Léo Meyer
email: leo.meyer@univ-orleans.fr

Magali Ribot

Romain Yvinec

A Lifschitz-Slyozov type model for adipocyte size dynamics : limit from Becker-Döring system and numerical simulation

Biological data show that the size distribution of adipose cells follows a bimodal distribution. In this work, we introduce a Lifshitz-Slyozov type model, based on a transport partial differential equation, for the dynamics of the size distribution of adipose cells. We prove a new convergence result from the related Becker-Döring model, a system composed of several ordinary differential equations, toward mild solutions of the Lifshitz-Slyozov model using distribution tail techniques. Then, this result allows us to propose a new advective-diffusive model, the second-order diffusive Lifshitz-Slyozov model, which is expected to better fit the experimental data. Numerical simulations of the solutions to the diffusive Lifshitz-Slyozov model are performed using a wellbalanced scheme and compared to solutions to the transport model. Those simulations show that both bimodal and unimodal profiles can be reached asymptotically depending on several parameters. We put in evidence that the asymptotic profile for the second-order system does not depend on initial conditions, unlike for the transport Lifshitz-Slyozov model.

Introduction

White adipose tissue is mainly composed of cells, called adipocytes, which store lipids in the body under the form of triglyceride droplets. Experiments in most animals [START_REF] Jo | Hypertrophy and/or Hyperplasia: Dynamics of Adipose Tissue Growth[END_REF][START_REF] Jo | Quantitative Dynamics of Adipose Cells[END_REF][START_REF] Soula | Modelling adipocytes size distribution[END_REF] show that the size distribution of adipocytes follows a striking bimodal distribution with a large peak for small adipocytes around the minimal radius, see Fig. 1. The changes in volume of an adipocyte are governed by two opposite phenomena : lipogenesis, that is to say size increase by triglyceride intake, and lipolysis, that is to say size decrease through the hydrolyze of triglycerides and the excretion of fatty acids. Modeling the dynamics of size evolution of adipocytes is of great interest in order to study metabolic disorders, such as obesity or type 2 diabetes. Correlation between such diseases and the size and metabolism of adipose cells has been well established in the biological literature. Indeed, in [START_REF] Varlamov | Single-Cell Analysis of Insulin-Regulated Fatty Acid Uptake in Adipocytes[END_REF], authors show that the size of an adipose cell has a strong correlation with its insulin sensitivity. As such, large cells are less sensitive, therefore a higher body weight leads to greater risks of type 2 diabetes. This study also shows that adipose tissue are very heterogeneous in terms Figure 1: Example of the distribution of adipocytes in a rat biopsy. Credits : H. Soula of size of cells. Those findings have also been described in [START_REF] Lee | Developmental and Functional Heterogeneity of White Adipocytes within a Single Fat Depot[END_REF], where the authors show that the adipose tissue is composed of cells that are different both molecularly and phenotypically. Some computational models have also been used to provide insights into the adipose tissue physiology. In [START_REF] Kim | A Computational Model of Adipose Tissue Metabolism: Evidence for Intracellular Compartmentation and Differential Activation of Lipases[END_REF], the authors use an ODE model to investigate the role of lipases in the biochemistry of lipids. They are able to show that determining the active metabolic subdomain in the tissue is the key for accurate simulations, as well as the different activation rates of lipases for diglyceride and triglyceride breakdown. The rate of lipid turnover has also been studied in [START_REF] Arner | Adipose Lipid Turnover and Long-Term Changes in Body Weight[END_REF], where a decrease of the lipid release rate is correlated with the age of the individual. Finally, there are strong links between the adipose tissue and its extracellular matrix, and in case of obesity, one may observe tissue fibrosis such as described in [START_REF] Divoux | Architecture and the Extracellular Matrix: The Still Unappreciated Components of the Adipose Tissue[END_REF]. In [START_REF] Peurichard | Simple mechanical cues could explain adipose tissue morphology[END_REF][START_REF] Peurichard | Extra-cellular matrix rigidity may dictate the fate of injury outcome[END_REF], adipose tissue is modeled by a 2D agent-based model which takes into account the mechanical interactions between adipose cells and fibers forming the extra-cellular matrix. The authors study the spatial distribution of adipocytes under the form of lobules or in the case of tissue regeneration.

However, only a few mathematical models have been proposed in order to describe the size dynamics of adipose cells and no previous work has tackled a mechanistic understanding of the bimodal feature of adipocyte size distribution.

A first model has been derived by Jo, Periwal et al. in [START_REF] Jo | Mathematical Models of Adipose Tissue Dynamics[END_REF][START_REF] Jo | Quantitative Dynamics of Adipose Cells[END_REF] using a PDE for the adipose cell growth with a phenomenological cell growth rate. They are able to recover the bimodal feature of distributions as well as to perform some curve fitting on biological data. In [START_REF] Soula | Modelling adipocytes size distribution[END_REF] and later in [START_REF] Soula | Model of adipose tissue cellularity dynamics during food restriction[END_REF], the authors describe the velocity of size change of adipocytes by biological considerations for lipogenesis and lipolysis, leading thus to some transport PDE models. They obtain bimodal distributions by using stochastic variations of the parameters. In [START_REF] Gilleron | Modeling and analysis of adipocytes dynamic with a differentiation process[END_REF], the authors perform the analysis and numerical simulations for a size-structured model describing the evolution of a set of adipocytes, including the creation of new adipocytes through differentiation processes from mesenchymal cells and preadipocytes, and accounting for a size velocity inversely proportional to the total surface of adipocytes. Finally, in [START_REF] Prana | Modeling the Effect of High Calorie Diet on the Interplay between Adipose Tissue, Inflammation, and Diabetes[END_REF], authors use an ODE model to investigate interplay and feedback loop between inflammatory response of bigger adipose cells and the immune system, which may lead to type 2 diabetes. The size of adipocytes is updated at each time step according to some probability of swelling and by a factor depending on the surplus of calories intake. However they do not concern themselves with the size distribution but with the whole tissue inflammation and the body wheight dynamic.

Transport equation for adipocyte size evolution

Following the work in [START_REF] Soula | Modelling adipocytes size distribution[END_REF], we first describe intake and release of lipids trough the cellular membrane, thus describing how the size of an adipose cell evolves. This will in turn allow us to build a model based on continuity equations.

Our first assumption will be the correlation between the amount of storage in an adipose cell and its radius. Cells shall be considered as spheres of a certain radius r, and the amount of lipids in the cell is denoted by x. Let us denote by r(x) the radius of a cell containing x amount of lipids, by V 0 the volume of an empty cell and by V lipids the molar volume of triglycerides.

We express the total volume of the cell in two different ways and we obtain the following relation: 3 , which leads to :

V lipids x + V 0 = 4 3 πr(x)
r(x) = 3 4π (V lipids x + V 0 ) 1 3
.

(

) 1
We also denote by L the amount of external lipids in the medium.

Henceforth, x will be considered as the size of our cell. Its variation dx dt depends on two flows : the intake of lipids by the cell from the medium and the release of lipids in the medium. As those two flows go through the membrane of the cell, they should be surface limited. We will also consider fast diffusion of the lipids in the medium so that the amount of lipids available for each cell is the same.

The intake term is a product of three factors :

• a term for a surface limited flow αr(x) 2 , where the constant α is the rate of this flow ;

• a Hill-like term with a radius cutoff ρ to describe resistance toward indefinite intake of lipids ρ n r(x) n + ρ n ;

• a term that accounts for the available amount of lipids in the medium, in the form of a Michaelis-Menten term L L + κ with a saturation effect when the amount of external lipids L is large, with κ giving the order of magnitude of the threshold.

The release is a product of two terms :

• a term with a basal level of release β and a surface limited flow γr(x) 2 , where the constant γ is the release equivalent of the constant α ;

• a Michaelis-Menten term for the available amount of lipids in the cell

x x + χ
, where χ is the equivalent of κ for the release.

The variation dx dt can therefore be expressed as the difference between intake and release as :

dx dt = αr(x) 2 ρ n r(x) n + ρ n L L + κ intake -(β + γr(x) 2 ) x x + χ release . (2)
We can now build a transport equation for the distribution f (t, x) of adipose cells by amount of lipids x ≥ 0 at time t. According to Eq. ( 2), the transport velocity will be given by :

v(x, L) = a(x) L L + κ -b(x), (3)
where

a(x) = αr(x) 2 ρ n r(x) n + ρ n (4) and b(x) = (β + γr(x) 2 ) x x + χ . (5)
Consequently, the function f satisfies the following transport equation :

∂ t f (t, x) + ∂ x (v(x, L)f (t, x)) = 0, x ≥ 0, t > 0. (6)
We now need to describe the behaviour of the available amount of lipids in the medium L. As per our assumption, the total quantity of lipids in our system, denoted by λ, should be constant. There are two types of lipids in the system : the ones contained in the cells, and the lipids in the medium and we therefore have the following equality :

L(t) + R+ xf (t, x)dx = λ. ( 7
)
Another hypothesis is that the number of adipocytes does not change in time. Thus, in regards to boundary conditions, we want to preserve the total population number and therefore we impose that :

R+ f (t, x)dx = R+ f 0 (x)dx = m for all t > 0. (8)
This leads for Eq. ( 6) to boundary condition

(v(x, L(t))f (t, x)) x=0 = 0 , for all t > 0 . (9)
Notice that by Eqs. ( 4)-( 5), we have b(0) = 0 and a(0) > 0. Hence, the boundary conditions ( 9) is equivalent to the Dirichlet boundary condition :

f (t, x) x=0 = 0 for all t > 0. (10)
To sum up, the transport model for adipose cells with initial conditions (f 0 , L 0 ), that will be called first-order Lifshitz-Slyozov model in the following, reads as:

               ∂ t f (t, x) + ∂ x (v(x, L(t))f (t, x)) = 0, L(t) + R+ xf (t, x)dx = λ, (v(x, L(t))f (t, x)) x=0 = 0, f (0, x) = f 0 (x) and L(0) = L 0 . ( 11a
) (11b) (11c) (11d)

New models for adipose tissue dynamics

In this subsection, we present the various models under consideration in this article. Starting from the description of lipogenesis and lipolysis as done in Eq. ( 4) and ( 5) following [START_REF] Soula | Modelling adipocytes size distribution[END_REF], we build sizestructured PDE model following the framework of Becker-Döring system [START_REF] Becker | Kinetische behandlung der keimbildung in übersättigten dämpfen[END_REF] and Lifshitz-Slyozov equations [START_REF] Ilya | The kinetics of precipitation from supersaturated solid solutions[END_REF] initially derived for polymerization.

The aim of this model is to reproduce the adipocyte size distributions observed experimentally and their bimodal structure. However, the transport equation ( 6) possesses asymptotic solutions as a linear combination of Dirac masses centered on the zeros of the asymptotic speed. To recover the bimodality, we are therefore lead to introduce a diffusion term in the equation : we can either add a diffusion term with a constant rate with no real biological meaning, or we can compute a time and space dependent diffusion term coming from the discrete nature behind the Lifshitz-Slyozov formalism [START_REF] Hariz | A Modified Version of the Lifshitz-Slyozov Model[END_REF][START_REF] Vasseur | The Beker-Döring System and Its Lifshitz-Slyozov Limit[END_REF]. For that purpose, we come back to a Becker-Döring system of ODEs giving the evolution with respect to time of the number of adipocytes with discrete sizes and from this, we derive a second order Lifshitz-Slyozov equation with a diffusion term.

Therefore, in the following, we will consider three different models for the size distribution of an adipocyte population, namely • the ODE system [START_REF] Gilleron | Modeling and analysis of adipocytes dynamic with a differentiation process[END_REF] with discrete sizes, a variant of the Becker-Döring model.

• the previously published transport equation [START_REF] Divoux | Architecture and the Extracellular Matrix: The Still Unappreciated Components of the Adipose Tissue[END_REF], also called first order Lifshitz-Slyozov equation,

• the transport-diffusion equation ( 17), the second order Lifshitz-Slyozov equation.

In all three models, the lipogenesis and lipolysis rate will be given by Eq. ( 4) and ( 5), respectively. Note that we impose in all three models two conservation laws: (i) the conservation of the total amount of lipids and (ii) the conservation of the total number of adipocytes. Therefore, all these models have a constant population number and are coupled with a lipid conservation equation which ensures that the sum of the lipids in the external medium and the lipids inside the cells is constant.

A brief insight in Becker-Döring and Lifshitz-Slyozov equations

Becker-Döring equations have been introduced in [START_REF] Becker | Kinetische behandlung der keimbildung in übersättigten dämpfen[END_REF] to model polymers undergoing aggregation and fragmentation. The Lifshitz-Slyozov model was introduced in [START_REF] Ilya | The kinetics of precipitation from supersaturated solid solutions[END_REF] and first used for nucleation in super-saturated solid solutions and polymerisation processes. A rigourous treatment of the mathematical properties of the Becker-Döring equations was given by [START_REF] Ball | The Becker-Döring cluster equations: basic properties and asymptotic behaviour of solutions[END_REF]. The relation between Becker-Döring equations and Lifshitz-Slyozov model goes back to [START_REF] Penrose | Growth of clusters in a first-order phase transition[END_REF]. For a detailed review of both models, see [START_REF] Hingant | Deterministic and Stochastic Becker-Döring Equations: Past and Recent Mathematical Developments[END_REF] and references therein.

Let us explain briefly the idea of these models for polymers. We denote by c i , the amount of polymers containing i monomers for i ∈ N * and hence c 1 stands for the amount of monomers. A polymer of size i denoted p i can gain one monomer and grow to p i+1 with rate a i or lose one monomer and shrink to p i-1 with rate b i . We may write as a system of ODEs for the time evolution of the number of polymers c i , one for each size i. Furthermore, the total amount of monomers, i.e free monomers and monomers within polymers, is assumed constant, which leads to a conservation equation. Stationary solutions of the Becker-Döring equations can be easily computed, and long time behavior has been characterized by [START_REF] Ball | The Becker-Döring cluster equations: basic properties and asymptotic behaviour of solutions[END_REF].

Another possibility for the modeling of polymerisation-fragmentation processes is to describe continuously the size of polymers through a variable x ∈ R. The distribution of polymers of size x at time t is therefore denoted by f (t, x) and the quantity of monomers at time t is denoted by L(t).

The distribution is classically transported as in Eq.( 6) with speed v(x, L) = a(x)L(t) -b(x) where a(x) is the rate of polymerisation for size x and b(x) is the rate of depolymerisation for size x. As previously, the total amount of monomers is conserved. Depending on the sign of a(0)L(t) -b(0), boundary conditions should be provided for the system, see [START_REF] Deschamps | Quasi steady state approximation of the small clusters in Becker-Döring equations leads to boundary conditions in the Lifshitz-Slyozov limit[END_REF] for example.

After an adapted rescaling, it has been shown in various papers [START_REF] Conlon | A Non-Local Problem for the Fokker-Planck Equation Related to the Becker-Döring Model[END_REF][START_REF] Deschamps | Quasi steady state approximation of the small clusters in Becker-Döring equations leads to boundary conditions in the Lifshitz-Slyozov limit[END_REF][START_REF] Laurençot | From the becker-döring to the lifshitz-slyozov-wagner equations[END_REF][START_REF] Niethammer | Macroscopic limits of the Becker-Döring equations[END_REF][START_REF] Schlichting | Macroscopic limit of the Becker-Döring equation via gradient flows[END_REF][START_REF] Vasseur | The Beker-Döring System and Its Lifshitz-Slyozov Limit[END_REF] that the solutions to Becker-Döring system tend to the solutions to Lifshitz-Slyozov model. Formally, the limit up to second order can be considered and gives rise to an advection-diffusion equation as computed in [START_REF] Hariz | A Modified Version of the Lifshitz-Slyozov Model[END_REF][START_REF] Vasseur | The Beker-Döring System and Its Lifshitz-Slyozov Limit[END_REF]. Existence of solutions is widely known for both models, see the seminal paper [START_REF] Ball | The Becker-Döring cluster equations: basic properties and asymptotic behaviour of solutions[END_REF] for the Becker-Döring model and [START_REF] Calvo | The Initial-Boundary Value Problem for the Lifshitz-Slyozov Equation with Non-Smooth Rates at the Boundary[END_REF][START_REF] Collet | On solutions of the Lifshitz-Slyozov model[END_REF] for the Lifshitz-Slyozov model..

Remark that Becker-Döring and Lifshitz-Slyozov equations have already been used in various contexts, for example modeling of biological phenomena, such as prions [START_REF] Doumic | Scaling limit of a discrete prion dynamics model[END_REF][START_REF] Laurençot | Well-posedness for a model of prion proliferation dynamics[END_REF][START_REF] Simonett | On the solvability of a mathematical model for prion proliferation[END_REF], [START_REF] Prigent | An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation[END_REF], [START_REF] Greer | A mathematical analysis of the dynamics of prion proliferation[END_REF] or modeling in oceanography, see [START_REF] Wurl | Formation and global distribution of sea-surface microlayers[END_REF][START_REF] Jackson | Aggregation in the marine environment[END_REF].

A Becker-Döring model for adipose cells

Now, let us explain how we adapt this formalism to derive new models for adipocyte size dynamics. The purpose of this construction is to investigate the classical convergence theorems from Becker-Döring to Lifshitz-Slyozov and deduce the form of a diffusion term to add in our model.

We mention the main differences with the classical Becker-Döring and Lifshitz-Slyozov systems for polymerisation. First, velocity (2), arising from biological considerations, possesses three zeros for a well-chosen range of parameters which leads to bimodal asymptotic distributions, whereas classical choices for a and b are constant or power laws of x, which yields the existence of a single positive root. See also [START_REF] Calvez | Prion dynamics with size dependency-strain phenomena[END_REF] for a polymerisation-fragmentation model without diffusion giving rise to bimodal asymptotics. Second, in our model, external lipids L cannot be assimilated to monomers c 1 and the conservation law [START_REF] José | Uniform moment propagation for the Becker-Döring equations[END_REF] is therefore not the same as in the usual polymerisation models. Moreover, the saturation term L L + κ is not common in polymerisation modeling. Finally, our model conserves the total population number due to the boundary condition (11c), which adds an additional conservation law compared to the classical Becker-Döring model.

We shall now consider that an adipose cell is a bundle of smaller vesicles of typical size Λ. Hence the size of a cell can be defined by the number of vesicles it contains. We denote by c i the number of cells of size i and by l the number of vesicles in the medium. A cell will aggregate a new vesicle with speed a i M (l),where M (l) = lΛ lΛ+κ following Eq. ( 2), and loose a vesicle at speed b i , following this reaction :

Λ + iΛ aiM (l) ----⇀ ↽---- bi+1 (i + 1)Λ.
Following a rescaling procedure, which is described in details in Annex 7, we define c ε = (c ε i ) i≥0 and J ε i (c ε , L ε ) the flow of the previous reaction given by :

J ε i (c ε , L ε ) = a ε i L ε L ε + κ c ε i -b ε i+1 c ε i+1 , i ≥ 0, (12)
where a ε i (resp. b ε i ) are discrete counterpart of the continous function a defined at Eq.(4) (resp. b defined at Eq.( 5)), see Sec.2 for more details.

Similarly as before, see Eq. ( 7), L ε will satisfy an equation accounting for conservation of the amount of lipids and we get the following ODE system :

                       dc ε i dt = 1 ε (J ε i-1 (c ε , L ε ) -J ε i (c ε , L ε )), ∀i ≥ 1, dc ε 0 dt = - 1 ε J ε 0 (c ε , L ε ), L ε (t) + ∞ i=0 iε 2 c ε i (t) = λ, ∀t ≥ 0, L ε (0) = L ε,0 , c ε i (0) = c ε,0 i , ∀i ≥ 1, (13a) (13b) (13c) (13d)
which is similar to Becker-Döring equations except for the definition of the flux J ε i (saturing fluxes of monomers), and the minimal size is 0 and not 1. Observe also that there is no 'boundary' flux, thus the quantity

m = ε i≥0 c ε i (t) is constant in time. ( 14
)
This is the discrete analogous to the previous conservation (8) of the zeroth order moment of f . A solution to the previous system exists according to Theorem 2.1 recalled in Sec.2. Now let us define the following step functions depending on both time and space :

f ε (t, x) = i≥0 1 Γ ε i (x)c ε i (t),
where

Γ ε i = [(i -1 2 )ε, (i + 1
2 )ε[, and (c ε i ) i≥0 is a solution to [START_REF] Gilleron | Modeling and analysis of adipocytes dynamic with a differentiation process[END_REF]. Convergence of function f ε when ε → 0 towards a solution f of the Lifshitz-Slyozov equation ( 11) is a classical result, see Theorem 2.3 recalled in Sec.2. In the present work, we prove that a similar convergence result hold in a stronger topology, and with a control of the speed of convergence, of order at least ε.

To that, we introduce the tail distributions :

F (t, x) = ∞ x f (t, y)dy, F ε (t, x) = ∞ x f ε (t, y)dy.
The main analytical result of this article is the following theorem, whose more rigorous statement will be specified later in Theorem 3.1:

Theorem (Convergence of tails of distributions). Under some hypotheses detailed in Sec.2 and 3, there exists some constant K > 0 and some time T > 0 such that for all t ∈ (0, T ) and for ε small enough :

|L ε (t) -L(t)| + R+ |F ε (t, x) -F (t, x)|dx ≤ εK .
This result provides a new approach for looking into convergence from Becker-Döring to Lifshitz-Slyozov. Contrary to more classical results where convergence towards a weak solution is achieved using Ascoli-Arzela's Theorem, this theorem yields convergence towards mild solutions and gives a bound of order ε on the speed of this convergence.

A second order Lifshitz-Slyozov model

Another goal in this article is to derive a new model with a diffusive term from Becker-Döring system [START_REF] Gilleron | Modeling and analysis of adipocytes dynamic with a differentiation process[END_REF]. One can see this diffusive term as a second order term emerging from the convergence theorem 3.1. There are various ways to yield this term, see for example [START_REF] Vasseur | The Beker-Döring System and Its Lifshitz-Slyozov Limit[END_REF], [START_REF] Schlichting | Macroscopic limit of the Becker-Döring equation via gradient flows[END_REF], [START_REF] Deschamps | Quasi steady state approximation of the small clusters in Becker-Döring equations leads to boundary conditions in the Lifshitz-Slyozov limit[END_REF]. The derivation of the diffusive term will be detailed in Section 4, but we present the model here for the sake of completeness.

The so-called second order Lifshitz-Slyozov model therefore takes the form of a transport-diffusion equation, with a diffusive term which depends both on x and L(t), i.e. :

∂ t g + ∂ x (vg) = ε 2 ∂ 2 x (dg), ∀x ≥ 0,
where

d : (x, L) ∈ R + × R + → d(x, L) = a(x) L L + κ + b(x). (15)
We need to complement this PDE with adapted boundary conditions. Since we want the conservation of the zeroth order moment denoted by R+ g(t, x)dx = m, we need to impose the following null-flux boundary condition :

(-vg + ε 2 ∂ x (dg)) x=0 = 0. (16)
Therefore, we consider the following system, which consists of the previous PDE and boundary conditions, complemented by previous constraint [START_REF] José | Uniform moment propagation for the Becker-Döring equations[END_REF] and initial conditions for g and L :

                   ∂ t g + ∂ x (vg) = ε 2 ∂ 2 x (dg), L(t) + R+ xg(t, x)dx = λ, -vg + ε 2 ∂ x (dg) x=0 = 0, g(0, x) = g 0 (x) and L(0) = L 0 . (17a) (17b) (17c) (17d)
We provide interesting numerical evidence of stationary solutions of the advection-diffusion model [START_REF] Hingant | Deterministic and Stochastic Becker-Döring Equations: Past and Recent Mathematical Developments[END_REF] following a bimodal distribution. The numerical simulations are performed using a well-balanced scheme developped in [START_REF] Goudon | Fokker-Planck Approach of Ostwald Ripening: Simulation of a Modified Lifshitz-Slyozov-Wagner System with a Diffusive Correction[END_REF]. We also demonstrate that to observe a bimodal asymptotics, parameters should be taken into an adapted parameter range.

Outline of the article

In Section 2, we will give some preliminary results on the existence of solutions to systems [START_REF] Gilleron | Modeling and analysis of adipocytes dynamic with a differentiation process[END_REF] and [START_REF] Divoux | Architecture and the Extracellular Matrix: The Still Unappreciated Components of the Adipose Tissue[END_REF]. In Section 3, we will show the convergence theorem thanks to the tail of distributions technique. Then, in Section 4, we derive formally the second-order Lifshitz-Slyozov model, that is to say system (17) and we give the expression for its stationary solutions. In Section 5, we display some numerical results and we show that bimodality of the stationary solution can be observed in well-chosen parameter range. Finally, we discuss our results in Section 6. We also recall the rescaling procedure of (13) in annex 7.

Preliminary results

In this section, we give the main already-known results of existence of solutions to systems ( 13) and [START_REF] Divoux | Architecture and the Extracellular Matrix: The Still Unappreciated Components of the Adipose Tissue[END_REF] and convergence of solutions to system (13) towards [START_REF] Divoux | Architecture and the Extracellular Matrix: The Still Unappreciated Components of the Adipose Tissue[END_REF]. Proofs have been easily adapted to our framework.

Existence results on Becker -Döring system

We consider first the Becker -Döring system (13) for fixed ε.

We assume that there exist some strictly positive constants A, B, C a , C b , K a , K b and δ such that for all i ≥ 0 :

a ε i ≤ C a and b ε i ≤ C b iε, (H'1)
|a ε i -a ε i+1 | ≤ K a ε and |b ε i -b ε i+1 | ≤ K b ε. (H'2)
We define the state space for Eq. ( 13) by

X := {x = (x i ) i≥0 ∈ R N + : +∞ i=0 ix i < ∞} , endorsed with the norm ∥x∥ X = +∞ i=0 i|x i |. We denote x ≥ 0 if x i ≥ 0 for all i ≥ 0, and X + := {x ∈ X : x ≥ 0}.
We give the following definition of solution to Eq. ( 13):

Definition 2.1. Let T > 0 and ε > 0. A solution (c ε , L ε ) of (13) in [0, T ) is a couple of a function L ε : [0, T ) → R and a sequence of functions c ε = (c ε i ) i≥0 , c ε i : [0, T ) → Xsuch that : (i) For all t ∈ [0, T ), L ε (t) ≥ 0 and c ε (t) ∈ X + , (ii) For all i ≥ 1, c ε i : [0, T ) → R is continuous and sup t∈[0,T ) ∥c ε (t)∥ X < +∞, (iii) L ε : [0, T ) → R is continuous and sup t∈[0,T [ |L ε (t)| < +∞, (iv) For all t ∈ [0, T ), t 0 +∞ i=0 a ε i c ε i (s)ds < ∞ and t 0 +∞ i=0 b ε i c ε i (s)ds < ∞, (v)
For all t ∈ [0, T ), for all i ≥ 1 :

c ε i (t) = c ε,0 i + 1 ε t 0 [J ε i-1 (c ε (s), L ε (s)) -J ε i (c ε (s), L ε (s))]ds, c ε 0 (t) = c ε,0 0 - 1 ε t 0 J ε 0 (c ε (s), L ε (s))ds, L ε (t) = L ε,0 -ε t 0 +∞ i=0 J ε i (c ε (s), L ε (s))ds
Well-posedness of solutions to ( 13) as defined at Def.2.1 can be shown by finite dimensional approximation, using the method developped in [START_REF] Ball | The Becker-Döring cluster equations: basic properties and asymptotic behaviour of solutions[END_REF] :

Theorem 2.1. Let T > 0 and ε > 0. Let L ε,0 ∈ R + et c ε,0 ∈ X + such that L ε,0 + +∞ i=0 iε 2 c ε,0 i = λ < ∞.
Assume that (H'1), (H'2) hold true. Then there exists a unique solution (c ε , L ε ) to Becker -Döring system (13) in the sense of Def.2.1 which satisfies initial conditions c ε (0) = c ε,0 and L ε (0) = L ε,0 .

The uniqueness and conservation properties of the solution are obtained using the following proposition that will be needed later on, see Sec.4. In particular, the following proposition states that any solution of the Becker -Döring system (13) preserves the first two moments for all times, and provides the starting point to compute any admissible moments for the solution of the Becker -Döring system. In [START_REF] Ball | The Becker-Döring cluster equations: basic properties and asymptotic behaviour of solutions[END_REF], we can find the following Theorem 2.5 that we reproduce here for the reader's convenience : Proposition 2.1. Let (ϕ i ) i≥0 be a given sequence. Let (c ε , L ε ) be the solution of (13) on [0, T ), 0 < T ≤ +∞.

Assume that for all

0 ≤ t 1 < t 2 < T , t2 t1 ∞ i=0 |ϕ i+1 -ϕ i |a ε i c ε i (t)dt < ∞
and that either of the following holds :

(a)

ϕ i = O(i) and t2 t1 ∞ i=0 |ϕ i+1 -ϕ i |b ε i+1 c ε i+1 (t)dt < ∞ or (b) ∞ i=0 ϕ i c ε i (t k ) < ∞, for k = 1, 2 and ϕ i+1 ≥ ϕ i ≥ 0 for i large enough.
Then :

∞ i=0 ϕ i c ε i (t 2 ) - ∞ i=0 ϕ i c ε i (t 1 )+ t2 t1 ∞ i=0 ϕ i+1 -ϕ i ε b ε i+1 c ε i+1 (t)dt = t2 t1 ∞ i=0 ϕ i+1 -ϕ i ε a ε i L ε (t) L ε (t) + κ c ε i (t)dt.

Lifschitz-Slyozov system and classical convergence result

Even though we have precise forms for the intake and release functions, for the sake of generality we make the following assumptions on functions a and b occurring in Eq. ( 3) :

a, b ∈ C 1 (R + , R + ), (H1)
a(0) > 0 and sup

x∈R+ |a(x)| = C a , (H2a) |b
(x)| ≤ C b x for all x ∈ R + and lim R→∞ sup x≥R b(x) x = 0, (H2b)
sup x∈R+ |a ′ (x)| = K a and sup x∈R+ |b ′ (x)| = K b , (H3) with C a , C b , K a , K b > 0.
We first define measured-valued solutions to the Lifschitz-Slyozov system [START_REF] Divoux | Architecture and the Extracellular Matrix: The Still Unappreciated Components of the Adipose Tissue[END_REF], following [START_REF] Collet | On solutions of the Lifshitz-Slyozov model[END_REF] Definition 2.2. Given an initial condition

(f 0 , L 0 ) ∈ C 0 (R + )∩L 1 (R + , (1+x)dx)×R + , a measured- valued solution to system (11) is composed of two functions f ∈ C(0, T ; M 1 (0, ∞) -weak - * )
and L ∈ C(0, T ) such that for all 0 < t < T and for all φ ∈ C 1 ([0, T ] × R + ) the following relations hold:

T 0 R+ (∂ t φ(t, x) + v(x, L(t))∂ x φ(t, x))f (t, dx) + R+ φ(0, x)f 0 (x)dx = 0, L(t) + R+ xf (t, dx) = λ.
Now, let us state the convergence of solutions to Becker-Döring system towards solutions to Lifschitz-Slyozov system. In order to compare solutions to Becker-Döring system to solutions to Lifschitz-Slyozov system, we need to define the following piecewise constant functions. Let

Γ ε i = [(i -1 2 )ε, (i + 1
2 )ε) and c ε i be solutions to ( 13), then we define

                   f ε (t, x) = i≥0 1 Γ ε i (x)c ε i (t) , a ε (x) = i≥0 1 Γ ε i (x)a ε i , b ε (x) = i≥1 1 Γ ε i (x)b ε i , (18a) (18b) (18c)
where we assume that :

a ε i = a(iε) and b ε i = b(iε), for all i ≥ 0 and ε > 0. ( H4
)
Given our definitions in Eq. ( 18), from Proposition 2.1 and with ϕ i = Γ ε i ϕ(x)dx, we deduce the following proposition, that is the starting point to study the convergence of the solution of the Becker-Döring system (13) towards solution of the Lifshitz-Slyozov equation [START_REF] Divoux | Architecture and the Extracellular Matrix: The Still Unappreciated Components of the Adipose Tissue[END_REF].

Proposition 2.2. Let ϕ ∈ L ∞ (R + ).
Then for every t ≥ 0, we have the following equality :

∞ 0 ϕ(x)(f ε (t, x) -f ε (0, x))dx = t 0 ∞ 0 (∆ ε ϕ(x)a ε (x) L ε (t) L ε (t) + κ -∆ -ε ϕ(x)b ε (x))f ε (t, x)dxdt,
where

∆ ε ϕ(x) = ϕ(x + ε) -ϕ(x) ε . (19)
Finally, we obtain the following convergence theorem from the Becker-Döring equations to the Lifshitz-Slyozov equations, as in [START_REF] Vasseur | The Beker-Döring System and Its Lifshitz-Slyozov Limit[END_REF]: Theorem 2.2. Consider an initial condition (L ε,0 , (c ε,0 i ) i≥0 ) and the corresponding solution (L ε , (c ε i ) i≥0 ) in the sense of Definition 2.1. We assume that there exists a constant K > 0 and 0 < s ≤ 1 such that :

• L ε,0 + ε 2 i≥0 ic ε,0 i = λ, • ε i≥0 c ε,0 i < K, • ε i≥0 (iε) 1+s c ε,0 i < K.
We also assume hypotheses H1 -H4 to hold. Then there exists a sequence ε n and a solution (f, L) to [START_REF] Divoux | Architecture and the Extracellular Matrix: The Still Unappreciated Components of the Adipose Tissue[END_REF] in the sense of Def. 2.2 such that :

f εn → f , xf εn → xf in C 0 ([0, +∞[; M 1 (0, +∞) -weak - * ), L εn → L uniformly in C 0 ([0, T ]).
Now, let us consider the existence of mild solutions to [START_REF] Divoux | Architecture and the Extracellular Matrix: The Still Unappreciated Components of the Adipose Tissue[END_REF]. For that purpose, we first define the characteristic curves.

Assume L ∈ C 0 (R + ) to be given. The characteristic curves associated to [START_REF] Divoux | Architecture and the Extracellular Matrix: The Still Unappreciated Components of the Adipose Tissue[END_REF] are solutions to :

∂ s X(s; t, x) = v(X(s; t, x), L(s)), X(t; t, x) = x.
Since v is C 1 in both x and L, the characteristics are uniquely defined and form an ordered family. We denote I t,x their maximal time interval and by X c (t) = X(t; 0, 0) the characteristic curve that is equal to 0 at time 0. Then, a mild solution to system [START_REF] Divoux | Architecture and the Extracellular Matrix: The Still Unappreciated Components of the Adipose Tissue[END_REF] is given by the following definition :

Definition 2.3. Given a smooth initial condition f 0 and L ∈ C 0 (R + ), a mild solution of      ∂ t f + ∂ x (v(x, L(t))f ) = 0, (v(x, L(t))f (t, x)) x=0 = 0, f (0, x) = f 0 (x),
is given by :

f (t, x) = f 0 (X(0; t, x)) exp - t 0 ∂ x v(X(s; t, x), L(s))ds 1 (Xc(t),∞) (x).
A couple (f, L) is said to be a solution of (11) iff f is a mild solution associated to L and L : R + → R + solves L(t) + R+ xf (t, x)dx = λ for all t ≥ 0. Remark. Since we impose null-flux boundary conditions on this system : v(x, L(t))f (t, x)| x=0 = 0, there is no term involving "incoming characteristics" 1 (0,Xc(t)) (x).

We follow the proofs in [START_REF] Collet | On solutions of the Lifshitz-Slyozov model[END_REF] and [START_REF] Calvo | The Initial-Boundary Value Problem for the Lifshitz-Slyozov Equation with Non-Smooth Rates at the Boundary[END_REF] and we obtain in a straightforward way the expected existence and uniqueness result : Theorem 2.3. Given an initial condition (f 0 , L 0 ) ∈ C 0 (R + )∩L 1 (R + , (1+x)dx)×R + and assuming hypotheses (H1)-(H3), Lifschitz-Slyozov system (11) has a unique solution on the interval [0, T ] in the sense of Def.2.3.

Note that the mild solution given by Theorem 2.3 is also a weak solution in the sense of Definition 2.2, see [START_REF] Calvo | The Initial-Boundary Value Problem for the Lifshitz-Slyozov Equation with Non-Smooth Rates at the Boundary[END_REF], and under hypotheses (H1)-(H3) both definitions coincide.

A new convergence result from Becker-Döring to Lifschitz-Slyozov equations

In this part of our work we introduce a different way to see the convergence from the Becker-Döring equations to the Lifschitz-Slyozov equations. Using tail distributions allows to reduce the non linearity of our system by pulling the speed of advection outside of the space derivative. Tail distributions were also found to be useful to obtain a quasi comparison principle in [START_REF] José | Uniform moment propagation for the Becker-Döring equations[END_REF] and to obtain refined uniqueness properties in [START_REF] Laurençot | Weak Solutions to the Lifshitz-Slyozov-Wagner Equation[END_REF][START_REF] Calvo | The Initial-Boundary Value Problem for the Lifshitz-Slyozov Equation with Non-Smooth Rates at the Boundary[END_REF]. The main idea is to use results on the tail of the distributions to show convergence. Finally, we note that out result uses the fact that a solution to system [START_REF] Divoux | Architecture and the Extracellular Matrix: The Still Unappreciated Components of the Adipose Tissue[END_REF] exists while the previous result also shows existence of solution of [START_REF] Divoux | Architecture and the Extracellular Matrix: The Still Unappreciated Components of the Adipose Tissue[END_REF], by showing a convergence to a measure valued function which turns out to be a solution of [START_REF] Divoux | Architecture and the Extracellular Matrix: The Still Unappreciated Components of the Adipose Tissue[END_REF].

Let (f ε , L ε ) be the solution of the Becker-Döring ODE system (13) and Eq. ( 18), and let (f, L) the mild solution of Lifshitz-Slyozov equations [START_REF] Divoux | Architecture and the Extracellular Matrix: The Still Unappreciated Components of the Adipose Tissue[END_REF]. We recall the tail distribution definition,

F (t, x) = ∞ x f (t, y)dy, F ε (t, x) = ∞ x f ε (t, y)dy , (20)
and introduce their difference

E(t, x) = F ε (t, x) -F (t, x) . (21)
We introduce the following additional hypotheses to use in our main theorem :

sup x∈R+ |a ′′ (x)| < ∞ and sup x∈R+ |b ′′ (x)| < ∞, ( H5
) i≥0 |c ε,0 i+1 -c ε,0 i | < +∞, (H6)
ε i≥0 i|c ε,0 i+1 -c ε,0 i | < +∞. ( H7
)
There exists some constant L > 0, such that inf

ε>0 L ε,0 ≥ L. ( H8
)
There exists some constant K > 0 such that sup

ε>0 c ε,0 0 < K. ( H9
)
We now state our main theorem.

Theorem 3.1. Let T > 0. Suppose that there exists some constant C init > 0 such that for all ε > 0, R+ |E(0, x)|dx ≤ εC init . Also assume that hypotheses (H1)-(H9) hold true. Then there exists some constant C(T ) > 0 such that for ε > 0 small enough and for all t ∈ [0, T ] :

|L ε (t) -L(t)| + R+ |E(t, x)|dx ≤ εC(T ).
The proof proceeds as follows. Taking inspiration from [START_REF] Laurençot | Weak Solutions to the Lifshitz-Slyozov-Wagner Equation[END_REF][START_REF] Calvo | The Initial-Boundary Value Problem for the Lifshitz-Slyozov Equation with Non-Smooth Rates at the Boundary[END_REF], we first note that owing to the total population number conservation, the lipid terms can be controlled by the tail, |L ε (t) -L(t)| ≤ R+ |E(t, x)|dx. The control on the tail relies on a Grönwall's lemma argument. For that purpose, we derive the equation followed by F ε (t, x) (Lemma 3.5). We point out that the case x < ε/2 has to be treated separately due to remaining boundary terms. This allows us to give a first estimate on the integral R+ |E(t, x)|dx. We then make use of the mild solution formulation to derive the partial differential equation followed by F and in turn the one followed by E (Lemma 3.6). The proof follows by bounding the terms in the estimate on R+ |E(t, x)|dx, and in particular we show that F ε (t, x) satisfies the same equation as F up to an order ε (Lemma 3.7). To this end, the key argument relies on refined estimates of the difference between the first order derivative of F ε (t, x) and its discrete analog. This estimate needs uniform control on the solutions c ε i of the Becker-Döring system and their increments c ε i+1 -c ε i (Subsection 3.1, Lemmas 3.1 to 3.4), which is new, up to our knowledge.

Hypotheses (H1) -(H4) are classical in the study of our model. However, other assumptions are less common but arise naturally from the result. Contrary to the classical convergence result, we work with mild solutions of the Lifshitz-Slyozov system. Hence, we need proper bounds on second order terms. We shall see in Section 4 that those terms lead us to the second order Lifshitz-Syozov model. Nonetheless, those terms involve second order derivatives of both a and b which leads us to hypothesis (H5). Hypotheses (H6) and (H7) simply tell us that the initial condition for the Becker-Döring system must have finite zeroth order moment and first moment increments. Lemma 3.4 shows that this property propagates in time. Additional assumptions have to be made to obtain our main theorem. The assumption (H8) on the initial condition L ε,0 is necessary since it leads to strict positivity of L ε in finite time, uniformly in ε. The assumption (H9) on the initial condition c ε,0 0 is technical and ensures that the proper boundary condition (11c) is satisfied for all times. Finally the assumption on R+ |E(0, x)|dx is made to conclude after using Grönwall's lemma at the very end of the proof. This assumption relates both initial conditions (c ε,0 i ) i≥0 and f 0 . A fair choice for the initial condition

(c ε,0 i ) i≥0 is c ε,0 i = f 0 (iε) for all i ≥ 0.
Then the assumption is verified as long as

(f 0 ) ′ ∈ L 1 (R + , xdx).
In all this section, we assume that hypotheses (H1) -(H9) hold true.

Preliminary results on Becker-Döring system

We start with a lemma that allows to control the lipid term away from 0, in the lines of previous results from [START_REF] Calvo | The Initial-Boundary Value Problem for the Lifshitz-Slyozov Equation with Non-Smooth Rates at the Boundary[END_REF].

Lemma 3.1. A solution (L ε , c ε ) of (13) with λ > 0 verifies that there exists C > 0 such that for all t > 0, inf

ε>0 L ε (t) ≥ L exp(-Ct). (22)
Proof. For all t > 0, we have, using the three first equations of system (13) :

dL ε (t) dt = -ε i≥0 J ε i (c ε (t), L ε (t)) = -ε i≥0 a ε i L ε (t) L ε (t) + κ c ε i (t) -b ε i+1 c i+1 ε(t) ≥ - L ε (t) L ε (t) + κ ε i≥0 a ε i c ε i (t),
and thus, because sup

x∈R+ |a(x)| = C a , and L ε L ε +κ ≤ 1 κ L ε : dL ε (t) dt ≥ - C a κ L ε +∞ -ε/2 f ε (t, x)dx
and by conservation of the moment [START_REF] Goudon | Fokker-Planck Approach of Ostwald Ripening: Simulation of a Modified Lifshitz-Slyozov-Wagner System with a Diffusive Correction[END_REF],

+∞ -ε/2 f ε (t, x)dx = ε i≥0 c ε i (t) = m, dL ε (t) dt ≥ - C a m κ L ε .
We conclude by Grönwall's lemma and using Hypothesis (H8).

We next state a lemma adapted from [START_REF] Deschamps | Quasi steady state approximation of the small clusters in Becker-Döring equations leads to boundary conditions in the Lifshitz-Slyozov limit[END_REF] that allows to obtain pointwise estimates of the density f ε near the boundary, through the uniform propagation of exponential moments. For x ∈ R + and t > 0, let

H ε (t, x) = i≥0 c ε i (t) e -ix . Lemma 3.2. Let x ∈ R * + .
Then there exist some constants K > 0 and ε * > 0 such that for all 0 < ε < ε * :

H ε (t, x) ≤ H ε (0, x) + K for all t > 0,
and in particular :

for all i ≥ 0, sup 0<ε<ε * sup t∈[0,T ] c ε i (t) ≤ ci < +∞. (23)
Proof. Using Lemma 3.1, and the assumption (H8) on L ε (0), we have that inf

ε>0 inf t∈(0,T ] L ε (t) ≥
L exp(-CT ). Thus we can find a constant c > 0 such that :

inf ε>0 inf t∈(0,T ] L ε (t) L ε (t) + κ ≥ c.
Now we choose δ > 0 such that c > 2δ. Using Taylor's expansion, we have a(iε) = a(0) + iεa ′ (0) + O((iε) 2 ). Then with hypotheses (H2b) and (H3) and for ε small enough, we find that a(iε) ≥ 3 4 (a(0) -iεK a ) > 0. Therefore we have that for ε small enough :

∀i ≤ 1 √ ε , a(iε) ≥ a(0) 2 . ( 24
)
In turn, by hypotheses (H2a), (H2b) and (H4), we have that for ε small enough and for all

i ≤ 1 √ ε : b ε i a ε i = b(iε) a(iε) ≤ 2C b √ ε a(0) ---→ ε→0 0.
Let x ∈ R * + . Hence, one can find ε * > 0 such that :

sup ε<ε * sup i≤ 1 √ ε | b ε i a ε i | ≤ δ e -x .
This gives us that for ε * small enough, ε < ε * and i ≤ 1

√ ε : L ε (t) L ε (t) + κ - b ε i a ε i e x ≥ 2δ -δ = δ. (25)
Now we proceed with the bound on H ε using Eq.( 13) and ( 12) :

ε∂ t H ε (t, x) = (e -x -1) i≥0 J ε i (c) e -ix
= (e -x -1)

  L ε (t) L ε (t) + κ a ε 0 c ε 0 (t) + i≥1 a ε i ( L ε (t) L ε (t) + κ - b ε i a ε i e x )c ε i (t) e -ix   .
Now we split the sum on the right depending on 1 √ ε with ε small enough as before. Note that since x > 0, we have that (e -x -1) < 0. The first sum is treated using [START_REF] Laurençot | Well-posedness for a model of prion proliferation dynamics[END_REF] and the bound [START_REF] Laurençot | From the becker-döring to the lifshitz-slyozov-wagner equations[END_REF] :

ε∂ t H ε (t, x) ≤ (e -x -1)   2δa ε 0 c ε 0 (t) + a(0) 2 δ ⌊ 1 √ ε ⌋ i=1 c ε i (t) e -ix -e x i≥⌊ 1 √ ε ⌋+1 b ε i c ε i (t) e -ix    .
The term in c ε 0 and the first sum are combined and using our choice of δ, it yields :

2δa ε 0 c ε 0 (t) + a(0) 2 δ ⌊ 1 √ ε ⌋ i=1 c ε i (t) e -ix ≥ a(0) 2 δ   H ε (t, x) - i≥⌊ 1 √ ε ⌋+1 c ε i (t) e -ix    .
Hence :

ε∂ t H ε (t, x) ≤ (1 -e -x )    a(0) 2 δ   -H ε (t, x) + i≥⌊ 1 √ ε ⌋+1 c ε i (t) e -ix    + e x i≥⌊ 1 √ ε ⌋+1 b ε i c ε i (t) e -ix    .
Observe that for ε small enough depending on x, for all i ≥ ⌊ 1 √ ε ⌋, we have :

(δ a(0) 2 + e x b ε i ) e -ix ≤ K(C a + C b )(1 + iε) e -ix ≤ Kε,
which leads to :

ε∂ t H ε (t, x) ≤ a(0) 2 δ(e -x -1)H ε (t, x) + (1 -e -x )Km.
We conclude by using Grönwall's lemma and K = 2Km δa(0) and ( 23) follows immediately. A direct consequence of Lemma 3.2 is the following refined estimate on c ε 0 which shows that at the limit ε → 0, the density f ε vanishes at the boundary, in agreement with the limiting boundary condition (11c): Lemma 3.3. There exist constants C 1 , C 2 > 0 such that for ε small enough and for all t ∈ (0, T ] :

c ε 0 (t) ≤ e -C 1 ε t c ε,0 0 + εC 2 . ( 26
)
Proof. As in the proof of Lemma 3.2, there exists ε small enough such that :

dc ε 0 (t) dt = 1 ε (b ε 1 c ε 1 (t) -a ε 0 L ε (t) L ε (t) + κ c ε 0 (t)) ≤ C b c1 - a(0) ε δc ε 0 (t),
thanks to hypothesis (H'1). Now applying Grönwall's lemma, we obtain :

c ε 0 (t) ≤ e -a(0)δt ε c ε 0 (0) + C b c1 ε δa(0) (1 -e -a(0) ε δt ),
which gives the desired result.

We end this section by a last lemma that will be useful to control the first spatial derivative of F ε . Lemma 3.4. We have, for all T > 0, and for ε small enough,

sup t≤T i≥0 |c ε i+1 -c ε i |(t) < ∞, (27)
sup t≤T ε i≥0 i|c ε i+1 -c ε i |(t) < ∞. (28)
Proof. Let u i = c ε i+1 -c ε i and let's estimate its time derivative. Then, for all i ≥ 1, we have from Eq.( 13) and Eq.( 12):

du i dt = L ε (t) L ε (t) + κ a ε i-1 ε u i-1 - a ε i-1 -a ε i ε c ε i - a ε i ε u i - a ε i+1 -a ε i ε c ε i+1 - b ε i ε u i - b ε i+1 + b ε i ε c ε i+1 - b ε i+1 ε u i+1 + b ε i+1 -b ε i+2 ε c ε i+2 = J ε i-1 (u, L ε ) -J ε i (u, L ε ) ε + L ε (t) L ε (t) + κ a ε i-1 -a ε i ε u i - a ε i-1 -a ε i ε + a ε i+1 -a ε i ε c ε i+1 + b ε i+1 -b ε i ε u i+1 + b ε i+1 -b ε i ε + b ε i+1 -b ε i+2 ε c ε i+2 .
We multiply the previous expression for i ≥ 1 by sign(u i ) on both sides, which gives :

d|u i | dt ≤ J ε i-1 (|u|, L ε ) -J ε i (|u|, L ε ) ε + |a ε i-1 -a ε i | ε |u i | + |a ε i-1 -2a ε i + a ε i+1 | ε c ε i+1 + |b ε i+1 -b ε i | ε |u i+1 | + |b ε i+2 -2b ε i+1 + b ε i | ε c ε i+2 .
Hence, thanks to hypotheses (H3) and (H5) and Lemma 3.2, there exists ε small enough such that for i ≥ 1:

d|u i | dt ≤ J ε i-1 (|u|, L ε ) -J ε i (|u|, L ε ) ε + ∥a ′ ∥ ∞ |u i | + ε∥a ′′ ∥ ∞ c ε i+1 + ∥b ′ ∥ ∞ |u i+1 | + ε∥b ′′ ∥ ∞ c ε i+2 . (29)
Now, for i = 0, we obtain :

du 0 dt = - 1 ε J ε 0 (u, L ε ) - a ε 1 -a ε 0 ε L ε (t) L ε (t) + κ c ε 1 + a ε 0 ε L ε (t) L ε (t) + κ c ε 0 + b ε 2 -b ε 1 ε c ε 2 - b ε 1 ε c ε 1 .
Since b ε 0 = b(0) = 0, we can treat the remaining terms in b as before by adding and removing the right terms for free. The terms with c ε 1 are bounded using Lemma 3.2 and the one with c ε 0 using Lemma 3.3. Hence, there exists ε small enough such that :

d|u 0 | dt ≤ - 1 ε J ε 0 (|u|, L ε ) + ∥a ′ ∥ ∞ c1 + a(0) ε e -C 1 ε t c ε,0 0 + a(0)C 2 + ∥b ′ ∥ ∞ |u 1 | + ε∥b ′′ ∥ ∞ c ε 2 .
We sum the previous estimates for all i ≥ 0 and we get :

d dt i≥0 |u i | ≤ (∥a ′ ∥ ∞ + ∥b ′ ∥ ∞ ) i≥0 |u i | + (∥a ′′ ∥ ∞ + ∥b ′′ ∥ ∞ )ε i≥0 c ε i + ∥a ′ ∥ ∞ c1 + a(0) ε e -C 1 ε t c ε,0 0 + a(0)C 2 .
We integrate the previous inequality over [0, t], for 0 < t < T :

i≥0 |u i |(t) ≤ i≥0 |u i |(0) + (∥a ′ ∥ ∞ + ∥b ′ ∥ ∞ ) t 0 i≥0 |u i |(s)ds + (∥a ′′ ∥ ∞ + ∥b ′′ ∥ ∞ )mT + a(0) C 1 c ε,0 0 + (a(0)C 2 + ∥a ′ ∥ ∞ c1 )T.
And finally Grönwall's lemma yields :

i≥0 |u i |(t) ≤ C u (T ) with C u (T ) =   i≥0 |u i |(0) + (∥a ′′ ∥ ∞ + ∥b ′′ ∥ ∞ )mT + a(0) C 1 c ε,0 0 + (a(0)C 2 + ∥a ′ ∥ ∞ c1 )T )   × exp((∥a ′ ∥ ∞ + ∥b ′ ∥ ∞ )T ),
which gives Eq. ( 27).

Using the definition (13c) of λ, estimate (29) and hypothesis (H'1), we obtain the following inequalities :

ε d dt i≥1 i|u i | ≤ ε i≥1 i J ε i-1 (|u|) -J ε i (|u|) ε + ε(∥a ′ ∥ ∞ + ∥b ′ ∥ ∞ ) i≥1 i|u i | + (∥a ′′ ∥ ∞ + ∥b ′′ ∥ ∞ )ε 2 i≥0 ic ε i ≤ i≥0 J ε i (|u|) + (∥a ′ ∥ ∞ + ∥b ′ ∥ ∞ )ε i≥1 i|u i | + (∥a ′′ ∥ ∞ + ∥b ′′ ∥ ∞ )λ ≤ C a i≥0 |u i | + (∥a ′ ∥ ∞ + ∥b ′ ∥ ∞ )ε i≥1 i|u i | + (∥a ′′ ∥ ∞ + ∥b ′′ ∥ ∞ )λ.
Integrating over [0, t] and using the previous bound on i≥0 |u i |, we conclude using Grönwall's lemma :

ε i≥1 i|u i |(t) ≤   ε i≥1 i|u i |(0) + C a C u (T )T + (∥a ′′ ∥ ∞ + ∥b ′′ ∥ ∞ )λT   exp((∥a ′ ∥ ∞ + ∥b ′ ∥ ∞ )T ),
which yields Eq. ( 28).

Proof of theorem 3.1

In this section, we make use of the lemmas from the previous section. As such, from then on, ε is taken small enough to apply those lemmas. We first derive the equation satisfied by the tail distribution F ε defined at Eq. [START_REF] Jo | Hypertrophy and/or Hyperplasia: Dynamics of Adipose Tissue Growth[END_REF]. Recall that operator ∆ ε is defined at Eq.( 19).

Lemma 3.5. For all x ≥ ε 2 and t ≥ 0 :

∂ t F ε (t, x) = - 1 ε x x-ε (a ε (y) L ε (t) L ε (t) + κ -a(x) L(t) L(t) + κ )f ε (t, y)dy -a(x) L(t) L(t) + κ ∆ -ε F ε (t, x) + 1 ε x+ε x (b ε (y) -b(x))f ε (t, y)dy + b(x)∆ ε F ε (t, x) (30)
and for all x < ε 2 and t ≥ 0 :

∂ t F ε (t, x) = 1 ε x -ε 2 a ε (y) L ε (t) L ε (t) + κ f ε (t, y)dy - 1 ε x+ε ε 2 b ε (y)f ε (t, y)dy. (31)
Remark. The function f ε is defined on [-ε 2 , +∞[ whereas f is defined on R + . However we will only concern ourselves with x ∈ R + in the following subsections. Hence we will treat the case x < ε/2 independently to accommodate for boundary terms that might be left off from f ε . We also point out that :

+∞ 0 f ε (t, x)dx = m - ε 2 c ε 0 (t) (32)
Owing to Lemma 3.2, the right hand side is bounded and tends to m as ε → 0.And for the first order we have an exact computation :

L ε (t) + +∞ 0 xf ε (t, x)dx = λ (33)
Proof. For all x ∈ R + and t ∈ [0, T ], it comes directly from the definitions ( 19) and ( 20) that the following equations hold true : for all x ≥ 0 and t ≥ 0,

∆ ε F ε (t, x) = - 1 ε x+ε x f ε (t, y)dy, (34)
and for all x ≥ ε 2 and t ≥ 0,

∆ -ε F ε (t, x) = - 1 ε x x-ε f ε (t, y)dy. (35)
Denote H x = 1 [x,+∞) . First observe that :

∆ ε H x (y) = 1 ε 1 [x,+∞) (y + ε) -1 [x,+∞) (y) = 1 ε 1 [x-ε,x) (y), ∆ -ε H x (y) = 1 ε 1 [x,x+ε) (y).
Then we use Proposition 2.2 for the Heaviside function. It yields that for all x ≥ ε 2 :

∂ t F ε (t, x) = R+ H x (y)∂ t f ε (t, y)dy = R+ ∆ ε H x (y)a ε (y) L ε (t) L ε (t) + κ -∆ -ε H x (y)b ε (y) f ε (t, y)dy = x x-ε 1 ε a ε (y) L ε (t) L ε (t) + κ f ε (t, y)dy - x+ε x 1 ε b ε (y)f ε (t, y)dy = 1 ε x x-ε (a ε (y) L ε (t) L ε (t) + κ -a(x) L(t) L(t) + κ )f ε (t, y)dy -a(x) L(t) L(t) + κ ∆ -ε F ε (t, x) - 1 ε x+ε x (b ε (y) -b(x))f ε (t, y)dy + b(x)∆ ε F ε (t, x)
The case for x < ε 2 follows from simple computation, using that

+∞ -ε/2
f ε (t, x)dx = m by conservation of the moment ( 14) :

∂ t F ε (t, x) = d dt +∞ -ε 2 f ε (t, y)dy - d dt x -ε 2 f ε (t, y)dy = -(x + ε 2 ) d dt c ε 0 (t) = 1 ε x -ε 2 a ε (y) L ε (t) L ε (t) + κ f ε (t, y)dy - 1 ε x+ε ε 2 b ε (y)f ε (t, y)dy.
We then derive an upper bound for the time derivative of R+ |E(t, x)|dx, where E is defined at Eq.( 21). Lemma 3.6. For all t ≥ 0, we have :

d dt R+ |E(t, x)|dx ≤ (∥a ′ ∥ ∞ + ∥b ′ ∥ ∞ ) R+ |E(t, x)|dx + R+ |∂ t F ε (t, x) + v(x, L(t))∂ x F ε (t, x)|dx + ε 2 a(0) L(t) L(t) + κ c ε 0 (t). (36)
Proof. From the definitions of the tail distributions in Eq. ( 20), the following equations hold true :

∂ x F (t, x) = -f (t, x) , ∂ x F ε (t, x) = -f ε (t, x), a.e. in R + .
By Def.2.3, we have :

F (t, x) = +∞ max(x,Xc(t)) f 0 (X(0; t, y))∂ y X(0; t, y)dy =          +∞ 0 f 0 (y)dy if x ≤ X c (t), +∞ X(0;t,x) f 0 (y)dy if x ≥ X c (t). Therefore if x ≤ X c (t), then ∂ t F (t, x) = 0 = ∂ x F (t, x).
And if x ≥ X c (t), the following expressions hold :

∂ t F (t, x) = -f 0 (X(0; t, x))∂ t X(0; t, x), ∂ x F (t, x) = -f 0 (X(0; t, x))∂ x X(0; t, x) = -f (t, x).
By properties of characteristics we have : ∂ t X(0; t, x) + v(x, L)∂ x X(0; t, x) = 0 and thus :

∂ t F (t, x) + v(x, L(t))∂ x F (t, x) = -f 0 (X(0; t, x))(∂ t X(0; t, x) + v(x, L)∂ x X(0; t, x)) = 0.
We then compute :

∂ t E(t, x) = ∂ t F ε (t, x) -∂ t F (t, x) = -v(x, L(t))∂ x (F ε -F )(t, x) + ∂ t F ε (t, x) + v(x, L(t))∂ x F ε (t, x).
We integrate the previous equality, we use the definition (3) of v with hypothesis (H3) and we find :

d dt R+ |E|dx = - R+ v∂ x |E|dx + R+ sign(E)(∂ t F ε + v(x, L(t))∂ x F ε )dx = -[v(x, L(t))|E(t, x)|] +∞ 0 + R+ ∂ x v|E|dx + R+ sign(E)(∂ t F ε + v∂ x F ε )dx ≤ v(0, L(t))|E(t, 0)| + (∥a ′ ∥ ∞ + ∥b ′ ∥ ∞ ) R+ |E|dx + R+ |∂ t F ε + v∂ x F ε |dx = (∥a ′ ∥ ∞ + ∥b ′ ∥ ∞ ) R+ |E|dx + R+ |∂ t F ε + v∂ x F ε |dx + ε 2 a(0) L(t) L(t) + κ c ε 0 (t).
The last equality is obtained since b(0) = 0 and [START_REF] Prana | Modeling the Effect of High Calorie Diet on the Interplay between Adipose Tissue, Inflammation, and Diabetes[END_REF].

|E(t, 0)| = | R+ (f ε -f )(t, x)dx| = ε 2 c ε 0 (t), see Eq.
Thanks to the equation on F ε given by Lemma 3.5, we control the second term in Eq. ( 36) in the next lemma. Lemma 3.7. There exist some constants C 1 , C 2 > 0 independent of ε such that for all t ∈ [0, T ]:

R+ |(∂ t F ε + v∂ x F ε )(t, x)|dx ≤ εC 1 + C 2 |L ε (t) -L(t)|. ( 37
)
Proof. First by construction of both a ε and b ε , and the fact that a and b are lipshitz continuous, one has for all x, y ∈ R + such that |y -x| ≤ ε:

| a ε (y) -a(x) ε | ≤ 2∥a ′ ∥ ∞
and similarly for b ε . Then using equation [START_REF] Peurichard | Simple mechanical cues could explain adipose tissue morphology[END_REF] and definition (3) of v, we find the following estimate for all x ≥ ε 2 :

|∂ t F ε + v∂ x F ε | ≤ 2∥a ′ ∥ ∞ x x-ε f ε + 1 ε a(x) κ |L ε (t) -L(t)| x x-ε f ε + 2∥b ′ ∥ ∞ x+ε x f ε + a(x) L(t) L(t) + κ |∂ x F ε -∆ -ε F ε | + b(x)|∂ x F ε -∆ ε F ε |.
Using equation [START_REF] Peurichard | Extra-cellular matrix rigidity may dictate the fate of injury outcome[END_REF], hypotheses (H'1) and (H2b) and Lemma 3.2, we find for all x < ε 2 :

|∂ t F ε + v∂ x F ε | ≤ | 1 ε x -ε 2 a ε (y) L ε (t) L ε (t) + κ f ε (t, y)dy -a(x) L(t) L(t) + κ f ε (t, x)| + | 1 ε x+ε ε 2 b ε (y)f ε (t, y)dy -b(x)f ε (t, x)| ≤ | x + ε 2 ε a(0) L ε (t) L ε (t) + κ c ε 0 (t) -a(x) L(t) L(t) + κ c ε 0 (t)| + | x + ε 2 ε b ε 1 c ε 1 (t) -b(x)c ε 0 (t)| ≤ a(0)c ε 0 (t) x + ε 2 ε 1 κ |L ε (t) -L(t)| + |a(x) - x + ε 2 ε a(0)|c ε 0 (t) + C b εc ε 1 (t) + C b ε 2 c ε 0 (t) ≤ a(0) c0 1 κ |L ε (t) -L(t)| + (a(0) + ε 2 ∥a ′ ∥ ∞ ) c0 + C b c1 ε + C b ε 2 c0 .
Now we can integrate |∂ t F ε + v∂ x F ε | over R + using the two previous estimates. Note that using Fubini's theorem and Eq.( 32), we have

+∞ ε 2 x x-ε f ε (y)dydx = +∞ -ε 2 f ε (t, y) y+ε max(y,ε/2) dxdy = ε +∞ ε 2 f ε (t, y)dy + ε 2 2 c ε 0 (t) = +∞ 0 f ε (t, y)dy ≤ εm.
Therefore, we get :

R+ |∂ t F ε + v∂ x F ε |dx ≤ ε(2∥a ′ ∥ ∞ + 2∥b ′ ∥ ∞ |)m + ∥a∥ ∞ κ |L ε (t) -L(t)|m + ∥a∥ ∞ L(t) L(t) + κ +∞ ε 2 |∂ x F ε -∆ -ε F ε |dx + +∞ ε 2 b(x)|∂ x F ε -∆ ε F ε |dx + ε 2 (a(0) c0 1 κ |L ε (t) -L(t)| + (a(0) + ε 2 ∥a ′ ∥ ∞ ) c0 + C b c1 ε + C b ε 2 c0 ). ( 38
)
We now compute the difference between the continuous and discrete derivatives on F ε . We denote by ⌊x⌉ the nearest integer function with the upper-rounding convention : ⌊0.5⌉ = 1. Figure 2 shows a representation of the cells Γ ε i-1 , Γ ε i and Γ ε i+1 as well as an example of the result of 35) and the fact that f ε is constant

⌊ x ε ⌉ for x ∈ Γ ε i . Let us first compute the integral of |∂ x F ε -∆ -ε F ε |, using Eq.(
(i -1)ε iε (i + 1)ε (i -3 2 )ε (i -1 2 )ε (i + 1 2 )ε (i + 3 2 )ε x x -ε ε⌊ • ε ⌉ Figure 2: Representation of the cells Γ ε i-1 , Γ ε i and Γ ε i+1 and of the value ε⌊ x ε ⌉ . equal to c ε i (t) on the cells Γ ε i : +∞ ε 2 |∂ x F ε -∆ -ε F ε |dx = +∞ ε 2 |f ε (t, x) - 1 ε x x-ε f ε (t, y)dy|dx = +∞ ε 2 1 ε | x x-ε (f ε (t, x) -f ε (t, y))dy|dx = +∞ ε 2 1 ε | ε(⌊ x ε ⌉-1 2 )
x-ε

(f ε (t, x) -f ε (t, y))dy|dx = +∞ ε 2 |c ε ⌊ x ε ⌉ (t) -c ε ⌊ x ε ⌉-1 (t)| ε(⌊ x ε ⌉ -1 2 ) -(x -ε) ε dx. Now, observe that for all x ∈ Γ ε i one has c ε ⌊ x ε ⌉ (t) -c ε ⌊ x ε ⌉-1 (t) = c ε i (t) -c ε i-1 (t), which gives : +∞ ε 2 |∂ x F ε -∆ -ε F ε |dx = i≥1 |c ε i (t) -c ε i-1 (t)| Γ ε i ε(⌊ x ε ⌉ -1 2 ) -(x -ε) ε dx = ε 2 i≥1 |c ε i (t) -c ε i-1 (t)|.
Hence, according to Lemma 3.4 there exists a constant C(T ) > 0 independent of ε such that :

+∞ ε 2 |∂ x F ε -∆ -ε F ε |dx ≤ εC(T ). (39)
We proceed similarly for the term

R+ b(x)|∂ x F ε -∆ ε F ε |dx: ∞ ε 2 b(x)|f ε (t, x) - 1 ε x+ε x f ε (t, y)dy|dx = ∞ ε 2 b(x) ε | x+ε x (f ε (t, x) -f ε (t, y))dy|dx = ∞ ε 2 b(x) ε | x+ε ε(⌊ x ε ⌉+ 1 2 ) (f ε (t, x) -f ε (t, y))dy|dx = ∞ ε 2 |c ε ⌊ x ε ⌉+1 (t) -c ε ⌊ x ε ⌉ (t)|b(x) x -ε(⌊ x ε ⌉ -1 2 ) ε dx = i≥1 |c ε i+1 (t) -c ε i (t)| Γ ε i b(x) x -ε(⌊ x ε ⌉ -1 2 ) ε dx.
Owing to hypothesis (H2b), we simply bound the last integral as follows :

Γ ε i b(x) x -ε(⌊ x ε ⌉ -1 2 ) ε dx ≤ C b ε 2 2 (i + 1 6
).

Hence, according to Lemma 3.4, there exists a constant C(T ) > 0 independent of ε such that :

+∞ ε 2 b(x)|∂ x F ε -∆ x F ε |dx < ε 6 C(T )(1 + ε 2 ). ( 40
)
We conclude from Eqs. ( 38)-( 39)-( 40) by regrouping together terms not depending on ε.

We now proceed with the proof of Theorem 3.1.

Proof of Theorem 3.1. Let T > 0 and consider t ∈ (0, T ]. We begin by integrating ( 36) over [0, t], using Lemma 3.7 :

R+ |E(t, x)|dx ≤ R+ |E(0, x)|dx + (∥a ′ ∥ ∞ + ∥b ′ ∥ ∞ ) t 0 R+ |E(s, x)|dxds + t 0 R+ |∂ t F ε (s, x) + v(x, L(s))∂ x F ε (s, x)|dxds + ε 2 a(0)T c0 ≤ R+ |E(0, x)|dx + (∥a ′ ∥ ∞ + ∥b ′ ∥ ∞ ) t 0 R+ |E(s, x)|dxds + εT C 1 + C 2 t 0 |L ε (s) -L(s)|ds + ε 2 a(0)T c0 .
Then observe that :

R+ xf ε (t, x)dx = R+ x i≥0 1 Γ ε i (x)c ε i (t)dx = i≥0 Γ ε i xdxc ε i (t) = i≥0 iε 2 c ε i (t).
Using conservation equations (13c) and ( 7) and Fubini's theorem, this leads to the bound :

|L ε (t) -L(t)| = | ∞ 0 x(f ε (t, x) -f (t, x))dx| = | ∞ 0 (F ε (t, x) -F (t, x))dx| ≤ R+ |E(t, x)|dx,
which finally yields :

|L ε (t) -L(t)| + R+ |E(t, x)|dx ≤ 2 R+ |E(0, x)|dx + 2(∥a ′ ∥ ∞ + ∥b ′ ∥ ∞ ) t 0 R+ |E(s, x)|dxds + 2εT C 1 + 2C 2 t 0 |L ε (s) -L(s)|ds + εa(0)T c0 .
By the assumption on E(0, x) and Grönwall's lemma, we finally conclude that :

|L ε (t) -L(t)| + R+ |E(t, x)|dx ≤ ε 2C init + 2C 1 T + a(0)T c0 exp(2(∥a ′ ∥ ∞ + ∥b ′ ∥ ∞ + C 2 )T ).
4 Derivation of second order model and stationary solutions

A second-order Lifshitz-Slyozov model with diffusion

Up to this point we have studied a Lifshitz-Slyozov model, that is to say a transport PDE. However, this model leads to stationary solutions which are combinations of Dirac masses centered at the zeros of velocity v. Hence, since we aim at obtaining asymptotically bimodal distributions, we would like to add a diffusion term to our model in order to smooth the stationary solutions. Unfortunately and up to our knowledge, no biological argument can be found to explain such a diffusive term or to give a proper way of deriving it from biological considerations. Nonetheless, one can see this diffusive term as a second order term emerging from the preceding convergence result, see for example [START_REF] Vasseur | The Beker-Döring System and Its Lifshitz-Slyozov Limit[END_REF], [START_REF] Schlichting | Macroscopic limit of the Becker-Döring equation via gradient flows[END_REF], [START_REF] Deschamps | Quasi steady state approximation of the small clusters in Becker-Döring equations leads to boundary conditions in the Lifshitz-Slyozov limit[END_REF].

We follow here the derivation of the diffusive term presented in [START_REF] Vasseur | The Beker-Döring System and Its Lifshitz-Slyozov Limit[END_REF] and we use the notation introduced at Section 3. Now, from Proposition 2.2, we can add and subtract the appropriate terms in ϕ to get :

∞ 0 (f ε (t, x) -f ε (0, x))ϕ(x)dx = t 0 ∞ 0 (∆ ε ϕ(x)a ε (x) L ε (t) L ε (t) + κ -∆ -ε ϕ(x)b ε (x))f ε (t, x)dxdt = t 0 ∞ 0 ϕ(x + ε) -ϕ(x -ε) 2ε (a ε (x) L ε (t) L ε (t) + κ -b ε (x))f ε (t, x)dxdt + ε 2 t 0 ∞ 0 ϕ(x + ε) -2ϕ(x) + ϕ(x -ε) ε 2 a ε (x) L ε (t) L ε (t) + κ f ε (t, x)dxdt + ε 2 t 0 ∞ 0 ϕ(x + ε) -2ϕ(x) + ϕ(x -ε) ε 2 b ε (x)f ε (t, x)dxdt = t 0 ∞ 0 ∆ε ϕ(x)(a ε (x) L ε (t) L ε (t) + κ -b ε (x))f ε (t, x)dxdt + ε 2 t 0 ∞ 0 ∆ 2 ε ϕ(x)(a ε (x) L ε (t) L ε (t) + κ + b ε (x))f ε (t, x)dxdt where ∆h ϕ(x) = ϕ(x+h)-ϕ(x-h) 2h and ∆ 2 h ϕ(x) = ϕ(x+h)-2ϕ(x)+ϕ(x-h) h 2
. This leads us to study the PDE (17).

Stationary solutions for the second-order Lifshitz-Slyozov model

In this section, we present the stationary solutions of system (11) without diffusion and system [START_REF] Hingant | Deterministic and Stochastic Becker-Döring Equations: Past and Recent Mathematical Developments[END_REF] with diffusion. We notice that stationary solutions are very different in nature from one model to the other. Eq.( 11) does not yield nontrivial smooth stationary functions, and we rather expect stationary solutions to be linear combinations of Dirac masses, located at roots of the asymptotic velocity.

We can compute explicitly stationary solutions of system [START_REF] Hingant | Deterministic and Stochastic Becker-Döring Equations: Past and Recent Mathematical Developments[END_REF], namely :

∂ t g = 0 ⇐⇒ ∂ x (vg) - ε 2 ∂ 2 x (dg) = 0.
Together with boundary conditions [START_REF] Hariz | A Modified Version of the Lifshitz-Slyozov Model[END_REF] this leads to stationary solutions denoted by M Lstat , depending on stationary L stat ∈ R + under the form :

M Lstat (x) = C(m, L stat ) d(x, L stat ) exp 2 ε x 0 v(y, L stat ) d(y, L stat ) dy , (41)
where the constant C(m, L stat ) is determined in order to satisfy

R+ M Lstat (t, x)dx = m, that is to say C(m, L stat ) = m R+ 1 d(x, L stat ) exp 2 ε x 0 v(y, L stat ) d(y, L stat ) dy dx
and L stat solves the constraint equation

L stat + R+ xM Lstat (x)dx = λ. ( 42
)
Note that function Φ :

L → L + R+ xM L (x)dx is continuous on R + . Moreover, straightforward
computations show that thanks to expression (41) and expressions for a and b that Φ(0) = 0 and Φ → L→+∞ +∞. Therefore, for all λ ≥ 0, there exists at least one value for L which satisfies Eq.( 42). Regarding unicity of stationary solutions, it would need to prove strict monotonicity of Φ, which is so far an open question. However, we may observe numerically that the application

Φ : L → L + xmax 0
xM L (x)dx seems strictly non-decreasing, see Figure 3. 4) and ( 5) and parameters given at Table 1.

Remark. In other modeling contexts, one may choose different functions a and b such that existence of stationary solutions may not be true for all value of λ. For example, a(x) = 1 and b(x) = x s with s ≤ 1 implies lim

L→0 + Φ(L) = λ 0 > 0.
Hence for values of λ such that λ < λ 0 , the system might not have smooth stationary solutions, see section 5.2.5 and Figures 14 and15.

In the following section, we will present some numerical simulations for system [START_REF] Hingant | Deterministic and Stochastic Becker-Döring Equations: Past and Recent Mathematical Developments[END_REF] and control that stationary solutions M Lstat follow a bimodal distribution for well-chosen parameters.

Numerical simulations

In this part, we use a finite volume well-balanced scheme introduced in [START_REF] Goudon | Fokker-Planck Approach of Ostwald Ripening: Simulation of a Modified Lifshitz-Slyozov-Wagner System with a Diffusive Correction[END_REF] to approximate time dependent solutions to Eq. ( 17). Afterwards, we explore numerically the solutions to system [START_REF] Hingant | Deterministic and Stochastic Becker-Döring Equations: Past and Recent Mathematical Developments[END_REF] for various sets of parameters. We also compare the Lifshitz-Slyozov diffusive equation with the transport equation [START_REF] Divoux | Architecture and the Extracellular Matrix: The Still Unappreciated Components of the Adipose Tissue[END_REF] and with the transport equation ( 11) with a constant diffusive term. We will finally explore the case when λ < λ 0 = lim L→0 + Φ(L) => 0 mentioned previously in the remark of Sec.4.2.

Note that in this section, unlike the previous ones, we are working on a bounded domain x ∈ [0, x max ] rather than on R + .

5.1 A well-balanced numerical scheme for system [START_REF] Hingant | Deterministic and Stochastic Becker-Döring Equations: Past and Recent Mathematical Developments[END_REF] In the following, we will need to compute some approximations for the stationary solutions M Lstat since we need them in the well-balanced scheme, see later on. Moreover, it will enable us to compare the asymptotic profiles with the stationary solutions in the numerical tests.

Let us recall that stationary solutions M Lstat are defined by an explicit expression given at Eq. ( 41) with L stat satisfying constraint equation (42). Therefore, to compute this stationary solution, a simple dichotomy method is implemented to find the solution to Φ(L) = λ, since the application Φ is increasing in the range of L that interests us, see Figure 3. We use the trapezoidal rule for the computation of the integrals.

Since we are interested in a conservative PDE, we use a finite volume scheme. We also aim at capturing correctly stationary solutions and for that purpose, we implement a well-balanced scheme introduced in [START_REF] Goudon | Fokker-Planck Approach of Ostwald Ripening: Simulation of a Modified Lifshitz-Slyozov-Wagner System with a Diffusive Correction[END_REF]. Let us detail the scheme here.

The scheme is based on a change of variables in the PDE [START_REF] Hingant | Deterministic and Stochastic Becker-Döring Equations: Past and Recent Mathematical Developments[END_REF] to obtain a symmetric operator. This will allow simpler calculations down the line. Denote D L the spatial operator in the PDE, i.e. :

D L g = ∂ x F (g; x, L) = ∂ x -v(x, L)g + ∂ x (d(x, L)g) .
We recall that the stationary solution associated with the value L is given by :

M L (x) = C(m, L) d(x, L) exp x 0 v(y, L) d(y, L) dy . (43)
This stationary solution satisfies D L M L = 0 and we can rewrite the operator D L in the following way :

D L g = ∂ x d(x, L)M L ∂ x ( g M L ) .

Then we perform the change of variable

h = g √ M L
and introduce the new operator DL , which is symmetric for the L 2 inner product :

DL h = 1 √ M L D L (h M L ) = 1 √ M L ∂ x d(x, L)M L ∂ x ( h √ M L
) .

Note that we use an implicit discretization in time in order to avoid a constraining time step for the diffusion operator.

Given a mesh of size ∆x > 0 in space, we discretize the interval [0, x max ] and consider N cells C j = [x j-1/2 , x j+1/2 ], 1 ≤ j ≤ N centered at point x j , with x j = j∆x and x j+1/2 = (j + 1/2)∆x. We also introduce a time step ∆t > 0 and the discretization times t n = n∆t, n ∈ N.

We denote by h n j an approximation of the average of function h on cell C j at time t n , that is to say h n j ∼ 1 ∆x Cj h(t n , x)dx. We also define M L n ,j as an approximation of stationary solution M L n defined at Eq. ( 43) at point x j with L = L n , and D n j+1/2 as an approximation of diffusion coefficient d(x j+1/2 , L n ) at point x j+1/2 with L = L n , see expression [START_REF] Greer | A mathematical analysis of the dynamics of prion proliferation[END_REF]. We denote by F n j+1/2 an approximation of flux d(x, L)

M L ∂ x ( h √ M L
) at the boundary x j+1/2 of cell C j at time t n . We therefore discretize Eq.(17a) as follows :

h n j+1 -h n j ∆t = 1 ∆x M L n ,j (F n j+1/2 -F n j-1/2 ) = 1 ∆x M L n ,j D n j+1/2 M L n ,j+1 M L n ,j h n+1 j+1 / M L n ,j+1 -h n+1 j / M L n ,j ∆x -D n j-1/2 M L n ,j M L n ,j-1 h n+1 j / M L n ,j -h n+1 j-1 / M L n ,j-1 ∆x .
Regarding boundary conditions, we want to preserve the zeroth-order moment xmax 0 g(t, x)dx = xmax 0 g 0 (x)dx, which implies to use the following null-flux boundary conditions :

-v(x, L(t))g(t, x) + ε 2 ∂ x (d(x, L(t))g(t, x))| x=0,xmax = 0.
In practice, those boundary conditions are implemented using ghost cells centered at points -∆x and x max + ∆x, and by setting the null-flux conditions F n -1/2 = F n N +1/2 = 0. In order to update the value of L, we derive Eq.(17b) with respect to time and we discretize the equation ∂ t L = -R+ x∂ t g(t, x)dx, which gives:

L n+1 = L n -∆x N i=1 x i (g n+1 i -g n i ).
This update leads to a restriction on the time step ∆t to preserve positivity of L n+1 as seen in [START_REF] Goudon | Fokker-Planck Approach of Ostwald Ripening: Simulation of a Modified Lifshitz-Slyozov-Wagner System with a Diffusive Correction[END_REF].

Numerical results

The previous numerical scheme enables us to explore the properties of system [START_REF] Hingant | Deterministic and Stochastic Becker-Döring Equations: Past and Recent Mathematical Developments[END_REF] as a model for adipocyte distribution evolution in time. Table 1 presents the value of most parameters for the simulations. Unless stated otherwise, these parameters shall be fixed for this section. Concerning values of parameters, a few of them are chosen in accordance with biological observations. V lipids and r 0 have fixed given values. The value of γ is taken from [START_REF] Soula | Model of adipose tissue cellularity dynamics during food restriction[END_REF]. Values of other parameters are chosen as to observe bimodal distributions. We refer the reader to [START_REF] Aubdebert | Mathematical modeling of adipose tissue size distributions: parameter identifiability study and estimation with measured adipocyte size distributions in rats[END_REF] for further investigation into the values of those parameters.

Bimodality vs unimodality

Since the main aim of the model we develop and study in this paper is to represent bimodality of the distribution on the stationary solution, we check if we effectively find some parameter ranges for which we observe this behaviour. In particular, we investigate the dependency with respect to λ. Note that, since λ is defined by expression [START_REF] José | Uniform moment propagation for the Becker-Döring equations[END_REF], we change λ by changing the initial conditions L 0 and g 0 , in the case of time evolution of the system, or by changing the value of L stat when considering stationary solutions. 1.

In Figure 5, we plot densities of adipocytes as a function of size x. It shows the result of the scheme starting from a Gaussian initial condition g 0 (x) = C exp -1 2

x -3 0.5 2 plotted in dashed blue line and L 0 such that λ = 3.5. The value of C is determined such that m = 1. The stationary solution is denoted g stat = M Lstat -in black full line -and the final result of the scheme at time t = t max is denoted g(t max , •) and represented in dotted yellow line. t max is determined such that the relative difference between the size distribution g(t max , •) and the stationary solution M Lstat is less than 5 × 10 -5 . We can observe that bimodality is obtained for the stationary solution as well as for the asymptotic profile of the adipocyte size distribution and that there is a good correspondence between the two functions. Up to some numerical error of order 10 -12 , both the initial number of cells m = xmax 0 g 0 (x)dx and the initial amount of lipids λ are conserved, as expected. In Fig. 6, we plot the time evolution of the solution : on the left, adipocyte density is displayed as a function of x for various times and on the right, the evolution with respect to time of external lipid concentration L is plotted. We observe that L tends to a stationary value and g to a stationary profile with bimodality as expected. Now we may investigate the behaviour of the stationary solutions and the asymptotic profiles with respect to λ. A first crucial information is that depending on λ, different types of modality can be observed. Figure 7 presents a case where the stationary solution is unimodal, obtained with Figure 6: On the left : time evolution of the size distribution with respect to size in the bimodal case; on the right : time evolution of the external lipid concentration. We observe that the asymptotic profile coincides with the computed stationary solution. Parameters of the system are given at Table 1. 1.

initial conditions g 0 (x) = C exp -1 2

x -6 0.5

2

, where C is choosen such that m = 1, and L 0 such that λ = 7. We remark that L also tends to a stationary value and densities converge towards a stationary distribution with unimodality. Biologically, we can relate this to the fact that if the amount of lipids in the system is higher, cells have a tendency to put into storage the maximum amount of lipids and thus cells are bigger in average. From a mathematical point of view, since the optima can be linked to the velocity zeros, this means that for bigger λ, two of the zeros of speed V -and therefore two optima -disappear and thus only one zero remains giving rise to a unimodal profile. More generally, we can investigate the profile modality with respect to the value of λ using the computation of stationary solutions. In Figure 8 on top, we present the type of modality of the stationary solutions as a function of λ. Left (resp. right) unimodality is labeled in green Y (resp. in black x) when a single mode concentrated on the left (resp. right) of the domain is observed. Central unimodal stationary solution is labeled in yellow + when the unique mode is concentrated inside the domain. Bimodality is labeled in red dot.

In Fig. 8 on bottom, a plot for each of the 4 types of modality is presented. On the top, we represent the stationary solution M Lstat with respect to x and on the bottom, we plot the stationary velocities with respect to x. Four different values of L stat corresponding to various λ are considered, namely L stat = 0.05 and λ = 0.191 for the left unimodality (top left, in blue), L stat = 0.075 and λ = 3.52 for the left bimodality (top right, in green), L stat = 0.1 and λ = 9.96 for the central unimodality (bottom left, in yellow) and L stat = 0.2 and λ = 14.9 for the left unimodality (bottom right, in black). For a biological interpretation, left modality is observed when the amount of lipids is too low and thus cells are of relative small sizes. Right modality is a consequence of the amount of lipids being too large and represents the whole cell population approaching its maximal volume. A mathematical interpretation is given by again considering the zeros of the velocity with an influence on the optima of the profile. Left (resp. right) modality is reached when zeros disappear and/or go outside the domain from the left (resp. from the right). The first mode in the bimodal case can also be localized at 0, the smallest zero of the velocity being outside the domain (on the left).

Influence of ε and comparison with a constant diffusion rate D

Figure 9: Different stationary solutions depending on the value of ε. We observe that bimodality holds for values of ε small enough. Parameters of the system are given at Table 1.

In this part, we explore the influence of parameter ε on the shape of stationary solutions. We can observe in Figure 9 that higher values of ε smoothen the two maxima of the solution, as expected. For smaller values of ε, the nadir (i.e. the local minimum between the two maxima) gets sharper and for very small ε this may result numerically in taking very small time and space steps. This is easily interpreted as the fact that when ε = 0, we consider the classical Lifshitz-Slyozov system where stationary solutions are sums of Dirac masses which is difficult to obtain numerically without a dedicated scheme.

The choice we made for the diffusion rate is supported by the convergence results from the Becker-Döring to Lifshitz-Syozov model and the behaviour of second order terms. However this choice is not motivated by biological observation. Hence one may make the assumption that the diffusion rate is constant in both time and space. This unfortunately results in quite different results as shown in Figure 10. We point out that to obtain bimodality some parameters need to be readjusted in this case. Hence comparing the solutions of the system under consideration (17) and the solutions with constant diffusion rate proves to be difficult because the behaviour of stationary solutions is heavily dependent on the choice of parameters.

Figure 10: Different stationary solutions depending on the value of diffusion rate D taken as constant in space and time. Parameters of the system are given at Table 1.

We still can make a few comments about the resulting solutions. The constant diffusion rate tends to smoothen the first maximum whereas in the non-constant case, the diffusion is relatively close to zero, leading to a sharper maximum. Our investigation of the available data for adipose cell distribution leads us to believe that non-constant diffusion rates have better chances of making the model fit with the data. We also point out that in the case of constant diffusion, each type of modality, as previously described, is obtainable. Stationary solutions for the first order Lifshitz-Slyozov model are not so easily computed theoretically. Nonetheless we can explore these solutions numerically as asymptotic profiles of the solutions Figure 12: Numerical solution for the first order Lifshitz-Slyozov model [START_REF] Divoux | Architecture and the Extracellular Matrix: The Still Unappreciated Components of the Adipose Tissue[END_REF] with same parameters and initial data as in Fig. 6. On the left : time evolution of the size distribution with respect to size; on the right : time evolution of the external lipid concentration. The solution to the first order Lifshitz-Slyozov model is expected to converge to a Dirac mass and is displayed for a time before reaching the asymptotic profile.

Comparison with the first order model

of system [START_REF] Divoux | Architecture and the Extracellular Matrix: The Still Unappreciated Components of the Adipose Tissue[END_REF]. For that purpose, we use a standard upwind scheme for transport equations, since the velocity is known. Figure 11 presents the result of an upwind scheme for the Lifshitz-Slyozov model with the same initial conditions and parameters as in Fig. 6. We expect singular stationary state for the first order Lifshitz-Slyozov model. We may interpret stationary state that concentrates at two points as a degenerate bimodal solution. Using the same parameters as in Fig. 6, we can see on Fig. 12 that the solution concentrates to a singular Dirac mass and that in this case we cannot recover bimodality, unlike the case of second-order Lifshitz-Slyozov model, see Fig. 11. We also point out that the asymptotic values of L are different in both cases.

By changing initial conditions and the parameter β to β = 100, we can nonetheless obtain a bimodal solution for the first order Lifshitz-Slyozov model [START_REF] Divoux | Architecture and the Extracellular Matrix: The Still Unappreciated Components of the Adipose Tissue[END_REF] as seen in Fig. 13 on the left. However, by changing the initial condition f 0 , we can see on Fig. 13 on the right that we do not obtain the same asymptotic solutions. This leads us to believe that in the case of the first order Lifshitz-Slyozov model the asymptotic solutions depend on the initial condition g 0 and not only on m and λ, unlike for second order Lifshitz-Slyozov model [START_REF] Hingant | Deterministic and Stochastic Becker-Döring Equations: Past and Recent Mathematical Developments[END_REF].

The case λ < Φ(0)

As explained in the remark of Sec.4.2, for different choices of functions a and b than those of the adipocyte model, we may find situations where lim L→0 + Φ(L) = λ 0 > 0. In this subsection, we explore the evolution of a solution for a value of λ such that 0 < λ < λ 0 , that is to say in a case when no smooth stationary solution exists. An example of choice for a and b is a(x) = 1 and b(x) = (x + 1) 2/3 and in Fig. 14, the function L → Φ(L) is displayed in that case.

We show in Figure 15 the time evolution of the density profile (on the left) and of the external lipid concentration L (on the right) computed numerically in a case where λ < λ 0 . We observe that, as expected, L tend to 0 asymptotically and that the adipocyte density seems to converge towards a Dirac mass centered at 0. Numerical simulations prove difficult because of the constraint on ∆t to enforce the stability of the numerical scheme. More precisely, this constraint induces that ∆t should be bounded above by L n . Hence as the computation time increases, we observe that the value of L n tends to zero, as the solution gets closer to the asymptotic profile and therefore that the time step eventually gets smaller than machine precision. In this case, the scheme fails to conserve both λ and m.

Conclusion

Our work provides a new approach for looking into convergence from Becker-Döring to Lifshitz-Slyozov, and numerical results indicating that the second order Lifshitz-Slyozov model is better suited to model adipocyte size distribution than previous approach relying on first order Lifshitz-Slyozov model.

The originality of this study lies in the following points :

• a new second order Lifshitz-Slyozov model [START_REF] Hingant | Deterministic and Stochastic Becker-Döring Equations: Past and Recent Mathematical Developments[END_REF] for adipocyte size distribution with a diffusion term derived from a discrete model,

• Becker-Döring and Lifshitz-Slyozov systems with an unusual velocity (3) -( 5) with three zeros and a saturation term in L, which leads to different types of stationary solutions,

• an additional conservation law [START_REF] Collet | On solutions of the Lifshitz-Slyozov model[END_REF] with respect to classical systems, enforcing uncommon boundary conditions, see Eq. ( 10) and ( 16),

• a new proof of convergence result from Becker-Döring solutions to Lifshitz-Slyozov solutions, using tails of distributions, that provides an upper bound on the speed of convergence.

• numerical results showing that bimodal distributions, as well as unimodal profiles, can be obtained asymptotically with system (17), according to the parameters,

• numerical results exploring the influence of parameter ε and comparing the diffusion term of system [START_REF] Hingant | Deterministic and Stochastic Becker-Döring Equations: Past and Recent Mathematical Developments[END_REF] with a time and space constant coefficient.

• numerical results shows that the second order system (17) provides universal asymptotic profile that does not depend on initial condition (but only on λ, m), contrary to first order system [START_REF] Divoux | Architecture and the Extracellular Matrix: The Still Unappreciated Components of the Adipose Tissue[END_REF].

We believe that the distribution tail approach could be further investigated to show convergence towards the solutions to the second order Lifshitz-Slyozov equation. The asymptotic behaviour of solutions to the second order Lifshitz-Slyozov model will be investigated in future works.

• T rescaling value of the time scale,

• λ rescaling value of λ.

We previously denoted by Λ the typical size of a vesicle. Hence it plays the role of a rescaling value and should be treated as so.

Recalling hypothesis (H4), we should make an analogous hypothesis on (a i ) i≥0 and (b i ) i≥0 so that after rescaling, hypothesis (H4) holds. Hence assume there exists A, B, δ > 0 such that : a i = Aa(iδ) and b i = Bb(iδ), for all i ≥ 0.

(H'4)

Now, we introduce the rescaled variables :

āi = a i Ā , ∀i ≥ 0, bi = b i B , ∀i ≥ 1, t = t T , ci ( t) = c i ( t T ) C , ∀i ≥ 0, L( t) = l( t T )Λ.
The quantity L therefore describes the total amount of lipids in the medium instead of the number of lipid vesicles. To have the proper convergence for the functions a and b later on, we relate the rescaling variables Ā and B with the constant in hypothesis (H'4) :

Ā = A and B = B.
We compute from equation ( 13) the derivative of ci for i ≥ 1 :

dc i dt ( t) = T C dc i dt ( t T )
= T C a i-1 l( t T )Λ l( t T )Λ + κ c i-1 ( t T ) -(a i l( t T )Λ l( t T )Λ + κ + b i )c i ( t T ) + b i+1 c i+1 ( t T ) = Ā T āi-1 L( t) L( t) + κ ci-1 ( t) -āi L( t) L( t) + κ ci ( t) -B T bi ci ( t) -bi+1 ci+1 ( t) .

The derivative of c0 writes as : dc 0 dt ( t) = -Ā T ā0 L( t) L( t) + κ c0 ( t) + B T b1 c1 ( t)

and the conservation equation for lipids as :

L( t) + CΛ i≥1 ic i ( t) = λ.
We now relate all the rescaling constants to a single variable ε > 0, such that :

Ā T = B T = 1 ε , CΛ = ε 2 and δ = ε.
At last, we drop the bar above the variables and replace it with ε as superscript to show the dependency of the solution on ε.

Figure 3 :

 3 Figure 3: Plot of function L → Φ(L) for L ∈ [10 -12 , 1] with functions a and b defined at Eq.(4) and[START_REF] Calvez | Prion dynamics with size dependency-strain phenomena[END_REF] and parameters given at Table1.

Figure 5 : 2 (

 52 Figure 5: Distributions of adipocytes with respect to size, i.e. amount of lipids, starting with λ = 3.5 and initial distribution g 0 (x) = C exp -1 2x -3 0.5 2

Figure 7 : 2 (

 72 Figure 7: On the left : distributions of adipocytes with respect to size, i.e. amount of lipids, starting with λ = 7 and initial distribution g 0 (x) = C exp -1 2x -6 0.5 2

Figure 8 :

 8 Figure 8: On top : plot of function λ → Φ -1 (λ), the inverse function of the one displayed at Fig.3, and type of modality of the stationary solution with respect to the value of λ. On bottom : Normalized stationary solutions for different values of λ, the stationary velocity as a function of x is represented bellow each solution. Top left : left unimodal; top right : bimodal ; bottom left : central unimodal ; bottom right : left unimodal.

Figure 11 :

 11 Figure 11: Numerical solution for the first order Lifshitz-Slyozov model (11) (in dotted blue line) compared to the stationary solution of the Lifshitz-Slyozov diffusive model (17) (in orange plain line) with the same parameters and same initial condition (displayed in black dashed -dotted line).The solution to the first order Lifshitz-Slyozov model is expected to converge to a Dirac mass and is displayed for a time before reaching the asymptotic profile.

Figure 13 :

 13 Figure 13: Asymptotic profiles for the first order Lifshitz-Slyozov model (11) with m = 1 and λ = 2. Left : f 0 (x) = C1 [∆x,1] (x). Right : f 0 (x) = C1 [0.5,1] (x). The difference in the initial conditions leads to different profiles. To observe bimodality the parameter β was changed to β = 100 in both cases.

Figure 14 :

 14 Figure 14: Plot of function L → Φ(L) with a(x) = 1 and b(x) = (x+1) 2/3 . In that case, lim L→0 + Φ(L) ∼ 0.025 > 0 and the existence of smooth stationary solutions for values of λ such that λ < λ 0 is not guaranteed.

Figure 15 :

 15 Figure 15: Case when a(x) = 1 and b(x) = (x + 1) 2/3 and λ < λ 0 . On the left : time evolution of the size distribution with respect to size; on the right : time evolution of the external lipid concentration.

To begin with, we check that the asymptotic profile obtained with the time evolution of the solution thanks to the previous described scheme coincides with the stationary solution of Sec.4.2.

First, one may assume that given an initial condition (g 0 , L 0 ), the asymptotic behaviour of the system is governed by the two parameters m and λ. This means that given two initial conditions (g 0 1 , L 0 1 ) and (g 0 2 , L 0 2 ) such that m 1 = m 2 and λ 1 = λ 2 , the stationary solutions are equal. In Figure 4, both initial conditions are Gaussian functions centered at x 1 = 1 and x 2 = 3 with m 1 = m 2 and initial values L 0 1 and L 0 2 are chosen so that λ 1 = λ 2 . We indeed observe that the asymptotic profile is the same for these two initial conditions. , from two initial conditions (g 0 1 , L 0 1 ) and (g 0 2 , L 0 2 ) such that m 1 = m 2 and λ 1 = λ 2 . Parameters of the system are given at Table 1. Note that the stationary solution and the two asymptotic distributions are superimposed.

Annex : the rescaling procedure

We start from Becker-Döring system, which may be written as :

The rescaling procedure we use is akin to the one in [START_REF] Deschamps | Quasi steady state approximation of the small clusters in Becker-Döring equations leads to boundary conditions in the Lifshitz-Slyozov limit[END_REF][START_REF] Vasseur | The Beker-Döring System and Its Lifshitz-Slyozov Limit[END_REF]. We introduce the following scaling constants :

• Ā rescaling value of (a i ) i≥0 ,

• B rescaling value of (b i ) i≥1 ,

• C rescaling value of (c i ) i≥0 ,