Léon Matar

Babacar Lèye

Babacar Leye

Analysis and numerical simulation of a polymerization model with aggregation and Ostwald ripening for the growth phenomenon

Keywords: Polymerization process, hyperbolic-parabolic coupling, coagulation operator, numerical simulation, anti-diffusive method, finite volume method Mathematics Subject Classification (2000) MSC 58F15, MSC 58F17, MSC 53C35

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Introduction

We are interested in the polymerization process when a population of polymers is immersed in a bath of monomers with possible growth by gain or loss of monomers and possible coalescence by a coagulation mechanism. This type of models for polymer (macro-particles) and monomer mixtures is theoretically based on the standard Lifshitz-Slyozov system [START_REF] Lifschitz | The kinetics of precipitation from supersaturated solid solutions[END_REF][START_REF] Landau | Cinétique Physique[END_REF] which is commonly used to understand the so called Ostwald ripening phenomenon. This phenomenon refers to the mecanism of the growth of larger polymers at the expense of smaller ones due to the minimization of the interfacial surface energy of the system which governs the processes of gain and loss of monomers. Initially, the standard Lifschitz-Slyozov system describes the dynamics of a solution of macro-particles, characterized by their size distribution function f (t, ξ), t, ξ ≥ 0, which interacts with monomers (free size particles) characterized by their concentration c(t), t ≥ 0. This interaction is governed by kinetic coefficients named a, b and represent the rate at which monomers are added to or removed from polymers. Of course the rates a and b depend on the considered processes of precipitation/dissolution. In [START_REF] Lifschitz | The kinetics of precipitation from supersaturated solid solutions[END_REF], Lifshitz and Slyozov propose the choice a(ξ) = ξ 1/3 and b(ξ) = 1 for the assumption that mass transfer in the interaction is based on monomer diffusion. Thanks to this choice, the growth rate V (t, ξ) = a(ξ)c(t) -b(ξ) vanishes at all time t ≥ 0 which means there exists a critical size, ξ crit , such that V (t, ξ crit ) = 0. So all polymers of size less than ξ crit shrink and those of size greater than ξ crit grow. Other choices of kinetic coeffients and their physical meaning can be found in [START_REF] Sagalovich | Diffusive decomposition of solid solutions[END_REF]. To go beyond the classical Lifshitz-Slyozov system in order to understand the asymptotic trend of the family of self-similar solutions for the macro-particles distribution functions, other works focus on variants of this system for instance when possible coalescence (coagulation) of large polymers is taken into account. It is the case in [START_REF] Laurençot | The Lifshitz-Slyozov Equation with encounters[END_REF] where in addition to the encounter process, the evolution equation for the volume (size) distribution function of macro-particle is coupled with a conservation of matter equation involving the monomers concentration. The author analyzes in [START_REF] Laurençot | The Lifshitz-Slyozov Equation with encounters[END_REF] the well-posedness of his model by performing compactness properties of the solutions to approximating equations for the weak topology L 1 (0, +∞). Other existence proofs exist like the one done in [START_REF] Collet | Lifshitz-Slyozov equations: the model with encounters[END_REF] using characteristic curves. One can refer to [START_REF] Lifschitz | The kinetics of precipitation from supersaturated solid solutions[END_REF][START_REF] Sagalovich | Diffusive decomposition of solid solutions[END_REF][START_REF] Collet | Lifshitz-Slyozov equations: the model with encounters[END_REF][START_REF] Herrmann | Self-similar solutions for the LSW model with encounters[END_REF][START_REF] Goudon | Simulations of the Lifschitz-Slyosov Equations: The role of coagulation terms in the asymptotic behavior[END_REF] for various modified Lifshitz-Slyozov system where coalescence of polymers is taken into account. For the parabolic correction of the Lifshitz-Slyozov model derived from the Becker-Döring model, an infinite system of ODEs when polymers have discrete sizes, see [START_REF] Vasseur | The Becker-Döring System and Its Lifshitz-Slyozov Limit[END_REF][START_REF] Joseph | On a Diffusive Version of the Lifschitz-Slyozov-Wagner Equation[END_REF][START_REF] Dady | Kinetics of Ostwald ripening[END_REF][START_REF] Hariz | A modified version of the Lifshitz-Slyozov model[END_REF][START_REF] Niethammer | A Scaling Limit of the Becker-Döring Equations in the Regime of Small Excess Density[END_REF]. In paper [START_REF] Goudon | The Lifshitz-Slyozov equations with space-diffusion of monomers[END_REF], the authors propose another variant derived from the Lifschitz-Slyozov standard system by assuming that monomers are subject to space diffusion. This assumption is motivated by the physical description of the sintering processes. They perform a rigorous proof of the existence and uniqueness of the polymers distribution function in the case of a free coagulation mechanism. In this paper, we wish to understand the behavior of the Lifschitz-Slyozov system when monomers are assumed to follow a diffusion equation as in [START_REF] Goudon | The Lifshitz-Slyozov equations with space-diffusion of monomers[END_REF] and polymers are allowed to merge (coalesce) with each other as in [START_REF] Goudon | Simulations of the Lifschitz-Slyosov Equations: The role of coagulation terms in the asymptotic behavior[END_REF]. Let us remind here that the main conjecture in [START_REF] Goudon | Simulations of the Lifschitz-Slyosov Equations: The role of coagulation terms in the asymptotic behavior[END_REF] concerning the coagulation effect is that it allows to keep the right expected behavior of the polymers distribution function. In our study, no diffusion is taken into account for the polymers distribution function. Some extensions where a diffusion term is added to take into account fluctuations exist (see [START_REF] Meerson | Fluctuations provide strong selection in Ostwald ripening[END_REF][START_REF] Niethammer | On Screening Induced Fluctuations in Ostwald Ripening[END_REF] and references therein). In our modeling, we deliberately choose a constant coagulation kernel and there is no physical motivation behind. All these variants of the Lifshitz-Slyozov standard system are motivated by the amount of applications related to the polymerization process which is often use in industrial processes for producing metallic alloys such as stainless steel or nickel alloys used in jewelry [START_REF] Landau | Cinétique Physique[END_REF][START_REF] Lifschitz | The kinetics of precipitation from supersaturated solid solutions[END_REF]; in population dynamics when for instance one is interested in the synthesis of DNA stand which is made by sequences of deoxyribonucleotides or when one is interested in the mechanism of Prion diseases where the polymerization of Prion protein is a central event [START_REF] Pierre | The Shape of the Polymerization Rate in the Prion Equation[END_REF][START_REF] Calvez | Prion dynamic with size dependency-strain phenomena[END_REF][START_REF] Calvez | Size distribution dependence of prion aggregates infectivity[END_REF][START_REF] Greer | A mathematical analysis of the dynamics of prion proliferation[END_REF][START_REF] Laurençot | Well-posedness for a model of prion proliferation dynamics[END_REF]. Recently several papers in Alzheimer disease modeling refer to such process of polymerization with aggregation (coagulation) in order to describe the ways of oligomerization and fibrillation which are considered to our knowledge as the central mechanisms of the evolution of the disease [START_REF] Helal | Alzheimers disease: analysis of a mathematical model incorporating the role of prions[END_REF][START_REF] Sakono | Amyloid oligomers: formation and toxicity of a oligomers[END_REF]. In our study, we denote as in [START_REF] Goudon | The Lifshitz-Slyozov equations with space-diffusion of monomers[END_REF] the size distribution function of polymers by f (t, x, ζ) with t ≥ 0, x ∈ Ω ⊂ R 2 and ζ ≥ 0 the time, space and size variables respectively and by c(t, x) the concentration of monomers at time t ≥ 0 and position x. So the considered model stands as follows:

               ∂ t f (t, x, ξ) + ∂ ξ ((a(ξ)c(t, x) -1)f (t, x, ξ)) = λQ(f )(t, x, ξ), t ≥ 0, x ∈ Ω, ξ ≥ 0, ∂ t c(t, x) + ∞ 0 ξf (t, x, ξ) dξ -∆ x c(t, x) = 0, t ≥ 0, x ∈ Ω, ∂ ν c = ∇c • ν = 0, on ∂Ω, (1.1)
where Ω is a smooth bounded domain, with boundary ∂Ω and ν(x) to be the outward unit normal vector at the point x ∈ ∂Ω. The parameter λ is a scale constant parameter for the coalescence process. The dynamics of the polymers is governed by the growth rate a(ξ)c(t, x) -b(ξ) where, in our case, we choose b(ξ) = 1.

The kinetic coefficients a and b are non negative functions and quantify the rates at which monomers are added to or removed from the polymers. Note that the precise expression of the coefficients depends on the modeling of mass transfer between monomers and polymers. In the case where mass transfer is based on monomer diffusion, [START_REF] Lifschitz | The kinetics of precipitation from supersaturated solid solutions[END_REF] proposes the choice a(ξ) = ξ 1/3 and b(ξ) = 1. One can refer to [START_REF] Sagalovich | Diffusive decomposition of solid solutions[END_REF] for other formulas for the kinetic coefficients. In any time, there exists a critical size ξ crit such that the growth rate vanishes. So all polymers of size smaller than ξ crit shrink while the larger ones grow.

The operator Q(f ) describes the encounters between polymers and is written

Q(f )(t, x, ξ) = 1 2 ξ 0 K(ξ -ξ , ξ )f (t, x, ξ -ξ )f (t, x, ξ ) dξ - ∞ 0 K(ξ, ξ )f (t, x, ξ)f (t, x, ξ ) dξ = 1 2 Q + (f )(t, x, ξ) -f (t, x, ξ) L f (t, x) where L f (t, x) = ∞ 0 K(ξ, ξ )f (t, x, ξ ) dξ . (1.2)
The first term in the right hand side of the equality in (1.2) represents the gain of polymers of size ξ when one of size ξ -ξ encounters another of size ξ smaller than ξ (ξ ≤ ξ) with the rate K(ξ -ξ , ξ ); while the second term represents the loss of polymers of size ξ when it encounters another of any size ξ with the rate K(ξ, ξ ). The coagulation kernel

K(ξ 1 , ξ 2 ) is a symmetric function from [0, ∞[ 2 into R + .
It represents the rate of coalescence between particles of size ξ 1 and ξ 2 . This coagulation operator is widely used for the dynamics of cluster growth especially in most coagulationfragmentation type models. For instance in [START_REF] Laurençot | On a class of continuous coagulation-fragmentation equations[END_REF], the author use the Smoluchowski operator to describe coalescence and breakage for a system of particles. He analyzed the existence solution for a class of coagulation and fragmentation kernels. We mention Norris, [START_REF] Norris | Smoluchowski's coagulation equation: Uniqueness, nonuniqueness and a hydrodynamic limit for the stochastic coalescent[END_REF], who obtains an existence result in the pure coagulation equation, by assuming that there exists a sub-linear function

φ on ]0, ∞[, such that K(ξ, ξ ) ≤ φ(ξ)φ(ξ ) and ∞ 0 φ 2 (ξ)f 0 (x, ξ)dξ < ∞.
He also investigates the case where K(ξ, ξ ) blows up as ξ → 0 or ξ → 0 and the case where any local regularity conditions is assumed on K. Of course this encounter mechanism doesn't change the total mass of the polymers ∞ 0 ξQ(f )(t, x, ξ) dξ = 0 and it decreases the number of polymers by collision

∞ 0 Q(f )(t, x, ξ) dξ ≤ 0.
Here for analytical convenience we assume that the encounter rate of polymers is constant.

We assume that the problem (1.1) is endowed with the following initial condition

f (t = 0, x, ξ) = f 0 (x, ξ) ≥ 0, c(t = 0, x) = c 0 (x).
In the following, we will refer to the continuity equation of (1.1) as (1.1) 1 and to the monomer diffusion equation as (1.1) 2 . The mathematical analysis of this kind of interactions between polymers and monomers is quite intriguing in view of the classical paper of Lifshitz and Slyozov [START_REF] Lifschitz | The kinetics of precipitation from supersaturated solid solutions[END_REF]. Many works discuss the existence and uniqueness results in different functional frameworks and one can refer to [START_REF] Collet | Some remarks on the large-time asymptotic of the Lifshitz-Slyozov equations[END_REF][START_REF] Laurençot | Weak solutions to the Lifschitz-Slyozov-Wagner equation[END_REF][START_REF] Laurençot | The Lifschitz-Slyozov-Wagner equation with conserved total volume[END_REF][START_REF] Niethammer | On the Initial-value Problem in the Lifshitz-Slyozov-Wagner Theory of Ostwald Ripening[END_REF][START_REF] Niethammer | Well-posedness for measure transport in a family of nonlocal domain coarsening models[END_REF] for more details. Numerical simulations exist for this kind of interaction without space variables and one can refer to [START_REF] Carrillo | A Numerical Study on Large-Time Asymptotics of the Lifshitz-Slyozov System[END_REF][START_REF] Chen | The dynamics of transient Ostwald ripening[END_REF][START_REF] Goudon | Simulations of the Lifschitz-Slyosov Equations: The role of coagulation terms in the asymptotic behavior[END_REF].

In our specific case we investigate the problem (1.1) and look to the influence of monomer spacediffusion and the encounter phenomenon on the evolution of the polymer size distribution. The aim of this paper is to analyze theoretically the well posedness and the numerical issues of (1.1).

Without loss of generality, we assume

Hypothesis 11 The data satisfy

• c 0 ∈ L ∞ (Ω), • f 0 ∈ L ∞ (Ω; L 1 ((0, ∞), (1 + ξ) dξ)).
The kinetic coefficients satisfy

• b = 1,
• a is non decreasing with a(0) = 0 and a(+∞) = +∞,

• a ∈ C 0 ([0, ∞)) ∩ C 1 ((0, ∞))
and for any ξ 0 > 0, there exists L a > 0 such that 0 ≤ a (ξ) ≤ L a for any ξ ≥ ξ 0 > 0.

One can make the obvious remark that the gain rate, a(ξ) = ξ 1/3 , proposed in [START_REF] Lifschitz | The kinetics of precipitation from supersaturated solid solutions[END_REF] fulfills the previous hypothesis. That is the case for the rates of the form a(ξ) = O( (ξ)) as used in [START_REF] Vasseur | The Becker-Döring System and Its Lifshitz-Slyozov Limit[END_REF]. Formally, by simple integration of the continuity equation (1.1) 1 against the size ξ, it is easy to remark that the diffusion equation can be recasted in the following simpler form In what follows we put our analysis in (1.1) where the diffusion equation (1.1) 2 is replaced by (1.3) and throughout the paper we investigate the following main result Theorem 1.1 From Hypothesis 11 and previous chosen properties for the coagulation operator Q(f ), the system (1.1) where the diffusion equation (1.1) 2 is replaced by

∂ t c(t, x) + c(t, x) ∞ 0 a(ξ)f (t, x, ξ) dξ = ∆ x c(t, x) + ∞ 0 f (t, x, ξ) dξ. ( 1
(1.3) has a unique "mild" solution (c, f ) with c ∈ L 2 ([0, T ]; H 1 (Ω)) ∩ C([0, T ]; L 2 (Ω)) and f ∈ L ∞ ([0, T ]; L 1 (R + , (1 + ξ)dξ)).
Most of the technical tools used in the proof of this main result comes from [START_REF] Goudon | The Lifshitz-Slyozov equations with space-diffusion of monomers[END_REF] with appropriate modifications. So we organize the paper as follows. In Section 2 we establish the general tools related to the monomer time evolution equation and polymer continuity equation by using a slight modification of Hypothesis 11 which consists in assuming the kinetic coefficient a to be Lipschitz continuous on [0, ∞). In Section 3 we first present the analysis of the well-posedness of our system by applying a double fixed point theorem in the case where a is Lipschitz continuous on [0, ∞) and we generalize the well-posedness in the conditions of hypothesis 11. Section 4 is devoted to the 2D numerical simulations of the model.

2 General tools related to polymer and monomer equations

Tools related to the monomer diffusion equation

The well-posedness of the equation (1.3) is obtained as a consequence of the following result on parabolic equations. Proposition 2.1 Let 0 < T < ∞. Consider non negative functions A, B ∈ L ∞ ((0, T ) × Ω). If we assume that there exists

N 0 ∈ R + such that 0 ≤ B(t, x) ≤ N 0 ∀ (t, x) ∈ (0, T ) × Ω, then for all c 0 ∈ L 2 (Ω) there exists a unique solution c ∈ L 2 ((0, T ); H 1 (Ω)) ∩ C([0, T ]; L 2 (Ω)) of the parabolic equation      ∂ t c(t, x) + A c(t, x) -∆ x c(t, x) = B(t, x), in (0, T ) × Ω, ∂ ν c = ∇c • ν = 0, on (0, T ) × ∂Ω , c(t = 0, x) = c 0 (x), on Ω .
(2.1)

In addition, for all 0 ≤ t ≤ T < ∞ there exists a constant C T such that

Ω |c(t, x)| 2 dx ≤ C T ; t 0 Ω |∇ x c(s, x)| 2 dx ds ≤ C T .
And, if 0 ≤ c 0 ∈ L ∞ (Ω), then the solution c satisfies the maximum principle relation

0 ≤ c(t, x) ≤ K T with K T = K T (N 0 , c 0 ∞ , T ).
Proof The proof of Proposition 2.1 is very classical and is based on the analog of the Lax-Milgram theorem for parabolic equations. For more details see [START_REF] Goudon | The Lifshitz-Slyozov equations with space-diffusion of monomers[END_REF].

Tools related to the polymer transport equation

Here we adopt the same reasoning as in the paper [START_REF] Goudon | The Lifshitz-Slyozov equations with space-diffusion of monomers[END_REF] by neglecting the space variable which appears only as a parameter in the continuity equation. So, making abstraction to the space variable "x" and taking λ = 1, we consider the following transport equation

   ∂ t f (t, ξ) + ∂ ξ (V f )(t, ξ) = Q(f )(t, ξ), t, ξ > 0, V (t, ξ) = a(ξ)c(t) -1, t, ξ > 0, f (t = 0, ξ) = f 0 (ξ), ξ > 0, (2.2)
such that the kinetic coefficients are required to satisfy for a given locally bounded c (0

≤ c ≤ K T ) the following assumptions • a ∞ ≤ L a and a ∈ L ∞ (R), • a(0) = 0,
where

M T = La K T .
With these assumptions, it is straightforward to deduce that V satisfies

V (t, 0) ≤ 0, V (t, ξ) ≤ M T ξ, |V (t, ξ)| ≤ M T (1 + ξ) and |V (t, ξ) -V (t, ξ )| ≤ M T |ξ -ξ |.
So we can introduce the characteristic curves associated to

V      d ds E (s; t, ξ) = V s, E (s; t, ξ) , E (t; t, ξ) = ξ. (2.3)
We define the mild formulation which arises by integrating (2.2) along the characteristic curves associated to V . So, the solution of (2.2) in its mild formulation is given by

f (t, ξ) = f 0 (E (0; t, ξ)) J(0; t, ξ) + t 0 Q(f )(s; E (s; t, ξ)) J(s; t, ξ) ds (2.4)
which is obtained by using the differential equation fulfilled by f (s, E (s; t, ξ)) and where J is the jacobian of the change of variables ξ → E (s; t, ξ) and is defined by

J(s; t, ξ) = ∂ ξ E (s; t, ξ) = exp - t s ∂ ξ V (σ, E (σ; t, ξ)) dσ ≥ 0.
The previous assumptions on the kinetic coefficients allows to infer the following results on the characteristics and the distribution function f . Lemma 2.1 If the previous hypothesis on the kinetic coefficients is satisfied, then

1. ∀ t ≥ 0, E (0; t, 0) ≥ 0, 2. ∀ t ≥ 0, lim ξ→+∞ E (0; t, ξ) = +∞, 3. ∀ 0 ≤ t ≤ T < ∞ and ξ ≥ 0, there exists L T > 0 such that E (t; 0, ξ) ≤ L T ξ.
If f is a solution of the characteristic view point, that is to say it satisfies equation (2.4), then 4. The proof of the well-posedness of the problem (1.1) is a straight adaptation of the one given in [START_REF] Goudon | The Lifshitz-Slyozov equations with space-diffusion of monomers[END_REF]. The main idea consists of performing a double fixed point strategy under the assumptions of Section 2.2. The first fixed point strategy is based on the polymers distribution function and the second one on the monomers diffusion equation.

f 0 ≥ 0 implies f (t, ξ) ≥ 0, 5. f 0 ∈ L 1 (0, ∞); (1 + ξ) dξ implies that ∀ t ≥ 0, f (t, •) ∈ L 1 (0, ∞); (1 + ξ) dξ and more precisely one has f ∈ C 0 ([0, T ]; L 1 (R + )), ∞ 0 f (t, ξ) dξ ≤ ∞ 0 f 0 (ξ) dξ and ∞ 0 ξf (t, ξ) dξ ≤ e M T T ∞ 0 ξf 0 (ξ) dξ + M T T ∞ 0 f 0 (ξ) dξ . 6. If we assume ∂ ξ V (t, ξ) = c(t)∂ ξ a(ξ) ≥ 0 ∀ ξ ≥ 0 and t ∈ [0, T ] then f 0 ∈ L 1 ((0, ∞)) implies that f ∈ L ∞ ((0, T ); L 1 (0, ∞)) and one can deduce f L ∞ (0,T ;L 1 (0,∞)) ≤ f 0 L 1 (0,∞) + Q L 1 ((0,T )×(0,∞)) . Proof For the proof of Lemma 2.

First fixed point strategy for polymers distribution function

Let us consider a given c such that c ∈ C 0 ((0, T ) × Ω), 0 ≤ c(t, x) ≤ K T . The choice of T is specified later. We define the application

F : g → F(g)(t, x, ξ) = f (t, x, ξ) such that ∂ t f + ∂ ξ (V f ) + K f = Q(g) + Kg, in (0, T ) × Ω, f |t=0 = f 0 , on Ω, (3.5)
where the parameter K is chosen so that

Q(g) + K g :≥ 0 when g ∈ f ∈ L ∞ ((0, T ) × Ω; L 1 (R + )); f ≥ 0, f (t, x, •) L 1 (R+) ≤ M := C T,Ω .
The choice of the constant K is possible and depend on the bound M . Indeed let

Q(g) + K g :≥ 0 ⇒ Q + (g) -L g g + K g ≥ 0 ⇒ Q + (g) + (K -L g )g ≥ 0.
Knowing that Q + (g) ≥ 0, then the implication is true if K -L g ≥ 0 which means K ≥ L g ≥ g L 1 (R+) . So we need to choose K such that K ≥ M .

The solution of (3.5) along the characteristic curves when neglecting the space variable (which acts just as a parameter) is written as

F(g)(t, ξ) = f 0 (E (0; t, ξ)) J(0; t, ξ)e -K t + t 0 Q(g)(s; E (s; t, ξ))+K g(s, E (s; t, ξ)) J(s; t, ξ)e -K(t-s) ds.
Looking at the L 1 norm we get thanks to the fact that J L ∞ ≤ 1,

F(g)(t, •) L 1 (R+) ≤ f 0 L 1 (R+) + t 0 Q(g)(s, •) L 1 (R+) + K g(s, •) L 1 (R+) ds ≤ f 0 L 1 (R+) + T g L ∞ ((0,T );L 1 (R+)) K + 1 2 g L ∞ ((0,T );L 1 (R+)) .
Knowing that F(g) = f , then we can fix M such as

f 0 L 1 (R+) ≤ M 2
and choose a small T such that

T (K + 2M ) ≤ 1 2 . So, one can choose M (f 0 ) so that 0 ≤ K ≤ 1 2T -2M and K ≥ M.
With this choice of K, one has

F(g) ∈ C T,Ω if g ∈ C T,Ω .
Consider f 1 and f 2 defined by f 1 =: F(g 1 ) and f 2 =: F(g 2 ) two solutions of the problem (3.5) associated to g 1 , g 2 ∈ C T,Ω and with the same initial condition. Following the characteristic curves, one deduces

(f 1 -f 2 )(t) L 1 (R+) ≤ t 0 (Q(g 1 ) -Q(g 2 ))(s) L 1 (R+) + K (g 1 -g 2 )(s) L 1 (R+) ds, (3.6)
since the coagulation operator satisfies, see [6, Lemma 3], the relation

Q(g 1 ) -Q(g 2 ) L 1 (R+) ≤ 1 2 g 1 L 1 (R+) + g 2 L 1 (R+) g 1 -g 2 L 1 (R+) .
From (3.6), we get:

(f 1 -f 2 )(t) L 1 (R+) ≤ T K + 1 2 ( g 1 L ∞ ((0,T );L 1 (R+)) + g 2 L ∞ ((0,T );L 1 (R+)) ) g 1 -g 2 L ∞ ((0,T );L 1 (R+)) .
The fact that g 1 and g 2 belong to C T,Ω implies that

(f 1 -f 2 )(t) L 1 (R+) ≤ T (K + 2M ) g 1 -g 2 L ∞ ((0,T );L 1 (R+)) .
So F is a contraction and this gives the existence and uniqueness of f satisfying

∂ t f + ∂ ξ (V f ) = Q(f ) and f ∈ L ∞ ([0, T ]; L 1 (R + )).
Here one remarks that there is a constraint on T depending implicitly on the bounds of the initial distribution function. To go further in time, the trick is to take f (T, x, •) as the new initial distribution function and reiterate the process on [T, 2T ] then on [2T, 3T ] and so on. This achieves the proof of the well-posedness for polymers continuity equation.

Remark 3.1 For initial data satisfying ξf 0 ∈ L 1 (R + ), using Proposition 2.1 one obtains ξf ∈ L ∞ ((0, T ); L 1 (R + )) and thanks to the hypothesis on the operator Q the following relation on the variation of polymers total mass holds

d dt ∞ 0 ξf dξ = ∞ 0 V f dξ.

Second fixed point strategy for monomers diffusion equation

This fixed point strategy follows exactly the same reasoning as in [START_REF] Goudon | The Lifshitz-Slyozov equations with space-diffusion of monomers[END_REF]. The analysis is based on equation (1.3) and the hypothesis 11.

Let

Q T = [0, T ] × Ω for fixed T . We note N 0 = sup x∈Ω ∞ 0 f 0 (x, ξ) dξ < ∞ and we put K T = c 0 L ∞ +
N 0 T the constant obtained in the maximum principle (Proposition 2.1). For the set

E T = c ∈ L 2 (Q T ) such that 0 ≤ c(t, x) ≤ K T and c ∈ C 0 (Q T ) ,
we define the application

C : E T -→ L 2 (Q T ) c -→ C (c) = c, with c satisfying        ∂ t c(t, x) + c(t, x) ∞ 0 a(ξ)f (t, x, ξ) dξ -∆ x c(t, x) = ∞ 0 f (t, x, ξ) dξ, in (0, T ) × Ω, ∂ ν c = 0, on (0, T ) × ∂Ω, c |t=0 = c init ,
on Ω where f solves

∂ t f (t, x, ξ) + ∂ ξ (a(ξ)c(t, x) -1)f (t, x, ξ) = Q(f )(t, x, ξ), in (0, T ) × Ω × (0, ∞), f |t=0 = f 0 , on Ω × (0, ∞).
We can state the following results on the well-posedness of the application C as in Lemma 3.1 below and on the continuity of C as in Proposition 3.1 below.

Lemma 3.1 The application C given above is well-defined and

C (E T ) is compact in L 2 (Q T ).
Proof The assumptions of Section 2.2 and c ∈ E T allow to apply the results in Section 2.1 by using the characteristics approach with the x variable acting as a space parameter. So one obtains f (t, x, ξ) = f 0 (x, E (0; t, x, ξ)) J(0, t, x, ξ)

+ t 0 Q(f )(s, x, E (s; t, x, ξ)) J(s, t, x, ξ) ds,
where E and J are associated to c. Thanks to Lemma 2.1 one has the relations

sup x∈Ω ∞ 0 f (t, x, ξ) dξ ≤ sup x∈Ω ∞ 0 f 0 (x, ξ) dξ = N 0 < ∞, and
sup x∈Ω ∞ 0 ξ f (t, x, ξ) dξ ≤ e M T T sup x∈Ω ∞ 0 ξf 0 (x, ξ) dξ + M T T sup x∈Ω ∞ 0 f 0 (x, ξ) dξ < ∞.
Then we deduce 0

≤ ∞ 0 a(ξ)f (t, x, ξ) dξ ∈ L ∞ (Q T ) and 0 ≤ ∞ 0 f (t, x, ξ) dξ ≤ N 0 .
So, the Proposition 2.1 gives the well posedness of the application C with

c = C (c) ∈ L 2 [0, T ]; H 1 (Ω) ∩ C [0, T ]; L 2 (Ω) and fulfilling 0 ≤ c(t, x) ≤ K T ⇒ C (E T ) ⊂ E T . Now we know that C (c) is bounded in L 2 [0, T ]; H 1 (Ω) and ∂ t c is also bounded in L 2 [0, T ]; H -1 (Ω) so the compact embedding H 1 (Ω) ⊂ L 2
(Ω) allows to apply J. Simon's compacity lemma [START_REF] Simon | Compact sets in the space L p (O, T ; B)[END_REF] and to deduce the compactness of Proof Using to the characteristic curves, one obtains

C (Q T ) in L 2 (Q T ).
f n (t, x, ξ) = f 0 (x, E n (0; t, x, ξ)) J n (0; t, x, ξ) + t 0 Q(s,
x, E n (s; t, x, ξ)) J n (s; t, x, ξ) ds, f (t, x, ξ) = f 0 (x, E (0; t, x, ξ)) J(0; t, x, ξ) + t 0 Q(s, x, E (s; t, x, ξ)) J(s; t, x, ξ) ds, where

J n (s; t, x, ξ) = exp - t s a E n (σ; t, x, ξ) cn (σ; x) dσ and J(s; t, x, ξ) = exp - t s a E (σ; t, x, ξ) c(σ; x) dσ .
In what follows, we give a sketch of the proof and let the readers refer to lemma 3.2 and lemma 3.3 of [START_REF] Goudon | The Lifshitz-Slyozov equations with space-diffusion of monomers[END_REF] for more details.

The first step is to show the following convergence results in L 2 (Q T ):

- ∞ 0 a(ξ)f n (t, x, ξ) dξ -→ ∞ 0 a(ξ)f (t, x, ξ) dξ when n → ∞, - ∞ 0 f n (t, x, ξ) dξ -→ ∞ 0 f (t, x, ξ) dξ when n → ∞.
These convergence results are obtained by quantifying the difference between two characteristic curves associated to two growth rates V 1 = a(ξ)c 1 (t, x) -1 and V 2 = a(ξ)c 2 (t, x) -1 for c 1 , c 2 ∈ E T . So, under the assumptions of Section 2.2 one deduces the following estimation

|E 1 -E 2 |(s; t, x, ξ) ≤ L T (1 + ξ) t s |c 1 -c 2 | 2 (σ, x) dσ 1/2 ∀ 0 ≤ s, t ≤ T < ∞. (3.7)
On one hand:

∞ 0 f n (t, x, ξ) dξ - ∞ 0 f (t, x, ξ) dξ = ∞ 0 f 0 (x, E n (0; t, x, ξ))J n (0; t, x, ξ) + t 0 Q(s, x, E n (s; t, x, ξ)) J n (s; t, x, ξ) ds dξ - ∞ 0 f 0 (x, E (0; t, x, ξ))J(0; t, x, ξ) - t 0 Q(s, x, E (s; t, x, ξ)) J(s; t, x, ξ) ds dξ = ∞ En(0;t,x,0) f 0 (x, δ) dδ + t 0 ∞ En(s;t,x,0) Q(s, x, δ) dδ ds - ∞ E (0;t,x,0) f 0 (x, δ) dδ - t 0 ∞ E (s;t,x,0) Q(s, x, δ) dδ ds = ∞ En(0;t,x,0) f 0 (x, δ) dδ - ∞ E (0;t,x,0) f 0 (x, δ) dδ + t 0 ∞ En(s;t,x,0) Q(s, x, δ) dδ ds - t 0 ∞ E (s;t,x,0) Q(s, x, δ) dδ ds . (3.8) It follows that for all s ∈ [0, T ] | ∞ 0 f n (t, x, ξ) dξ - ∞ 0 f (t, x, ξ) dξ|(t, x) ≤ |
En(0;t,x,0) E (0;t,x,0)

f 0 (x, δ) dδ| + | t 0 En(s;t,x,0) E (s;t,x,0) Q(s, x, δ) dδ ds| ≤ f 0 L ∞ (Q T ) |E n -E |(0; t, x, 0) + T Q L ∞ (Q T ×R+) |E n -E |(s; t, x, 0). Simple integration along Ω yields ∞ 0 (f n -f )(t, x, ξ) dξ 2 L 2 (Ω) ≤ 4 f 0 2 L ∞ (Q T ) Ω |E n -E | 2 (0; t, x, 0) dx + 4T 2 Q 2 L ∞ (Q T ×R+) Ω |E n -E | 2 (s; t, x, 0) dx.
Using the result (3.7) we obtain

∞ 0 (f n -f )(t, x, ξ) dξ 2 L 2 (Ω) ≤ 4L 2 T t 0 Ω | cn -c| 2 (σ, x) dx dσ f 0 2 L ∞ (Q T ) + 4T 2 L 2 T Q 2 L ∞ (Q T ×R+) t s Ω | cn -c| 2 (σ, x) dx dσ ≤ 4L 2 T f 0 2 L ∞ (Q T ) + 4T 2 L 2 T Q 2 L ∞ (Q T ×R+) t 0 Ω | cn -c| 2 (σ, x) dx dσ, since s ≥ 0.
On the other hand, we obtain using the characteristic curves

| ∞ 0 a(ξ)f n (t, x, ξ) dξ - ∞ 0 a(ξ)f (t, x, ξ) dξ| = | ∞ En(0;t,x,0) a E n (t; 0, x, δ) f 0 (x, δ) dδ + ∞ En(s;t,x,0) t 0 a E n (t; s, x, δ) Q(s, x, δ) ds dδ - ∞ E (0;t,x,0) a E (t; 0, x, δ) f 0 (x, δ) dδ - ∞ E (s;t,x,0) t 0 a E (t; s, x, δ) Q(s, x, δ) ds dδ| ≤ | En(0;t,x,0) E (0;t,x,0) a E n (t; 0, x, δ) f 0 (x, δ) dδ| + ∞ E (0;t,x,0) | a(E n ) -a(E ) (t; 0, x, δ)|f 0 (x, δ) dδ + | En(s;t,x,0) E (s;t,x,0) t 0 a E n (s; t, x, δ) Q(s, x, δ) ds dδ| + ∞ E (s;t,x,0) t 0 | a(E n ) -a(E ) (t; s, x, δ)||Q(s, x, δ)| ds dδ, where a(E n ) -a(E ) (t; s, x, δ) = a E n (t; s, x, δ) -a E (t; s, x, δ) , ∀ t, s.
This implies that

| ∞ 0 a(ξ)f n (t, x, ξ) dξ - ∞ 0 a(ξ)f (t, x, ξ) dξ| ≤ L a | En(0;t,x,0) E (0;t,x,0) E n (t; 0, x, δ)f 0 (x, δ) dδ| + L a ∞ E (0;t,x,0) |E n -E |(t; 0, x, δ)f 0 (x, δ) dδ + L a | En(s;t,x ,0)
To continue our computation we need estimations on all terms in the right hand side of the previous relation. So we write:

|E n (t; 0, x, δ)| = |E n (t; 0, x, δ) -E n (t; 0, x, E n (0; t, x, 0)) =0 | = | En(0;t,x,0) δ ∂ ξ E n (t; 0, x, ξ) dξ| = | En(0;t,x,0) δ exp - 0 t a (E n (σ; t, x, ξ)) cn (σ, x) dσ dξ| ≤ L T |δ -E n (0; t, x, 0)|. (3.10) The fact that δ ∈ [min(E n , E ), max(E n , E )] at the point (0; t, x, 0) implies that |E n (t; 0, x, δ)| ≤ L T |E n -E |(0; t, x, 0) with L T = exp(-T M T ). Then we deduce | En(0;t,x,0) E (0;t,x,0) E n (t; 0, x, δ)f 0 (x, δ) dδ| ≤ L T |E n -E |(0; t, x, 0) ∞ 0 f 0 (x, δ) dδ. (3.11)
Thanks to (3.7) we also have |Q(s, x, δ)| dδ ds

∞ E (0;t,x,0) |E n -E |(t; 0, x, δ)f 0 (x, δ) dδ ≤ L T t 0 | cn -c| 2 (σ, x) dσ 1/2 ∞ 0 (1 + δ)f 0 (x,
≤ L T t 0 |E n -E |(t; s, x, 0) ∞ 0 |Q(s, x, δ)| dδ ds ≤ L T sup s∈[0,t] (|E n -E |(t; s, x, 0)) ∞ 0 t 0 |Q(s, x, δ)| dδ ds ≤ L 2 T t s | cn -c| 2 (σ, x) dσ 1/2 Q L ∞ (Q T ) . (3.13)
For the last term of equation (3.9) we use the following majoration

∞ E (s;t,x,0) t 0 |E n -E |(t; s, x, δ) |Q(s, x, δ)| ds dδ ≤ sup s∈[0,T ] |E n -E |(t; s, x, δ) ∞ 0 t 0 |Q(s, x, δ)| ds dδ ≤ L T t s | cn -c| 2 (σ, x) dσ 1/2 t 0 ∞ 0 (1 + δ)|Q(s, x, δ)| dδ ds.
| ∞ 0 a(ξ)f n (t, x, ξ) dξ - ∞ 0 a(ξ)f (t, x, ξ) dξ| ≤ L a L 2 T t 0 | cn -c| 2 (σ, x) dσ 1/2 f 0 L ∞ (R+) + L a L T t 0 | cn -c| 2 (σ, x) dσ 1/2 f 0 L ∞ (R+,(1+ξ) dξ) + L a L 2 T t s | cn -c| 2 (σ, x) dσ 1/2 Q L ∞ (Q T ) + L a L T t s | cn -c| 2 (σ, x) dσ 1/2 Q L ∞ (Q T ,(1+ξ) dξ) ≤ L a L 2 T f 0 L ∞ (R+) + Q L ∞ (Q T ) t 0 | cn -c| 2 (σ, x) dσ 1/2 + L a L T f 0 L ∞ (R+,(1+ξ) dξ) + Q L ∞ (Q T ,(1+ξ) dξ) t s | cn -c| 2 (σ, x) dσ 1/2 ,
which achieves the proof of following convergence result

∞ 0 a(ξ)f n (t, x, ξ) dξ n→∞ -→ L 2 (Q T ) ∞ 0 a(ξ)f (t, x, ξ) dξ. Right now the convergence c n = C ( cn ) towards c = C (c) in L 2 (Q T )
space is obtained by an energy estimate based on the following system:

               ∂ t c n (t, x) + c n (t, x) ∞ 0 a(ξ)f n (t, x, ξ) dξ -∆ x c n (t, x) = ∞ 0 f n (t, x, ξ) dξ in (0, T ) × Ω, ∂ t c(t, x) + c(t, x) ∞ 0 a(ξ)f (t, x, ξ) dξ -∆ x c(t, x) = ∞ 0 f (t, x, ξ) dξ in (0, T ) × Ω, ∂ ν c n = 0 ; ∂ ν c = 0 on (0, T ) × ∂Ω, c n | t=0 = c |t=0 = c init on Ω,
what allows to obtain the relation

d 2 dt Ω (c n -c) 2 (t, x) dx + Ω |∇ x (c n -c)| 2 (t, x) dx + Ω (c n -c) 2 (t, x) ∞ 0 a(ξ)f (t, x, ξ) dξ dx = - Ω c n (c n -c) ∞ 0 a(ξ)f n - ∞ 0 a(ξ)f (t, x) dx + Ω (c n -c) ∞ 0 f n - ∞ 0 f (t, x) dx.
Applying the Cauchy and Young inequalities and the fact that c n ∈ E T one obtains that

d 2 dt Ω (c n -c) 2 (t, x) dx + Ω |∇ x (c n -c)| 2 (t, x) dx + Ω (c n -c) 2 (t, x) ∞ 0 a(ξ)f (t, x, ξ) dξ dx ≤ Ω (c n -c) 2 (t, x) dx + K 2 T 2 Ω ∞ 0 a(ξ)f n (t, x, ξ) dξ - ∞ 0 a(ξ)f (t, x, ξ) dξ 2 dx + 1 2 Ω ∞ 0 f n (t, x, ξ) dξ - ∞ 0 f (t, x, ξ) dξ 2 dx.
The gronwall lemma yields that:

Ω (c n -c) 2 (t, x) dx ≤ cte t 0 Ω | ∞ 0 a(ξ)f n dξ - ∞ 0 a(ξ)f dξ| 2 (s, x) dx ds + t 0 Ω | ∞ 0 f n dξ - ∞ 0 f dξ| 2 (s, x) dx ds , where the constant cte = cte(T, C 0 , f 0 L ∞ (Q T )
). This gives us the convergence result. So by the Schauder fixed point theorem, the application C admits a fixed point c = C (c) satisfying the equation

∂ t c(t, x) + c(t, x) ∞ 0 a(ξ)f (t, x, ξ) dξ -∆ x c(t, x) = ∞ 0 f (t, x, ξ) dξ.
The proof of the uniqueness of the solution c is obvious if one applies the same reasoning as in the proof of the continuity of C . The existence and uniqueness of the couple (c, f ) solution of the problem (1.1) is then proved in the case where the kinetic coefficient a is Lipschitz continuous on [0, ∞).

For the general case stated in Hypothesis 11 where the polymerization rate is singular, one remarks that the Lipschitz constant L a is not involved in the above estimations. So we can adapt the previous proof to the particular case of a non smooth kinetic coefficient a(ξ) as indicated in the hypothesis. This consideration allows to take into account the kinetic coefficients, a(ξ) = ξ 1/3 , used by Lifshitz and Slyozov [START_REF] Lifschitz | The kinetics of precipitation from supersaturated solid solutions[END_REF] to describe the kinetics of precipitation from supersaturated solid solutions. For the adaptation of this general case, we consider a sequence of smooth kinetic coefficients a n (ξ) which converges pointwise to a(ξ). So, if (f n , c n ) is the associated solution of the system with a n (ξ), then there exists a subsequence which converges to (f, c) associated to the kinetic coefficients a(ξ). Otherwise, from the pointwise convergence of a n , we can pass to the limit when n tends to infinity and obtain (f, c) as a weak solution with the kinetic coefficients a(ξ). This method of solution extension is explained in [START_REF] Laurençot | Weak solutions to the Lifschitz-Slyozov-Wagner equation[END_REF] for the case of homogeneous Lifshitz-Slyozov equation. The proof of the uniqueness follows the same reasoning as in [START_REF] Goudon | The Lifshitz-Slyozov equations with space-diffusion of monomers[END_REF] (see this reference for more details). Let us give some hints for the proof of the uniqueness. From two couples of solutions (c (1) , f (1) ) and (c (2) , f (2) ) with the same initial data of the system (1.1) 1 -(1.3) we have to show that c (1) = c (2) and f (1) = f (2) for a.e (t, x) ∈ (0, T ) × Ω and ξ > 0. One needs to derive an L 1 estimation of the monomer concentration by introducing the approximation, S (z) = z √ +z 2 , for the sign function. So, Z (z) = z 0 S (s) ds approaches |z| as tends to 0. From (1.3) we have (2) where

(∂ t -∆ x + A (1) )(c (1) -c (2) ) = B (1) -B (2) + (A (2) -A (1) )c
A (i) = ∞ 0 a(ξ)f (i) (t, x, ξ) dξ and B (i) = ∞ 0 f (i) (t, x, ξ) dξ i = {1, 2}
, we apply the composition with Z then integrate over Ω and obtain when → 0 the following

L 1 estimation Ω |c (1) -c (2) |(t, x) dx ≤ Ω |c (1) 0 -c (2) 0 |(t, x) dx + t 0 Ω |B (1) -B (2) |(s, x) dx (3.15) +K T t 0 Ω |A (2) -A (1) |(s, x) dx.

Knowing that c

(1)

0 = c (2)
0 , from the hypothesis of the same initial data, then the first term at the right hand side of the inequality vanishes. To prove that the two other terms vanish, one can adapt the same argument as in [START_REF] Laurençot | Weak solutions to the Lifschitz-Slyozov-Wagner equation[END_REF] for the problem without diffusion and coalescence. More details for this adaptation can be found in [START_REF] Goudon | The Lifshitz-Slyozov equations with space-diffusion of monomers[END_REF].

Numerical resolution of the equations

In this section we present a numerical scheme to simulate the modified Lifschitz-Slyosov model (1.1). The construction of the scheme gives some hints concerning stability issues. For calculating the approximate solution, we give a discretization in time, space and size of the polymers. We choose to discretize the monomers by an implicit finite difference scheme in time and space. To solve the equation of polymers, the time derivative is discretized by the finite differences method, and the length one by a finite volume scheme.

The discretization provides a set of points (t n ) n=1,...,Nt , N t ∈ N in the time interval ]0, T [, (ζ l ) l=1,...,N l , N l ∈ N in the size range ]0, L[, and nodes (x i,j ) i,j=1,...,N , N ∈ N of the domain Ω. For simplicity a constant discretization in length and space is used. Let ∆x be the space step, ∆ζ the length step and ∆t n the time step. We set

t 0 = 0, t n = t n-1 + ∆t n , t n+ 1 2 = t n + ∆t n 2 for n = 1, ..., N t , ζ l = l∆ζ, ζ l+ 1 2 = (l + 1 2
)∆ζ for l = 1, ..., N l .

Let us consider the discrete cells

C l =]ζ l-1/2 , ζ l+1/2 [ centered on ζ l .
We compute an approximative solution (c, f ) to the problem (1.1), denoted

c n i,j c(t n , x i,j ), f n i,j,l 1 ∆ζ C l f (t n , x i,j , ζ)dζ.
The scheme is based on the following time-splitting between advection and aggregation:

1. First, the homogeneous advection equation is solved:

∂ t f + ∂ ξ (V f ) = 0. ( 4.1)
2. Second, the solution obtained from the advection equation is used to build the aggregation operator:

∂ t f = λ Q(f ). (4.2)
Scheme for the advection of macro-particles The updating of the particles distribution follows by integrating the advection equation over the finite volume cells C l :

f n+1 i,j,l = f n i,j,l -ν n V n i,j,l+ 1 2 f n i,j,l+ 1 2 -V n i,j,l-1 2 f n i,j,l-1 2 , (4.3)
with V n i,j,l V (t n , x i,j , ζ l ) and ν n = ∆tn ∆ζ . It remains to define the interface fluxes f n i,j,l+ 1 2 . When investigating the large time behavior of solutions to transport equations, one faces the classical drawback of usual stable schemes: numerical diffusion, in its general sense, that is, the deterioration of profiles, the spreading of the initial data and the loss of the dependence cone. High-order schemes (such as WENO ...) are a cure, but are known to spread all the same for any initial data, in very long time. On the other hand, the anti-diffusive, limited downwind scheme of Ref [START_REF] Goudon | The Lifshitz-Slyozov equations with space-diffusion of monomers[END_REF], has the advantage of avoiding this spreading, keeping in arbitrary large time the profiles. We recall its basic principles in the case of advection with constant velocity. The idea is to take a downwind flux, under some stability constraints. The stability constraints are those that lead to a local numerical maximum principle. We adapt this idea to the present problem and refer to [START_REF] Goudon | Simulations of the Lifschitz-Slyosov Equations: The role of coagulation terms in the asymptotic behavior[END_REF] for more details.

Let us consider the following notation:

-

m n i,j,l+ 1 2 = min f n i,j,l , f n i,j,l+1 , M n i,j,l+ 1 2 = max f n i,j,l , f n i,j,l+1 .
-

If V n i,j,l , V n i,j,l+ 1 2 , V n i,j,l-1 2 > 0, then b n i,j,l+ 1 2 = 1 ν n |V n i,j,l | (f n i,j,l -M n i,j,l-1 2 ) + M n i,j,l-1 2 , B n i,j,l+ 1 2 = 1 ν n |V n i,j,l | (f n i,j,l -m n i,j,l-1 2 ) + m n i,j,l-1 2 , B n i,j,l+ 1 2 =          min B n i,j,l+ 1 2 , ν n m n i,j,l-1 2 |V n i,j,l-1 2 | + f n i,j,l ν n |V n i,j,l+ 1 2 | , if m n i,j,l-1 2 ≥ 0, B n i,j,l+ 1 2
, otherwise.

-

If V n i,j,l , V n i,j,l+ 1 2 , V n i,j,l-1 2 < 0, then: b n i,j,l+ 1 2 = 1 ν n |V n i,j,l | (f n i,j,l -M n i,j,l+ 1 2 ) + M n i,j,l+ 1 2 , B n i,j,l+ 1 2 = 1 ν n |V n i,j,l | (f n i,j,l -m n i,j,l+ 1 2 ) + m n i,j,l+ 1 2 , B n i,j,l+ 1 2 =          min B n i,j,l-1 2 , ν n m n i,j,l+ 1 2 |V n i,j,l+ 1 2 | + f n i,j,l ν n |V n i,j,l-1 2 | , if m n i,j,l+ 1 2 ≥ 0, B n i,j,l+ 1 2 , otherwise.
-

If the V n i,j,l , V n i,j,l+ 1 2 , V n i,j,l-1 2
do not have the same sign, we set

b n i,j,l+ 1 2 = B n i,j,l+ 1 2 =      f n i,j,l if V n i,j,l+ 1 2 > 0, f n i,j,l+1 if V n i,j,l+ 1 2 < 0. - µ n i,j,l+ 1 2 = max(m n i,j,l+ 1 2 , b n i,j,l+ 1 2 ), M n i,j,l+ 1 2 = min(M n i,j,l+ 1 2 , B n i,j,l+ 1 2
).

The following statement clarifies the principles on which the construction of the fluxes is based.

Proposition 4.1 [START_REF] Goudon | Simulations of the Lifschitz-Slyosov Equations: The role of coagulation terms in the asymptotic behavior[END_REF] Assume that the Courant-Friedrichs-Levy stability condition:

CF L = ν n max i,j,l (|V n i,j,l |, |V n i,j,l+ 1 2 |) ≤ 1 (4.4) holds, then [µ n i,j,l+ 1 2 , M n i,j,l+ 1 2 ] = ∅, for all l = 1, .., N l . Moreover if f n i,j,l+ 1 2 ∈ [µ n i,j,l+ 1 2 , M n i,j,l+ 1 2
], ∀l = 1, ..., N l , the following assertions hold:

1. The scheme (4.3) is consistent with (4.1).

2. If ∆ζ is sufficiently small then f n i,j,l ≥ 0 implies f n+1 i,j,l ≥ 0. Otherwise a restricted condition CF L ≤ 1 2 is requested to ensure the non-negativity.

Let us consider

f n * i,j,l = f n i,j,l -ν n V n i,j,l (f n i,j,l+ 1 2 -f n i,j,l- 1 2
), ∀l = 1, ..., N l .

(a) If V n i,j,l ≥ 0, then m n i,j,l-

1 2 ≤ f n * i,j,l ≤ M n i,j,l-1 2 . (b) If V n i,j,l ≤ 0, then m n i,j,l+ 1 2 ≤ f n * i,j,l ≤ M n i,j,l+ 1 2
.

The goal now consists in defining f n i,j,l+ 1 2 so that on the one hand, Proposition 4.1 holds, and on the other hand the numerical diffusion is reduced as possible. To this end, we adopt a downwinding approach. When V n i,j,l-1 2 > 0, V n i,j,l > 0 and V n i,j,l+ 1 2 > 0, we choose the closest value to f n i,j,l+1 that fulfills the requirements of Proposition 4.1. Namely f n i,j,l+ 1 2 will be the solution of the following problem: Minimize

|f n i,j,l+ 1 2 -f n i,j,l+1 | under the constraint f n i,j,l+ 1 2 ∈ [µ n i,j,l+ 1 2 , M n i,j,l+ 1 2
]. By assuming that V n i,j,l-1 2 > 0, V n i,j,l > 0 and V n i,j,l+ 1 2 > 0, the minimization problem leads to the following three cases:

f n i,j,l+ 1 2 =              µ n i,j,l+ 1 2 if f n i,j,l+1 ≤ µ n i,j,l+ 1 2 , f n i,j,l+1 if µ n i,j,l+ 1 2 ≤ f n i,j,l+1 ≤ M n i,j,l+ 1 2 , M n i,j,l+ 1 2 if f n i,j,l+1 ≥ M n i,j,l+ 1 2 .
In the case of locally positive velocity, the stability constraint involves f n i,j,l-1 and f n i,j,l that are upwind values for f n i,j,l-1 2 ; it justifies the name of downwind flux under upwind constraint.

Treatment of the coagulation operator How the coagulation term Q modifies the asymptotic behavior of the solutions is investigated numerically. For the Smoluchowski operator treatment, two main approaches are discussed in [START_REF] Matar | Analyse mathématique et numérique de modèles de coagulation-fragmentation[END_REF]. The first "naive" approach and the second one suggested by F. Filbert and P. Laurençot [START_REF] Filbert | Mass-conserving solutions and non-conservative approximation to the smoluchowski coagulation equation[END_REF] are discussed in [START_REF] Matar | Analyse mathématique et numérique de modèles de coagulation-fragmentation[END_REF]. The "naive" approach preserves the solution positiveness, but is very cost in CPU time. In our case, we use the approach suggested by F. Filbert and P. Laurençot for its good performances. To this end, the treatment of the collision term thanks to the approach developed in [START_REF] Laurençot | Weak solutions to the Lifschitz-Slyozov-Wagner equation[END_REF] is incorporated in the time-splitting algorithm.

The starting point of the method consists in rewriting ∂ t f = λQ(f ) as follows:

ζ∂ t f = λζQ(f ) = -λ∂ ζ J(f ), with J(f )(t, x, ζ) = ζ 0 ∞ ζ-s sf (t, x, s)f (t, x, r)drds.
The next step relies on the approximation of the integrals that define J(f ) and the necessary truncation, embodied into a parameter 0 < R < ∞, of the infinite integration domain.

The approach designed is the "Conservative method", which consists in replacing J(f ) by

J R (f )(t, x, ζ) = ζ 0 R-s ζ-s sf (t, x, s)f (t, x, r)drds, for 0 < ζ < R < ∞. Note that J R (f )(t, x, R) = J R (f )(t, x, 0) = 0. Consequently, the solution f R of ζ ∂f R ∂t = -λ ∂J R ∂ζ (f R ), 0 < ζ < R < ∞
satisfies the preservation of the first-order moment:

R 0 ζJ R (f )(t, x, ζ)dζ = R 0 ζJ R (f )(0, x, ζ)dζ = 0.
One can see [START_REF] Filbet | Numerical simulation of the Smoluchowski coagulation equation[END_REF] for a thorough analysis of the method, in particular for the convergence analysis when R → ∞. The problem addressed in [START_REF] Filbet | Numerical simulation of the Smoluchowski coagulation equation[END_REF] is essentially concerned with the capture of the gelation phenomenon, that is a loss of mass in finite time, a typical feature of certain coagulation equations. Here, the situation is different and it turns out that the conservation of the first moment by the process of encounter is crucial for the accuracy of the scheme and the evaluation of the monomers concentration in the last step of the splitting. That is the reason why we work here with the conservation method.

For the discrete expression of the operator J R (f ), it is convenient to introduce the change of variables w = s + r to obtain

J R (f )(t, x, ζ) = ζ 0 R ζ sf (t, x, s)f (t, x, w -s)dwds.
The numerical scheme is written

ζ l f n+1 i,j,l = ζ l f n+ 1 2 i,j,l -λν n J n+ 1 2 i,j,l+ 1 2 -J n+ 1 2 i,j,l-1 2 ,
where the numerical flux J

n+ 1 2 i,j,l+ 1 2 is the approximation of J R (f )(t n+ 1 2 , x i,j , ζ l+ 1 2 ).
Monomers scheme For updating the monomers concentration, an implicit finite difference scheme is used to solve equation (1.1 3 ). This numerical scheme is the following:

c n+1 i,j -c n i,j ∆t n - c n+1 i-1,j + c n+1 i,j-1 -4c n+1 i,j + c n+1 i+1,j + c n+1 i,j+1 ∆x 2 = - N l l=1 ζ l f n+1 i,j,l -f n i,j,l ∆t n ∆l, ∀ i, j = 2, ..., N -1, ∀n = 1, ..., N t , c 0 i,j = c init (x i,j ), ∀ i, j = 1, ..., N.
Boundary conditions On ∂Ω, the Neumann boundary conditions are approximated by the following relationships:

-For internal nodes to boundary edges:

c 1,j = c 2,j , c N,j = c N -1,j for j = 2, ..., N -1, c i,1 = c i,2 , c i,N = c i,N -1 for i = 2, ..., N -1. (4.5)
-At the corner of the domain:

c 1,1 = c 2,2 , c N,1 = c N -1,2 , c N,N = c N -1,N -1 , c 1,N = c 2,N -1 . (4.6)

Numerical results

The geometric domain, defined by Ω = [0, 1] 2 , is discretized using a structured mesh with a space step size h equal to 0.02. The polymers size domain is [0, 1] with 50 subdivisions. The CFL number is set to 1.

Evolution of a mixed solution of monomers and polymers

The Coagulation coefficient λ is equal to 1. The diffusion coefficient of the monomers is equal to 0.001. The monomers are initially randomly distributed in space. The polymers are also randomly distributed is size and space. Only polymers whose size are lower than 0.2 are initially present in the domain.

The evolution of the monomer spatial distribution is illustrated by Figure 5.1. The diffusion leads to a smoothing of the density. After a while, the monomers are more concentrated at the bottom and right sides of the domain.

The mean density of polymers with size included in the intervals [0.4, 0.6] and [0.8, 1] are respectively displayed in Figures 2 and3. The polymer density evolution exhibits at the first step a growth caused by the presence of the monomers. After a while the lack of monomers causes a depolymerization of polymers. The evolution in time of the number of polymers according to their size is illustrated by Figure 4, where one can observe the growth of the polymers. The evolution of the mass of monomers and polymers is illustrated by Figure 5. One can observe first a polymerization process, followed by a depolymerization.

Sensitivity analysis according to monomers diffusion

Figure 6 displays the evolution in time of the number of polymers; according to their size, for different values of the monomer diffusion coefficient. One can see that the growth of the number of polymers increases with the diffusion coefficient.

Sensitivity analysis according to the Coagulation coefficient

The evolution of the number of polymers, according to their size, for different values of Coagulation coefficient is displayed in Figure 7. The diffusion coefficient is set to 0.001. In this case, the polymers growth dynamic seems to be very insensitive to the caogulation coefficient change.

Conclusion and perspectives

We described in this study a model of polymerization where polymers are immersed in a bath of monomers. We assumed the interaction to be driven by mass transfer based on the space diffusion of monomers. We added in the model the possible lengthening of polymers when one polymer coalesces with another thanks to the Smoluchowski coagulation operator. With realistic assumptions we prove that our coupling system of hyperbolic-parabolic type is well defined and the numerical approximation based on the 2D generalization of the anti-dissipative scheme [START_REF] Goudon | Simulations of the Lifschitz-Slyosov Equations: The role of coagulation terms in the asymptotic behavior[END_REF] is very suitable for the behavior of the solution even when the coagulation arises. Our numerical results show that monomer diffusion stimulate polymers growth. In order to go deeper in the understanding of the polymerization mechanism with others related sub-processes, we need to investigate several directions for future work. One of the interesting directions is to study the behavior of the polymerization process when in addition polymers are subject to possible coagulation and fragmentation events. In particular, coagulation and fragmentation do they regularize the asymptotic behavior of the solution toward on profile or not? An other direction not far from this previous one is to allow polymers to follow also a space diffusion equation and then investigate the well-posedness and numerical simulations of the system.

  1 one can refers to [13, Lemma 2.1 and Proposition 2]. 3 Analysis of the well-posedness of problem (1.1)

Proposition 3 . 1

 31 Let (f n ) n∈N and f solutions of the problem with the same initial data f (t = 0, x, ξ) = f n (t = 0, x, ξ). Let define a sequence ( cn ) n∈N ∈ E T which converges toward c in L 2 (Q T ). Then c belongs to C (E T ). More precisely C ( cn ) → C (c) in L 2 (Q T ) when n tends to infinity.

(3. 14 )

 14 Combining (3.11)-(3.14) we deduce from (3.9):

Fig. 1

 1 Fig.1Density of monomers at times 0, 0.009, 0.03, 0.14, 0.22, 0.36, from left to right and from top to bottom.

Fig. 2

 2 Fig. 2 Mean density of polymers with size included in the interval [0.4, 0.6] at times 0, 0.08, 0.16, 0.24, 0.36, 0.58, from left to right and from top to bottom.

Fig. 3

 3 Fig. 3 Mean density of polymers with size included in the interval [0.8, 1] at times 0, 0.16, 0.36, 0.57, 0.79, 1, from left to right and from top to bottom.

Fig. 4

 4 Fig. 4 Evolution of the number of polymers according to their size at times 0, 0.02, 0.1, 0.18, 0.27, 0.36, from left ro right and from top to bottom.

Fig. 5

 5 Fig. 5 Evolution in time of monomers and polymers masses.

Fig. 6

 6 Fig. 6 Evolution of the number of polymers according to their size for different values of the diffusion coefficient 0.001, 0.01, 0.1, 1, at times 0, 0.11, 0.4, 0.53, from left to right and from top to bottom.

Fig. 7

 7 Fig. 7 Evolution of the number of polymers according to their size for different values of the coagulation coefficient 0.001, 0.01, 0.1, 1, at times 0, 0.11, 0.4, 0.53, from left to right and from top to bottom.

Acknowledgements

The authors gratefully acknowledge the "BQR" (Bourse Qualité Recherche) grant of University Lyon 1. The authors thanks Thomas Lepoutre for his precious help and corrections.


Timothy M Flemming
email: timothy.flemming@gmail.com

Roger K R Thompson

Joêl Fagot

Baboons, like humans, solve analogy by categorical abstraction of relations

Keywords: Analogy, Analogical reasoning, RMTS, Monkey, Same/different, Abstract thinking, Representation

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

As a measure of abstract thinking, reasoning by analogy proves to be one of the most complex cognitive processes. To solve an analogical problem, one must determine the relational concept instantiated between or among one set of stimuli (e.g., icons, words, symbols, problem spaces or even related/unrelated individuals) and subsequently search for that same structural or functional relation among sets of novel and unfamiliar exemplars. As a foundation for critical thinking [START_REF] Gentner | Analogy[END_REF], success in analogical reasoning entails judging and mapping conceptually abstract relations-between-relations rather than mere surface perceptual features [START_REF] Hofstadter | Analogy as the core of cognition[END_REF][START_REF] Gentner | Structure-mapping: a theoretical framework for analogy[END_REF]. In humans this ability occurs at an early age and, importantly, is facilitated by the emerging capacity to represent abstract relations in concrete symbolic terms via linguistic labeling, as is the case also for language-trained or symbolusing chimpanzees (e.g., [START_REF] Premack | Minds with and without language[END_REF][START_REF] Rattermann | More evidence for a relational shift in the development of analogy: children's performance on a causal mapping task[END_REF][START_REF] Thompson | Language-naı ¨ve chimpanzees (Pan troglodytes) judge relations between relations in a conceptual matching-to-sample task[END_REF]; see also Clark and Thornton 1997;[START_REF] Flemming | What meaning means for same and different: analogical reasoning in humans (Homo sapiens), chimpanzees (Pan troglodytes), and rhesus monkeys (Macaca mulatta)[END_REF]. Crucial to analogical reasoning, whether by human or by nonhuman, thus are (1) examination of underlying structural similarities rather than perceptual features and (2) re-encoding or reinterpretation of the abstract relations employed.

sample and comparison stimuli (see Fig. 1). In principle, this task could be solved either by considering the abstract (same/different) relation instantiated by these stimuli, or by matching the stimuli on the basis of their perceptual entropy. Our results provide unique evidence for the categorical representation of abstract relations on a RMTS task by both human and nonhuman primates, implicating continuity of underlying cognitive processes for analogical reasoning.

Method

Participants and apparatus

Eight Guinea baboons (Papio papio, two female) aged 3-7 years participated in computerized tasks. Eight human (Homo sapiens) participants (six female) aged 23-34 years were also recruited from the CNRS field site in Rousset, France. All participants were naı ¨ve to testing hypotheses and were not familiar with any related topics being investigated.

Baboon participants belong to a large social group of 29 animals living in 670 m 2 enclosure at the CNRS primatology station (Rousset-sur-ARC, France). All baboons had a radio frequency identification (RFID) microchip implanted in each forearm that served for auto-identification within the test boxes. The baboons were tested with a new test apparatus named ''Automatic Learning Device for Monkeys'' [ALDM, see their detailed description in [START_REF] Fagot | Automated testing of cognitive performance in baboons maintained in social groups[END_REF] and [START_REF] Fagot | Automated testing of cognitive performance in monkeys: use of a battery of computerized test systems by a troop of semi-free-ranging baboons (Papio papio)[END_REF]]. Each of the 10 identical ALDM test systems used in the research consisted of an experimental chamber (0.7 9 0.7 9 0.8 m) freely accessible from the living enclosure. Upon entering the test chamber, a baboon faces an opaque panel within which a (7 9 7 cm) viewing port and two (8 9 5 cm) hand ports are positioned to see and touch a 19-inch LCD touch screen monitor (model ET1739L from Elotouch; Berwyn, IL) at eye level 25 cm behind the view port. Each ADLM system had a network-controlled PC computer (Dell precision 67, 3.2-GHz) and comprised a ''homemade'' dispenser delivering dry-wheat reinforcers (www.ebly.co.uk) inside the test chamber. The test program, written in E-prime (V2 professional, Psychology Software Tools, Pittsburgh), automatically identified an animal via the microchip when it placed its forearm within an arm port and determined its last stopping point in trial presentations, in order to assign the independent variables to be experienced during the trial. The stimulus choices were recorded in each trial. With this system, the participants could experience identical test programs at their own pace, independently of the test system they chose. Participants were neither food nor water deprived.

Recent theoretical perspectives on analogical reasoning assert that it is not only a hallmark of higher-order reasoning, but also a uniquely human cognitive trait [START_REF] Penn | Darwin's mistake: explaining the discontinuity between human and nonhuman animal minds[END_REF]. In this view, the symbolic reinterpretation of analogical relations by other species, including all nonhuman primates, is impossible due to their lack of a generalized abstract representational capacity (cf., [START_REF] Thompson | Categorical perception and conceptual judgments by nonhuman primates: the paleological monkey and the analogical ape[END_REF], and therefore, their abstract conceptual abilities are grounded solely on directly perceptible surface features. Hence, relational-matching behavior by nonhuman animals would reflect their expert sensitivity to perceptual characteristics, rather than analogical relations per se.

Supporting this claim is evidence from same/different discrimination tasks that distinguishing same from different in multiple-icon arrays by both pigeons and monkeys is profoundly affected by the magnitude of perceptual variability (i.e., entropy) in a visual array [START_REF] Young | Entropy detection by pigeons: response to mixed visual displays after same-different discrimination training[END_REF]; for reviews see [START_REF] Wasserman | Same-different discrimination: the keel and backbone of thought and reasoning[END_REF][START_REF] Zentall | Concept learning in animals[END_REF]. Entropy here refers to the amount of visual diversity in the sample calculated from the number of different bits of perceptual information present. If relational-matching performance is based on entropy/perceptual variability alone, then judgments of the concepts same and different would thus be derived from a perceptually grounded continuous scale (from mostly sameness to mostly difference) rather than categorically (same or different) as presumably they are by humans [START_REF] Smith | The comparative psychology of same-different judgments by humans (Homo sapiens) and monkeys (Macaca mulatta)[END_REF]. Hence, if animals match relations solely in terms of similar or dissimilar levels of perceptual variability, they cannot be said to be representing abstract concepts categorically.

Recent evidence for reliable success by Old World monkeys in generalized 2 9 2 item relational matchingto-sample (RMTS) tasks (e.g., match AA with BB, not CD, and match EF with CD, but not BB) by monkeys (e.g., [START_REF] Fagot | Generalized relational matching by Guinea baboons (Papio papio) in two by two-item analogy problems[END_REF][START_REF] Flemming | Analogical reasoning and the differential outcome effect: transitory bridging of the conceptual gap for rhesus monkeys (Macaca mulatta)[END_REF] suggests that nonsymbol/language-trained primates are indeed capable of rudimentary analogical reasoning. However, as yet there is no unequivocal evidence of commonalities or differences between the cognitive processes underlying the RMTS performances of these Old World monkeys and humans.

Do non-linguistic monkeys accomplish the RMTS task by categorically representing or ''reinterpreting'' relations as is the case for humans or alternatively, is their performance mediated via rudimentary perceptual judgments? The present study aimed to settle the above debate. In our study, we presented eight baboons (Papio papio) and eight humans (Homo sapiens) with a computerized relationalmatching task involving different combinations of fouritem geometrical shapes (e.g., of the type AAAA, ABBA and ABCD) as stimuli, with no item sharing between All baboons in the present experiment had recently participated in a 2 9 2 relational-matching task [START_REF] Fagot | Generalized relational matching by Guinea baboons (Papio papio) in two by two-item analogy problems[END_REF]) that utilized the same geometric shapes as stimuli and reliably performed this task above chance. Individuals from the social group who previously performed at or below chance on relational matching were not presented with the current task.

Procedure

During initial training, all participants completed relational-matching trials with uniform same/different arrays each composed of four geometrical shapes. When converted to letters for expository purposes, this task can be formalized as ''match AAAA with BBBB, not CDEF.'' The baboons completed trials until they reached 80 % or greater accuracy in two consecutive 100-trial blocks. Correct choices resulted in the delivery of a wheat-grain reward; incorrect choices resulted in no reward and the presentation of a green screen for 1 s (ITI) before the initiation of the next trial. Baboons completed this basic training after M = 400 trials, SD = 129.1.

Human participants followed a parallel procedure with two slight modifications. Humans completed the trials until they reached 80 % or greater accuracy in two consecutive 12-trial blocks. Feedback was given following each trial with a display of ''OUI'' (yes) or ''NON'' (no). The 80 % criterion was reached in training after M = 34.5 trials, SD = 16.3.

In test sessions, mixed-array probe trials of three kinds were inserted randomly within a trial block and rewarded nondifferentially: AAAB (entropy = 0.81), AABB (entropy = 1.0) and AABC (entropy = 1.5), with the location of each item randomized across trials. Choice arrays on both baseline and probe trials were comprised of all same (DDDD: entropy = 0) or all different (EFGH: entropy = 2.0) stimuli. See Fig. 1 for a depiction of trial types. Our rationale was that an entropy-based strategy should be demonstrated by a greater frequency of same responses than that of different responses to the 0.81 entropy arrays, because the degree of variability within those arrays is much closer to all same (entropy = 0) than to all different (entropy = 2) arrays. Similarly, no response bias should be observed in response to the arrays with an entropy of 1.0, if the subject used an entropy-based strategy, because that value of entropy falls exactly in between that of the all same (entropy = 0) and all different (entropy = 2) arrays. Probe trials were neither determinately ''correct'' nor ''incorrect'' as their purpose was to discover how the participants spontaneously matched mixed arrays that could be classified as either same or different, dependent upon cognitive mechanism utilized (perceptual/conceptual) and how relations were defined (continuous/categorical).

Each trial began with the presentation of a sample fouritem array (uniform baseline or mixed probe). This sample remained centered in the upper third of the computer screen until touched (see Fig. 1, top left). After the participant touched the array, it was removed from the screen and two four-item choice arrays (all baseline type, e.g., AAAA vs. BCDE) were presented in the lower third of the screen at the left and right extremes (see Fig. 1, top right). Arrays were composed of simple white geometric shapes from a set of 10 items. By using a small stimulus set, the opportunity for confounding matching strategies was high, providing more support for a relational match if observed. Baseline trials were differentially rewarded in a manner identical to training, and probe trials were nondifferentially rewarded as described above.

Eighty percent of randomly selected probe trials were rewarded regardless of response (correct or incorrect) to reflect the levels of reward during baseline trials. Baboons completed 10 blocks of 124 trials (24 probe trials each block), resulting in approximately 240 total probe trials for each subject. Human participants completed two blocks of 100 trials (60 probe trials each block), resulting in 120 total probe trials of completely trial-unique configuration for each subject, making rote memorization unlikely.

Results

As illustrated in Fig. 2, the selection of choice arrays provides evidence of categorical representation of relations by baboons, wherein only baseline identity samples (AAAA) were reliably matched to same (binomial z = 46.89, p \ 0.001). Regardless of entropy level (i.e., perceptual variability), all other trial types were matched to different. As a group, the baboons responded ''different'' significantly more often than chance on all probe-trial types [entropy = 0.81 (AAAB), 1.0 (AABB), 1.5 (AABC) and 2.0 (baseline ABCD)], binomial z-scores = 4. 85, 8.99, 15.62 and 16.40, respectively, all p \ 0.01. No notable individual differences between subjects were observed.

Human participants matched according to an identical pattern of responding, and the correlation between the humans' and baboons' data was very close to 1, r = 0.998, p \ 0.0001. Humans reliably matched baseline identity samples to same (binomial z = 14.39, p \ 0.001). As was the case for baboons, all other display types (including 2.0 baseline and mixed probe arrays from entropy 0.81-1.5) were reliably matched to different. Human participants responded ''different'' significantly more often than chance on all probe-trial types [entropy = 0.81 (AAAB), 1.0 (AABB), 1.5 (AABC) and 2.0 (baseline ABCD)], binomial z-scores = 3. 43, 7.96, 11.58 and 13.91, respectively, all p \ 0.01, see Fig. 2.

Whereas an effect of entropy within different classifications was observed across all trial types for baboons, the effect was driven largely by significant differences between the all same (entropy = 0) and all other types. Using logittransformed proportion of different responses, an ANOVA revealed a significant effect of entropy value, F(4,35) = 92.54, p \ 0.01. Tukey's HSD post hoc tests (a level 0.01) revealed significant differences in responding between all entropy levels, except between 0.81 and 1.0, and between entropy level 1.5 and 2.0. A similar pattern of results was obtained for human participants, as a main effect of entropy level was also observed, F(4,35) = 37.22, p \ 0.01. Tukey's HSD post hoc tests (a level 0.01) revealed significant differences between all entropy levels, except between entropy levels within the different category (i.e., 0.81-2.0). Critically, no reliable differences between percentage of different selection for baboons and humans at any entropy level were observed: entropy = 0.0 (baseline AAAA), 0.81 (AAAB), 1.0 (AABB), 1.5 (AABC) and 2.0 (baseline ABCD), z-scores = 1.74, 0.14, 1.68, 0.69 and 1.94, respectively, p [ 0.05.

At entropy level 0.81 (e.g., AAAB, ABAA), wherein matches based on perceptual variability would predict classification as same, we observe categorical different selection for both humans and baboons. However, whereas at the group level all probe trials of this type were matched to different, responding to configuration types with repeated identity (AAAB and BAAA) was at the levels of chance for both humans and baboons, binomial z-scores 1.47 and 1.22, respectively, p [ 0.05, see Fig. 3a. This pattern of responding suggests for the two species a ''fuzzy'' boundary for different when presented with higher levels of repeated identity.

Because AABB trial types fall at the midpoint between same and different, responding at this trial type is predicted to be random if driven by mechanisms of perceptual variability comparison as is observed in pigeons (see Young and Wasserman 1997, their Fig. 4). Contrary to this hypothesis, and in accordance with the above-reported findings at the 0.81 entropy level, we find responding at the 1.0 entropy level to be categorically defined as different. Regardless of configuration (AABB, ABAB and ABBA), Whereas we observed a small increase in responding same to mixed arrays of entropy 0.81 (e.g., AAAB), matches to different (e.g., CDEF) were still significantly above chance at this and all samples of greater variability. Even when a display contains mostly identical (e.g., AAAB) or an equal number of identical and nonidentical (e.g., AABB) items, both humans and baboons reliably classify relations as different, implying a categorical abstraction and representation of relations. Analyses of the different spatial arrangements of the shapes within each level of entropy further stressed markedly similar responding in humans and baboons.

Discussion

That both human and baboon follow a remarkably similar categorical pattern of responding to same and different when matching relations provides unique support for relational abstraction by nonhuman primates. Importantly, these results provide good evidence of continuity in the evolution of the cognitive requisites for analogical reasoning in primates. This would not have been the case had the monkeys' response decisions been determined predominantly by the perceptual strategy of matching the degree of stimulus variability instantiated by the different levels of entropy.

Admittedly, both baboons and humans perceived and were secondarily sensitive to changes in entropy. Interestingly, predominant effects of entropy as a basis for relational matching have been observed in up to 20 % of human participants in a similar task [START_REF] Young | Entropy and variability discrimination[END_REF] and have repeatedly been found in both monkeys [START_REF] Fagot | Discriminating the relation between relations: the role of entropy in abstract conceptualization by baboons (Papio papio) and humans (Homo sapiens)[END_REF][START_REF] Flemming | Conceptual thresholds for same and different in old-(Macaca mulatta) and new-world (Cebus apella) monkeys[END_REF][START_REF] Flemming | Disconnect in concept learning by rhesus monkeys: judgment of relations and relations-between-relations[END_REF]) and pigeons (for review, see Wasserman and Young 2010). Importantly, however, both monkeys and humans in the present task were primarily predisposed to reinterpret relations in an abstract categorical manner.

The present observed effect of entropy within the different category for both humans and baboons suggests that same and different are ''quantifiable'' in that some mixed displays are relationally more same or more different than others, as [START_REF] Smith | The comparative psychology of same-different judgments by humans (Homo sapiens) and monkeys (Macaca mulatta)[END_REF] suggest. Nevertheless, even the quantifiable nature of these relations does not preclude the predominance of abstract categorical representation, as observed in the present experiment for both human and baboon.

Because the explicit employment of concepts (rather than percepts) is essential to analogy-making by even a strict definition (i.e., [START_REF] Penn | Darwin's mistake: explaining the discontinuity between human and nonhuman animal minds[END_REF], we provide here the first conclusive demonstration that analogical relational matching is carried out by such means in nonhuman animals other than apes (cf., [START_REF] Gillian | Reasoning in the chimpanzee: I. Analogical reasoning[END_REF][START_REF] Oden | Can an ape reason analogically? Comprehension and production of analogical problems by Sarah, a Chimpanzee (Pan troglodytes)[END_REF]. More broadly, the present results implicate an abstract rule-based approach to relational matching by monkeys suggestive of their capacity to re-encode and abstractly represent relations categorically. This cognitive capacity further brings into question the extent and limits of abstract thinking in the absence of language or symbolic codes and the nature of the cognitive scaffolding effects they facilitate.

Given this evidence of cognitive continuity between human and nonhuman animal in the evolution of the requisite cognitive foundation for analogical reasoning, we can better entertain and investigate other contexts under which nonhuman animals may or may not similarly represent their world abstractly. This capacity for abstract reasoning, for example, may well have its evolutionary origins in the social domain wherein animals have to deal with complex networks of relations-between-relations [START_REF] Bateson | Trading spaces: computation, representation, and the limits of uniformed learning[END_REF][START_REF] Dasser | Mapping social concepts in monkeys[END_REF]). However, if so, it is Fig. 3 Percent different matches for various configuration types at entropy levels 0.81 and 1.0 for baboons and humans, respectively. At entropy = 0.81 (a), lead/end refers to configuration types wherein the unique stimulus either led or ended the array (BAAA and AAAB). Imbedded configuration types were those wherein the unique object was in a middle position (ABAA or AABA). Dashed horizontal line represents chance selection noteworthy that more than transitory expression of this cognitive ability is all the more difficult when the relational exemplars are dissociated from the animals' natural environment during initial acquisition training (cf., [START_REF] Fagot | Generalized relational matching by Guinea baboons (Papio papio) in two by two-item analogy problems[END_REF][START_REF] Flemming | Analogical reasoning and the differential outcome effect: transitory bridging of the conceptual gap for rhesus monkeys (Macaca mulatta)[END_REF]).

Fig. 1

 1 Fig. 1 Depiction of baseline and probe-trial types in fouritem relational matchingto-sample task completed by Homo sapiens and Papio papio. Each sample configuration was presented first on a black background centered on the computer screen (top left) and disappeared upon touch allowing for the subsequent presentation of two choice arrays (top right). Exemplary sample arrays at each level of entropy level (0-2.0) are depicted along bottom

Fig. 2

 2 Fig. 2 Proportion of different responses for humans and baboons for all trial types (baseline and probe) combined. Error bars indicate standard error (SE); dashed horizontal line represents chance performance. All matches to different at entropy levels 0.81-2.0 are significantly above chance levels of responding

Acknowledgments Timothy M. Flemming was supported by Fyssen Foundation. Joe ¨l Fagot was supported by project ANR-2010-BLANC-1908-01 and European Project EuprimNet-II. Roger Thompson was supported by the Franklin and Marshall College Faculty Research and Development Fund.