THE LIFSCHITZ-SLYOZOV EQUATION WITH SPACE-DIFFUSION OF MONOMERS

. The Lifschitz–Slyozov system describes the dynamics of mass ex-changes between macro–particles and monomers in the theory of coarsening. We consider a variant of the classical model where monomers are subject to space diﬀusion. We establish the existence–uniqueness of solutions for a wide class of relevant data and kinetic coeﬃcients. We also derive a numerical scheme to simulate the behavior of the solutions.

1. Introduction. We are interested in the mathematical modeling of the late stage of the precipitation kinetics in supersaturated metastable solid solutions. The problem is for instance motivated by the design of industrial processes for producing metallic alloys (stainless steel, a mixture of iron, nickel, and chromium, or gold and nickel alloys used in jewelry to name a few), based on theoretical and numerical predictions. The supersaturated alloys under consideration are made of two phases: solute monomers and coarsening precipitates. Existing models phrase the time-evolution of the solution in terms of a PDEs system for the monomers concentration and the distribution function of the precipitate in size space. The latter gives the concentration of solute clusters of a given size. The kinetic equations for cluster concentrations are driven by the rates at which clusters absorb and lose a solute atom. The attachment and detachment processes are actually governed by basic principles of overall reduction of the interface energy, where volume effects, which favor growth, compete with surface effects, which favor dissolution. It turns out that minimizing the total interfacial surface energy of the system leads to the growth of larger particles at the expense of smaller particles, which are thus 326 THIERRY GOUDON, FRÉDÉRIC LAGOUTIÈRE AND LÉON M. TINE assigned to become still smaller and the overall process results in an increase in the average particle radius, a phenomenon referred to as Ostwald ripening. The Lifschitz-Slyozov equations are commonly used in an effort to understand ripening from a theoretical perspective.
The standard Lifschitz-Slyozov system, as introduced in [20,21], describes the evolution of a solution of polymers. In this model, macro-particles, or polymers, interact with free particles, or monomers. The macro-particles are described by their size distribution function f (t, ξ), with t ≥ 0 and ξ ≥ 0 the time and size variables respectively, while the monomers are described by their concentration c(t). Therefore ζ ζ f (t, ξ) dξ is interpreted as the number of polymers having at time t their size between ζ and ζ while ζ ζ ξf (t, ξ) dξ is proportional to the corresponding mass. The dynamics is governed by the growth rate with a, b given non negative functions: these kinetic coefficients represent the rates at which monomers are added to or removed from the macro-particles with size ξ. The precise expression of the coefficient relies on the modeling of the precipitation/dissolution processes; in [21], assuming that mass transfer is based on monomer diffusion, the following expression is proposed We refer to [32] for other relevant formulae for the kinetic coefficients. In this paper we shall assume the following Hypothesis 1.1. The kinetic coefficients a, b are required to satisfy: i) b = 1, ii) a is non decreasing with a(0) = 0 and a(+∞) = +∞, iii) a ∈ C 0 ([0, ∞)) ∩ C 1 ((0, ∞)) and for any ξ 0 > 0 there exists L a,0 > 0 such that 0 ≤ a (ξ) ≤ L a,0 for ξ ≥ ξ 0 > 0.
Note that these assumptions cover the case of the Lifschitz-Slyozov coefficients (1). As a matter of fact we remark that at any time t ≥ 0 the size space splits into two parts: 0 ≤ ξ ≤ ξ c (t) and ξ ≥ ξ c (t) where ξ c (t) is the unique positive number verifying a(ξ c (t)) = 1/c(t). It agrees with the basic principle of ripening where larger particles grow and smaller particles shrink. In this model, direct collisions of clusters are not considered. Therefore, the dynamics of the precipitation process is embodied into the transport equation (2) coupled to the mass conservation relation a given positive constant. Eq. (2) is a conservation law for the polymer concentration in size space, while (3) expresses the fact that the total mass is conserved, the solute material being accounted for either as dissolved particles or as macroparticles. We point out that for ξ = 0, the growth rate V (t, 0) = −1 is negative so that we do not need a boundary condition. Despite its apparent simplicity the Lifschitz-Slyozov system is quite intriguing for the mathematical analysis, and some aspects remain unclear even for physicists. We refer to [7,17,19,25,27] for existence-uniqueness results in various functional frameworks (bounded, integrable or measure-valued solutions). While crucial for the applications, the understanding of the large time behavior is highly challenging, definitely far from the asymptotic trend to a universal profile, as derived in [21]. The Lifschitz-Slyozov system admits a family of self-similar solutions for the clusters distribution function. All of the profiles have compact support and can be parameterized by the value of the logarithmic derivative at the edge of the support which determines the asymptotic value of the (rescaled) monomers concentration. The selection of the correct self-similar solution which defines the profile for large time associated to a given initial data is highly non-trivial: it is precisely determined by the behavior of the data at the tip of its support! We refer on these questions to the analysis performed in [9,24,26] and the numerical simulations in [2,4,34]. These results are in apparent contradiction to most of experimental results which appear to show strong selection, insensitive to initial conditions. However, recent microgravity experiments have also revived the controversy from a practical perspective and have shown that the subject deserves thorough investigation [31]. Finding selection principles appeals to go beyond the classical Lifschitz-Slyozov model.
To this end, modifications of the Lifschitz-Slyozov system (2)-(3) have been discussed in the literature. The addition of a coagulation operator is suggested in [21,32] to account for possible coalescence of large clusters formed on the late stages of the process. Analysis of the Lifschitz-Slyozov equation with such a coagulation term can be found in [6,18]. The impact of the coalescence terms on the selection of the asymptotic profile is analyzed in [16]; we also refer to the numerical experiments in [34]. Other variants of the Lifschitz-Slyozov system (2)-(3) are obtained by introducing parabolic corrections. Such corrections can be motivated through suitable asymptotic arguments, deriving the Lifschitz-Slyozov model from the Becker-Döring system, an infinite system of ODEs where clusters' size is a discrete variable. The corrected model is intended to share more basic features with the discrete Becker-Döring model, in particular concerning selection mechanisms of the large time asymptotics. This aspect has been detailed in various ways by many authors, for instance we can refer to [8,10,11,15,22]. Another diffusive correction is discussed in [29], based on a deep mean field analysis.
In this paper we wish to discuss another relevant version of the Lifschitz-Slyozov equations by assuming that monomers are also subject to space diffusion. The model we wish to investigate is derived in details in [21,Section 4], with the formation of vacancies (which are the clusters in this framework) in a crystal as a specific application, in connection to the description of sintering processes. It is likely that considering diffusion of monomers induce spatial correlations in the Lifschitz-Slyozov model, which in turn can modify the asymptotic trend. This question is beyond the scope of the paper. Here, as a preliminary step, we restrict to investigate the well-posedness issues, dealing with physically relevant functional spaces, while the preliminary numerical study we propose gives some hints on the qualitative behavior of the solutions. Let Ω ⊂ R N be a smooth bounded domain, with boundary ∂Ω; given x ∈ ∂Ω we denote ν(x) the outward unit normal vector at point x. Then, we are interested in the following variant of (2)-(3), where now the unknowns also depend on the space variable and monomers are subject to diffusion: To start with, the diffusion equation is endowed with homogeneous Neumann boundary condition ∂ ν c = ∇c · ν = 0, on ∂Ω, (5) but we will detail below adaptations to Dirichlet conditions (see Section 4.4) and to the free-space problem (Section 4.5). Finally, the problem is completed by initial conditions In view of the physical interpretation it appears quite natural to assume By using the conservation equation for f and integrating by parts, we observe that It allows to rewrite the equation for the monomers concentration in the more familiar fashion Owing to (5), the system preserves the total mass: we have We point out that a coupling with the stationary diffusion equation is derived in [23] through homogenization arguments, the model being further analyzed in [28]. In this paper we shall establish the following well-posedness statement for the system (4)-(6).
The difficulty of course arises from the non-linear coupling which involves PDEs of different types acting on different variables. This work is organized as follows. In Section 2, we briefly set up the necessary material on transport and diffusion equations. Then, in Section 3 we make use of a fixed point strategy to obtain the existence-uniqueness of solutions associated to bounded initial data when the kinetic coefficients are globally Lipschitz. Section 4 extends the result in several directions: more general data, singular coefficients, Dirichlet boundary conditions and free-space problem can be dealt with as well. Finally, in Section 5 we introduce a numerical scheme for the simulation of (4)-(6) and we conclude with some commented numerical experiments.

2.
Basic results on diffusion and transport equations. In this Section we collect some statements on diffusion and transport equations which will be useful for our purposes. We start with the following claim. Proposition 1. Let 0 < T < +∞. Let A and B be non negative functions in L ∞ ((0, T ) × Ω). Suppose that 0 ≤ B(t, x) ≤ C 0 < ∞ for almost every (t, x). Then, for any c init ∈ L 2 (Ω), there exists a unique c ∈ L 2 (0, T ; H 1 (Ω)) ∩ C([0, T ]; L 2 (Ω)) with ∂ t c ∈ L 2 (0, T ; (H 1 (Ω)) ) solution of for some constant C T depending on C 0 , T, Ω and c init L 2 (Ω) .
Proof. The existence result is a direct consequence of a general statement on parabolic equation. Indeed, the bilinear form (Ω) and it verifies the continuity estimate: Furthermore, we also have the coercivity property We can therefore apply the analog of the Lax-Milgram theorem for parabolic equations, see e. g. [1, Theorem X.9, p. 218], and we get the existence uniqueness statement in Proposition 1.
In order to prove the uniform estimate, we proceed as follows. Consider a function G ∈ C 1 (R + ) such that: • There exists M 0 > 0 such that |G (s)| ≤ M 0 for any s ∈ R; • The function s → G(s) is increasing on (0, +∞); • G(s) = 0 on (−∞, 0]. We start by checking that c(t, x) ≥ 0. We set In particular, we observe that since tG(t) ≥ 0 and G (t) ≥ 0. We conclude that ϕ(t) = 0 and thus H(−c(t, x)) = 0 for a.e. (t, x) It implies c(t, x) ≥ 0 a.e. Next, we prove the bound from above. To this end, we set We have ϕ(0) = 0 and It follows that ϕ(t) = 0 and thus H c(t, x) − K(t) = 0 for a. e. t ≥ 0, x ∈ Ω which implies 0 ≤ c(t, x) ≤ K(T ) a.e. on (0, T ) × Ω. The last estimate follows from standard energy estimates and application of the Grönwall lemma.
Let us now recall a few facts about transport equations. For the time being we neglect the space variable which appears only as a parameter in the equation for the size density. Thus, we are concerned with the problem on t ≥ 0 and ξ ≥ 0 where the function (t, ξ) → V (t, ξ) is required to satisfy Hypothesis 2.1. We have V (t, ξ) = a(ξ)c(t) − b(ξ) with continuous and non negative functions a, b, c, such that a(0) = 0, b(0) > 0. We suppose that c is locally bounded while a and b belong to L ∞ (R). Accordingly, for any 0 ≤ t ≤ T < ∞, there exists M T such that for any ξ, ξ ≥ 0, we have Remark that V (t, x, ξ) = a(ξ)c(t, x) − 1 satisfies the requirements in Hypothesis 2.1, uniformly with respect to the parameter x ∈ Ω, as far as the kinetic coefficient a has a globally bounded derivative (see Hypothesis 3.1 below, ||a || ∞ ≤ L a ) and satisfies a(0) = 0, and the monomers concentration satisfies the Owing to Hypothesis 2.1, we can solve (9) by means of integration along characteristics. Indeed, we can define the characteristic curves solutions to the ODE Ξ(t; t, ξ) = ξ. .
the Jacobian of the change of variables ξ → ζ = Ξ(s; t, ξ). The fundamental properties on the characteristics that are needed for our analysis are summarized in the following claim. (We refer to [7] for similar considerations and details.) Proof. Derivating with respect to the initial time, we obtain Since J ≥ 0 and V (t, 0) ≤ 0, we deduce that t → Ξ(0; t, 0) is non decreasing and thus i) holds. Next, we have Ξ(σ; t, ξ) dσ and the Grönwall lemma yields With s 2 = t we have e M T (s1−t) ξ ≤ Ξ(s 1 ; t, ξ) which allows to conclude for ii) by letting ξ go to ∞. The third item is a direct consequence of the Grönwall lemma.
Proposition 2. Let Hypothesis 2.1 be fulfilled. Let f be the solution of (9), as given by (11). Then, the following assertions hold ) and the following estimates hold for any t ≥ 0 and, for any with L T depending on M T in Hypothesis 2.1. ii

THIERRY GOUDON, FRÉDÉRIC LAGOUTIÈRE AND LÉON M. TINE
Proof. We simply integrate (11) and use Lemma 2.1 to obtain 3. Existence-uniqueness for bounded data and smooth coefficients. In this Section, we restrict to the case where the data are bounded and the coefficients are globally Lipschitz. To be more specific we strengthen Hypotheses 1.1 and 1.2 as follows Hypothesis 3.1. Additionally to the requirements in Hypotheses 1.1 and 1.
We wish to prove the well-posedness of the non homogeneous Lifschitz-Slyozov equation in this framework. We will explain in the next section how these additional assumptions can be relaxed.
Theorem 3.1. Suppose that Hypothesis 3.1 is fulfilled. Then, there exists a unique weak solution (c, f ) of (4)-(6) with, for any 0 < T < ∞, The proof uses the Schauder fixed point theorem, see [13, Corollary 3.6.2]. We set Q T = [0, T ] × Ω for a fixed 0 < T < ∞. Let us denote We associate to this quantity the constant K T as defined in the proof of Proposition Then, we define the mapping where the coefficients A, B are given by From now on we adopt the convention to denote by L T > 0 a constant that depends on T , C 0 , c init ∞ , and on the Lipschitz constant L a of a, even if the precise value of the constant might change from a line to another. Conversely, we will denote by C T a constant which depends only on T , C 0 and c init ∞ but not on L a (like K T ). According to Hypothesis 1.1, for anyc ∈ C T , the rate V (t, x, ξ) = a(ξ)c(t, x) − 1, which is now parametrized by x ∈ Ω, satisfies the estimates required in Hypothesis 2.1, uniformly with respect to x ∈ Ω. (Namely M T in Hypothesis 2.1 is L a K T .) Up to a slight abuse with regularity issues we can therefore appeal to the results established in Section 2. Indeed, within the functional framework adopted here, for fixed x ∈ Ω, t →c(t, x) cannot be considered as a continuous function of the time variable. The classical theory of characteristics with C 1 solutions of the ODE (10) does not apply. The alternative to circumvent the difficulty is as follows. The first option consists in dealing with a less regular notion of characteristics. The standard Picard iteration scheme actually shows that admits a continuous solution assuming only integrability of V with respect to the time variable and all the necessary estimates on Ξ hold in this framework (see [5, Theorem 1.1, p. 43] for an existence theorem without regularity in time). The second option consists in replacingc in the convection term by ζ t,xc , with ζ a convenient sequence of mollifiers. Again, all the necessary estimates are not affected by the regularization process and are uniform with respect to . Accordingly, the compactness arguments detailed below apply to pass to the limit as goes to 0. We do not detail further this issue, adopting the slight abuse of working with the characteristics Ξ, parametrized by the space variable x, without any further precision. Hence, we can apply Proposition 2: f reads with Ξ and J defined by the characteristics equation associated toc. In particular, It follows that A(t, x) ≥ 0 lies in L ∞ ((0, T ) × Ω), and 0 ≤ B(t, x) ≤ C 0 . Coming back to Proposition 1 we conclude that T is well defined with c = T (c) ∈ L 2 (0, T ; H 1 (Ω)) ∩ C([0, T ]; L 2 (Ω)) and furthermore 0 ≤ c(t, x) ≤ K T . In other words T (C T ) ⊂ C T .

THIERRY GOUDON, FRÉDÉRIC LAGOUTIÈRE AND LÉON M. TINE
Let us now show that T (C T ) is a compact set in L 2 (Q T ). In fact Proposition 1 also shows that The equation satisfied by c finally tells us that Since H 1 (Ω) embeds compactly in L 2 (Ω), we can therefore apply the compactness results in [33,Corollary 4] It remains to establish the continuity of T in the sense of the L 2 (Q T ) norm. To this end, let us consider a sequence c n n∈N in C T which converges to somec (strongly) in L 2 (Q T ). Clearlyc ∈ C T . We define f n and f as to be the solution of the transport equations The first step of the the proof consists in establishing the following claim Then, A n and B n tend to A and B, respectively, in L 2 (Q T ).
In order to establish this property, we need an estimate on the distance between characteristic curves associated to different rates. Lemma 3.3. We assume that Hypothesis 3.1 is fulfilled. Let c 1 and c 2 in C T and We denote by Ξ 1 and Ξ 2 the associated characteristics. Then, we have for any 0 ≤ s, t ≤ T < ∞ Proof. We detail the proof for the case 0 ≤ s ≤ t ≤ T , the other situation follows by the same argument. By using the equation for the characteristics, we arrive at On the one hand, since a(0) = 0, we have On the other hand, we remark that holds. The Grönwall lemma then yields the estimate It follows that holds. Therefore, we obtain Applying the Grönwall lemma again leads to (17).

Indeed, we have
by using Hypothesis 1.1. Since we are concerned with y restricted to the interval defined by Ξ n (0; t, x, 0) and Ξ(0; t, x, 0) we have

It yields
Moreover, Lemma 3.3 allows to estimate Combining (18) et (19) we are led to Therefore, we deduce that It finishes the proof of Lemma 3.2.
We are left with the task of proving that c n = T (c n ) converges to c = T (c) in L 2 (Q T ). We remind that c n and c are the solutions of We obtain the following energy estimate It can be recast as We make use of the Cauchy-Schwarz and Young inequalities, together with the fact that c n ∈ C T to obtain Eventually, an application of the Grönwall lemma yields and with initial data f |t=0 = f init . This ends the proof of the existence of solution to the system (4)- (6). What we did can be used to justify the uniqueness of the solution as well. Indeed let us assume that (c 1 , f 1 ) and (c 2 , f 2 ) are solutions of (4)-(6) for the same initial data (c init , f init ). Reproducing the arguments for proving the continuity of T , we arrive at Now, coming back to the proof of Lemma 3.2, we can estimate the right hand side so that The Grönwall lemma then implies that c 1 = c 2 .
As a concluding remark of this section, we observe that holds. (It follows by integrating by parts, we refer to [7, Lemma 3] for details.) Thus, with the Neumann boundary condition (5), the obtained solution satisfies the mass conservation relation 4. Further existence-uniqueness results.

4.1.
Existence for general initial data. In this section we wish to relax the assumptions on the initial data, requiring only and removing the finiteness of the uniform norm of f init which could be physically questionable. To justify the existence of solution in this framework, we appeal to approximation and compactness arguments. To this end, we consider a sequence f n init made of bounded functions which converge to f init in L 1 (Ω × R + , (1 + ξ) dξ dx): , (with C n possibly tending to +∞; for instance we can define the approximate initial data by f n init (x, ξ) = 1 0≤finit(x,ξ)≤n f init (x, ξ)). According to the previous Section we can associate to f n init the solution of the system We can collect the following estimates, on with C T a finite constant depending on c init L 2 (Ω) , C 0 and T . Accordingly, Therefore, ∂ t c n is bounded in L 2 (0, T ; H −1 (Ω)). We can apply the compactness statement in [33] which implies that, possibly at the price of extracting a subsequence, c n → c strongly in L 2 (Q T ) and a. e.
We can also show that c n converges to c in C 0 ([0, T ]; L 2 (Ω) − weak). Next, we discuss further estimates on f n . From the uniform integrability of (f n ) n and by using De La Vallée Poussin's lemma, see [12, p. 38], there exists a non negative function Φ satisfying Using characteristics, we show that the property extends to the solution f n . Indeed, we have, with obvious notation, f n (t, x, ξ) = f n init x, Ξ n (0; t, x, ξ) J n (0; t, x, ξ). Since 0 ≤ J n (0; t, x, ξ) ≤ 1 and Φ(0) = 0, the convexity of Φ yields Φ f n (t, x, ξ) ≤ Φ f n init x, Ξ n (0; t, x, ξ) J n (0; t, x, ξ). Integrating leads to the following uniform estimate Since moreover the first moment with respect to ξ of f n is controlled, the Dunford-Pettis theorem, see e. g. [ , implies that f n is relatively compact in L 1 ((0, T ) × Ω × (0, ∞)) for the weak topology. We can thus assume that f n f weakly in L 1 ((0, T ) × Ω × (0, ∞)). Furthermore, we can apply the De La Vallée Poussin Lemma again to exhibit a non negative function Ψ such that This is the De La Vallée Poussin Lemma applied to the function (ξ → ξ) ∈ L 1 (Ω × (0, ∞), f init dξ dx Integrating the equation satisfied by f n we get d dt We evaluate the right hand side by separating small and large sizes: let ξ 0 > 0 and write where C 0 is defined in Equation (15). Hence applying the Grönwall lemma yields the uniform estimate , Ω and T . Therefore, for any function ϕ such that |ϕ(ξ)| ≤ C(1 + ξ), we can show that As a consequence A n and B n converge weakly to A(t, x) = ∞ 0 a(ξ)f (t, x, ξ) dξ and B(t, x) = ∞ 0 f (t, x, ξ) dξ in L 1 ((0, T ) × Ω), respectively. Since c n is uniformly bounded and converges a.e. to c, a classical application of the Dunford-Pettis and Egoroff theorems (see [14,Lemma 7.62]) proves that c n f n converges weakly to cf in L 1 ((0, T ) × Ω × (0, ∞)). Similarly A n c n converges weakly to Ac in L 1 ((0, T ) × Ω).

4.2.
Existence for singular coefficients. We remark that in the arguments developed above, the estimates do not involve the Lipschitz constant L a that appears in Hypothesis 3.1. Therefore, we can adapt straightforwardly the proof to deal with non smooth coefficients a(ξ), as stated in Hypothesis 1.1, including the physical case a(ξ) = ξ 1/3 . It suffices to consider a sequence of smooth coefficients a n (ξ) which converges pointwise to a(ξ). Let (c n , f n ) be the associated solution of the Lifschitz-Slyozov system. We prove that a subsequence admits a limit (c, f ), with convergences as stated in the previous section. Furthermore, we show that as a consequence of the pointwise convergence of a n , the growth assumption on a for large ξ's, and the weak convergence of f n . Finally, we are able to justify by letting n go to ∞ that (c, f ) is a weak solution with the coefficient a. We refer to [17] for such an extension in the context of the homogeneous Lifschitz-Slyozov equation.
Since |S η (z)| ≤ 1, S η (z) ≥ 0 and zS η (z) ≥ 0, we arrive at the following estimate The next step of the proof of uniqueness relies on an adaptation of the reasoning and estimates in [17] for the homogeneous case. We associate to f (k) (k = 1, 2) the repartition function As a matter of fact, we have ∂ ξ F (k) = −f (k) , and We need to introduce ξ T > 0 such that for any 0 ≤ ξ ≤ ξ T , we have a(ξ)K T − 1 ≤ a(ξ T )K T − 1 < 0, which makes sense owing to Hypothesis 1.1. Furthermore, we can pick r > 1 large enough such that In what follows, L a,T will stand for the Lipschitz constant of a on [ξ T , ∞). We will use weighted L 1 estimate, which relies of defining the auxiliary function We have and thus Up to a regularization argument we deduce the following inequality (obtained formally by multiplying the previous relation by |W T (ξ)| r sgn(F (1) − F (2) ) and integrating over ξ ∈ (0, ∞)).
The last integral in the right hand side can be recast as When 0 ≤ ξ ≤ ξ T , the integrand is dominated by according to the definition of ξ T and the choice of r. When ξ ≥ ξ T we can simply use the fact that a (ξ) is bounded far away from ξ = 0. Note that we can also dominate, for some ξ 0 > 0, Finally, we are led to the following estimate init |(t, x, ξ) dξ where C T is the bound on ∞ 0 ξf (k) (t, x, ξ) dξ. We combine the obtained relations, bearing in mind that W T is bounded from below and above and that B (k) (t, x) = F (k) (t, x, 0). Let λ > 0 to be precised. By using (24) and (25), we are led to It remains to discuss the last integral of the right hand side. We split as follows We now rearrange terms to obtain 1 (a(ξ T ) + 1) r Thus, we pick λ > 0 so that 1 (a(ξ T ) + 1) r > λ > 0 and a(ξ T ) + 1/K T (a(ξ T ) + 1) r+1 > λ > 0. It suffices to apply the Grönwall lemma to conclude with a continuity estimate where with a suitable constant Γ T > 0.

Dirichlet boundary condition.
Let us consider the same problem, but we replace (5) by the Dirichlet condition This is precisely the case presented in [21]. The total mass conservation does not hold because there is a diffusion current from the boundary. Nevertheless, the general strategy of proof can be adapted to this case. Let us indicate where the main modifications are, within the arguments. The discussion of Section 2 adapts readily using the space H 1 0 (Ω) instead of H 1 (Ω). Therefore, we can repeat the arguments of Section 3; the derivation of all the necessary estimates works exactly as before, except (21). However, (20) can still be used to estimate the first order moment of the cluster distribution, since we infer and, by using Hypothesis 1.1, Applying the Grönwall lemma, we conclude again that ∞ 0 ξf (t, x, ξ) dξ ≤ C T holds for x ∈ Ω and 0 ≤ t ≤ T < ∞. This estimate, which does not require a global Lipschitz estimate on the kinetic coefficient a, allows to deal with general initial data and singular coefficients as we did in the previous subsections.

4.5.
Free-space problem. It is finally worth to investigate the situation where Ω is replaced by the whole space R N thus neglecting any boundary effect. Technically, it introduces new difficulties because we are working in an unbounded domain and the compactness argument does not work directly. Hence, we need to establish some weighted estimates. In the context of Proposition 1, the estimates for the diffusion equation can be obtained as follows By using the Grönwall lemma and repeating the arguments of Section 2, we are thus led to the following analog to Proposition 1.
Proposition 3. Let 0 < T < +∞. Let A and B be non negative functions in then the solution c satisfies 0 ≤ c(t, x) ≤ K T with K T a constant depending on C 0 , c init ∞ and T . We also have for some constant C T depending on C 0 , T and 1 + |x| 2 c init L 2 (R N ) .
The estimate on A and B can be deduced from the transport equation: since the space variable is only a parameter, we have (16) as well as It allows to apply the same reasoning as in Section 3 once it has been remarked that the set {φ ∈ H 1 (R N ), |x|φ ∈ L 2 (R N )} embeds compactly in L 2 (R N ). Finally we can make use of the mass conservation to extend the result to unbounded data and singular coefficients, as we did in Section 4.1 and 4.2. The first order moment is bounded independently on the (global) Lipschitz constant of a, and we can show that the sequence of approximations f n (resp. A n and B n ) is weakly compact in L 1 ((0, T ) × B(0, R) × (0, ∞)) (resp. L 1 ((0, T ) × B(0, R)) for any 0 < T, R < ∞. Details are left to the reader, which lead to the following statement.
Then, there exists a weak solution (c, f ) of (4) associated to the initial condition (c init , f init ), with, for any 0 5. Numerical simulations. In this Section we present a numerical scheme to simulate the behavior of the density of particles and monomers concentration, when monomers are subject to space diffusion, namely we design a scheme for (4). The construction of the scheme takes care of the mass conservation and we give some hints concerning stability issues. Note that adding the space variable considerably increases the computation cost in comparison to the homogeneous case. The scheme is satisfactory to investigate transient states, but, definitely, it seems difficult to expect relevant numerical experiments of the large time behavior. For the sake of simplicity we consider the problem set on the one-dimensional slab x ∈ (0, L) with homogeneous Neumann boundary conditions, but the extension to higher dimension and other boundary conditions is straightforward. 5.1. Presentation of the algorithm. We consider time, space and size steps ∆t > 0, ∆x = L/i max > 0 (with i max ∈ N \ {0}), and ∆ξ > 0, respectively. We define discrete times t n = n∆t for n ∈ N, discrete sizes ξ j = (j + 1/2)∆ξ and ξ j−1/2 = j∆ξ for j ∈ N, and positions x i = i∆x for i ∈ {1, . . . , i max }. We consider the discrete cells C j = (ξ j−1/2 , ξ j+1/2 ) centered on ξ j . The discrete unknowns c n i and f n i,j are intended to be approximations of c(t n , x i ) and 1 ∆ξ Cj f (t n , x i , ζ) dζ, respectively. The scheme is based on the following time-splitting: • The updating of the particles distribution follows by integrating the advection equation over the finite volume cells C j ; for any fixed i ∈ {1, . . . , i max }, we set which requires a suitable definition of the numerical fluxes at the interfaces ξ j±1/2 . In practice, the index j spans a finite set {0, ..., j max }. In our simulation we use the Rusanov scheme where for all fixed space indices i. Then we have the following approximation Furthermore, we need fictitious mesh points, where data and unknowns are defined as follows: The stability of the scheme is guaranteed by the CFL condition ∆t ≤ ∆x L n i . We point out that we tried other classical finite volume schemes like WENO (Weighted Essentially Non-Oscillatory method) or the ADM (Anti Dissipative Method) method described in [34] but we did not observe any substantial changes in the results (for short times).
• For updating the monomers concentration, we use the following numerical finite difference approximation of the second equation of (4) for any i ∈ {2, . . . , i max − 1} or the implicit version for any i ∈ {2, . . . , i max − 1}, with obvious and classical modifications for i = 1 and i = i max (see below the Laplacian matrix with Neumann boundary conditions). Of course it is possible to discretize (8) instead, but we prefer the above discretization because it guarantees the total mass conservation on the discrete level: see Equation (28) below. The scheme can be written in matrix form with C n = (c n i ) i∈{0,...,imax} , r n+1/2 = ∆ξ and either A 1 = I, the identity matrix, A 2 = ∆t ∆x 2 A for scheme (E) or A 1 = I− ∆t ∆x 2 A, A 2 = I for scheme (I). The stability of the explicit scheme (E) requires the CFL condition ∆t ≤ ∆x 2 /2. Since this condition is usually more restrictive than the one obtained at the previous step, it can be efficient to use a subcycling method where we perform one time step ∆t adv for f while several time steps ∆t diff ∆t adv for c. Anyway, the parabolic CFL condition leads to a prohibitive computational cost for multi-dimension simulations where the implicit scheme (I) will be preferred. It requires the inversion of the sparse positive definite matrix I − ∆t ∆x 2 A, that can be done by using performing algorithms like the conjugate gradient method. In numerical simulations we do not observe significant discrepancies between results obtained by either the explicit or the implicit scheme. The numerical results in the next section are provided by the explicit one. Owing to the Neumann boundary conditions, the discrete mass conservation relation holds. We check numerically that this quantity is indeed exactly conserved.

5.2.
Numerical results. The numerical simulations are performed in the slab x ∈ [0, 100] with 10 points by length unit. The size variable is truncated to ξ ∈ [0, 100] meshed with 20 points by size unit. The initial data are defined by c init (x) = 0.5 1 x∈ [20,35] , f init (x, ξ) = 0.01 1 x∈ [20,35] × 1 ξ∈ [30,35] . Figure 1 shows the initial data f init (x, ξ), while on Figure 2, the solution f (T, x, ξ) at the final time T = 20 can be compared to the solution obtained by getting rid of the diffusion term in the monomers equation. We clearly observe the influence of the diffusion of monomers on the space repartition of the macro-particles.     The monomers concentration in the same situations is displayed in Figure 3 (diffusion case vs. diffusion-free case). As said above, the simulations also show a numerical evidence of the conservation of the total mass. The time evolution of the monomers concentration can be found in Figure 4. As expected the support of the concentration spreads as time increases, by contrast to the diffusion free case. Note however that the maximum of c seems unchanged between the two cases. Of course, since the space repartition of monomers is modified, it influences the dynamics of the whole system. In Figure 5 we show the time evolution of the mean value of c and f over space, that is compared to the usual solutions of the Lifschitz-Slyozov system. It clearly shows that, even considering only mean values, space diffusion changes the behavior of the solutions, for both the monomers concentration and the particles distribution function.
As explained in the Introduction, many questions arise with the large time behavior of the solutions of the Lifschitz-Slyozov equations (2)-(3), and capturing the correct asymptotic profile is numerically challenging. Similar questions can be addressed for the modified model with space diffusion of monomers. Like for the standard model a numerical difficulty comes from the formation of particles with large sizes. As time goes, the support of f (t, x, ξ) might reach the largest size of the numerical domain, which then induces a fictitious loss of mass. Increasing the size domain leads to a considerable increase of the computational cost because f now also depends on the space variable. Therefore, the present method is restricted to quite short times of simulations.