptions on the coagulation kernel are quite general, with isotropic power law bounds. The proof relies on a particular measure concentration estimate and on the control of asymptotic scaling of the solutions which is allowed by previously derived estimates on the mass current observable of the system.

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Section: Stationary Injection Solutionsmentioning

confidence: 84%

Section: Constant Flux Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Localization in Stationary Non-equilibrium Solutions for Multicomponent Coagulation Systems

Ferreira

Lukkarinen

Nota

et al. 2021

Commun. Math. Phys.

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Coagulation Equations for Aerosol Dynamics

Ferreira¹

2021

Trails in Kinetic Theory

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Binary coagulation is an important process in aerosol dynamics by which two particles merge to form a larger one. The distribution of particle sizes over time may be described by the so-called Smoluchowski's coagulation equation. This integrodifferential equation exhibits complex non-local behaviour that strongly depends on the coagulation rate considered. We first discuss well-posedness results for the Smoluchowski's equation for a large class of coagulation kernels as well as the existence and nonexistence of stationary solutions in the presence of a source of small particles. The existence result uses Schauder fixed point theorem, and the nonexistence result relies on a flux formulation of the problem and on power law estimates for the decay of stationary solutions with a constant flux. We then consider a more general setting. We consider that particles may be constituted by different chemicals, which leads to multi-component equations describing the distribution of particle compositions. We obtain explicit solutions in the simplest case where the coagulation kernel is constant by using a generating function. Using an approximation of the solution we observe that the mass localizes along a straight line in the size space for large times and large sizes.

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Long-time asymptotics for coagulation equations with injection that do not have stationary solutions

Cristian,

Ferreira,

Franco

et al. 2023

Arch Rational Mech Anal

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In this paper we study a class of coagulation equations including a source term that injects in the system clusters of size of order one. The coagulation kernel is homogeneous, of homogeneity $$\gamma < 1$$ γ < 1 , such that K(x, y) is approximately $$x^{\gamma + \lambda } y^{-\lambda }$$ x γ + λ y - λ , when x is larger than y. We restrict the analysis to the case $$\gamma + 2 \lambda \ge 1 $$ γ + 2 λ ≥ 1 . In this range of exponents, the transport of mass toward infinity is driven by collisions between particles of different sizes. This is in contrast with the case considered in Ferreira et al. (Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire, 2023), where $$\gamma + 2 \lambda <1$$ γ + 2 λ < 1 . In that case, the transport of mass toward infinity is due to the collision between particles of comparable sizes. In the case $$\gamma +2\lambda \ge 1$$ γ + 2 λ ≥ 1 , the interaction between particles of different sizes leads to an additional transport term in the coagulation equation that approximates the solution of the original coagulation equation with injection for large times. We prove the existence of a class of self-similar solutions for suitable choices of $$\gamma $$ γ and $$\lambda $$ λ for this class of coagulation equations with transport. We prove that for the complementary case such self-similar solutions do not exist.

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Stationary Non-equilibrium Solutions for Coagulation Systems

Cited by 13 publications

References 39 publications

Localization in Stationary Non-equilibrium Solutions for Multicomponent Coagulation Systems

Localization in Stationary Non-equilibrium Solutions for Multicomponent Coagulation Systems

Coagulation Equations for Aerosol Dynamics

Long-time asymptotics for coagulation equations with injection that do not have stationary solutions

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