On solutions of the Lifshitz–Slyozov model

Jean-François Collet† and Thierry Goudon†

†Laboratoire J. A. Dieudonné, UMR 6621, Université Nice-Sophia Antipolis, Parc Valrose, F-06108 Nice cedex 2

Abstract. We derive global existence and uniqueness of solutions for the Lifshitz– Slyozov system, which appears in the modelling of precipitation from supersatured solutions. Mathematically, the system consists of a transport equation for the density function, coupled to an integral equation which expresses conservation of the total mass. The existence is obtained by a fixed point technique, based on the expression of the density function in terms of the monomer concentration, as given by the method of characteristics. For some particular cases we obtain an explicit expression of the solution by using the Laplace transform. Finally in some more general cases, we prove, if a solution exists, convergence to a stationnary solution for large times.

AMS classification scheme numbers: 35, 82

Published in: Nonlinearity, 13, 1239-1262 (2000).

1. Introduction

In this paper we consider the Lifshitz–Slyozov system [6]

\{ \begin{matrix}∂_{t}f(t, x) + ∂_{x}((k(x)c(t) − q(x))f(t, x)) = 0, & for & t ∈ (0, T), x > 0, \\ c(t) + ∫_{0}^{∞} xf(t, x) dx = ρ, & for & t ∈ (0, T). \end{matrix} (1.1)

This system models precipitation from a solution, viewed as the continuous addition of monomers to large clusters. Here c(t) denotes the monomer concentration at time t and f(t, x) is the size density of the cluster population. The variable x represents the volume of the clusters. The first equation is the kinetic equation for the cluster density, while the second expresses the conservation of the total mass ρ of the population. The first derivation of this system was given in the seminal paper [6], but recently some attempts have been made to obtain this system as a continuous limit of some more microscopic models. In this connection we refer the reader to [7] for a derivation of (1.1) by homogenization methods, and to [1] for a derivation from the Becker-Döring system, of a parabolic analog of (1.1).

The kinetic coefficients k(x) and q(x) represent respectively the rates at which monomers are added to or removed from clusters of size x. The exact dependence of these coefficients on the variable x is dictated by the physics of the precipitation/dissolution process. In [6], the mass transfer mechanism is monomer diffusion. In this case the kinetic coefficients are given by:

k(x) = 3x^{\frac{1}{3}} , q(x) = 3c_{∞}x^{\frac{1}{3}} + α, (1.2)

where c_{∞} denotes the monomer equilibrium concentration for clusters of infinite radius, and α is, within a constant multiplicative factor, the inter-phase surface tension. A detailed analysis of the large time asymptotics is then performed, and it is argued on physical grounds that the density f approaches (after suitable rescaling) a universal profile. The argument presented in [6] is in part formal, and recently, a number of workers have been trying to rigorously address the mathematical properties of the Lifshitz-Slyozov system ([8, 9, 5]). In this connexion, the following mathematical questions arise naturally:

• Under what assumptions on the coefficients k and q does one have existence and uniqueness of the solution (c, f), with f in some given class of functions or measures ?

• If existence and uniqueness are known, does the solution stabilize for large times to some universal profile ?

Clearly the answers to these questions depend in a crucial way on the regularity and asymptotic behaviour of the kinetic coefficients, and these properties in turn depend on the physics of the mass transfer mechanism. For a review on various mass transfer mechanisms and corresponding coefficients, we refer the reader to [12]. In this paper we will restrict ourselves to the case of C^{1} coefficients, with bounded derivative. This will enable us to use the method of characteristics, thereby getting a precise control of the various moments of the density f. By using this method we obtain existence and uniqueness of the solution for integrable or measure-valued data. Analysis of the case with coefficients (1.2) and integrable data is performed in [5]. In [2] some results of this paper are extended to a model taking into account the effect of encounters between particles. Another very interesting contribution is given in [8] in the case of compactly supported initial data. However, an essential difference between the model studied therein and (1.1) is that the mass conservation equation does not include the term c(t). The resulting system is simpler in the sense that it has an invariant scaling (as noted in [12]), and therefore lends itself to a detailed mathematical analysis.

The issue of large time behaviour of the Lifshitz Slyozov system is of central importance in physical chemistry. In this connection, the existence of a critical size is one of the factors which drive the system to a universal behaviour. The critical size is, at any time t, that unique value of x at which the Lagrangian growth rate of clusters changes sign. In fact this is the essence of the so-called Ostwald ripening: subcritical clusters tend to shrink, whereas supercritical ones tend to grow. Roughly stated, the universal behaviour obtained in [6] is the convergence of the density to a universal profile, after rescaling x by the critical size. A forthcoming publication [3] will address this issue. For related results on a variant of (1.1), we refer the reader to [9]. In this paper we investigate the asymptotic behaviour in a much simpler situation, in which both k(x) and q(x) are given by the same power of x. Physically this case does not lead to Ostwald ripening, since there is no critical radius: at any time, all clusters either shrink or grow, depending on the value of c(t). For more general types of interaction (the complete fragmentationcoagulation equations), such power rates have been studied in [11] by using a suitably defined Laplace transform of the solution. Given the simplicity of the dynamics here, one might think that either the monomer concentration, or the total amount of clusters, should converge to zero for large times. However by using the same method as in [11] we explicitly obtain convergence to some intermediate stationary state. The interesting physical feature obtained is that the asymptotic behaviour of the system is uniquely determined by the undersaturated or supersaturated nature of the data, so that in some loose sense the Lifshitz Slyozov model may be thought of as a model of phase transition.

The paper is organized as follows: in Section 2 we give the statement of our main results. In Section 3, we show the existence of integrable solutions for system (1.1). The proof uses a classical fixed point technique based on Schauder’s theorem. The required properties of the corresponding map are all derived from the expression of f in terms of c, as given by the method of characteristics. In Section 4 we establish a uniqueness result. In Section 5, we extend the results to the context of measure-valued solutions. Finally in Section 6, in some cases where the kinetic coefficients have a simple x−dependence (such as powers), we obtain explicit solutions by the use of the Laplace transform, and we give some results about the asympotic behaviour of the solution.

2. Main results

First of all, let us specify our technical assumptions on the kinetic coefficients. In all that follows, the functions q and k are supposed to be of class C^{1} on the interval [0, ∞[, and to satisfy the following:

∀x ∈ [0, ∞[: |k′(x)|, |q′(x)| ≤ K,  (2.1)
∀x ∈ [0, ∞[: k(x) ≥ 0, q(x) > 0, (2.2)
q(0), k(0) ≤ K, k(0)ρ − q(0) ≤ 0.  (2.3)

It is worth pointing out that assumption (2.3) is not only a technical assumption which implies that no boundary condition for f is needed at x = 0, but means that clusters of ”null size” are completely dissolved and fall out of consideration, as explained in [6]. Therefore, the characteristic curves (defined in (3.9) below) which intersect the axis x = 0 never return to the domain x > 0.

Let us now precise how solutions to the system will be understood. A pair (c, f) with c ∈ C^{0}([0, T]) and xf ∈ L^{∞}(0, T; L^{1}([0, ∞[)) is said to be a solution of the problem if the mass conservation equation holds for all t ∈ [0, T], and if f is the mild solution (see Appendix A for a definition and formula (3.21) below) of the transport equation. Our main existence result is the following:

Theorem 1 Assume that the kinetic coefficients satisfy (2.1)-(2.3). Let the initial data f_{0} be nonnegative and satisfy

∫_{0}^{∞} f_{0}(x) dx < ∞, ∫_{0}^{∞} xf_{0}(x) dx ≤ ρ.

Then the system (1.1) has a unique solution defined for all positive time.

When replacing (2.3) by the stronger condition k(0) = 0 = q(0), the boundary x = 0 becomes characteristic: any characteristic curve which intersects the line x = 0 is identically 0. In this case, our method of proof still works (actually in a simpler way) and allows us to weaken the requirements on the data f_{0}. Namely one has

Theorem 2 Assume that the kinetic coefficients satisfy k(0) = 0 = q(0) and (2.1)- (2.2). Let the initial data f_{0} be nonnegative with xf_{0}(x) ∈ L^{1}(R^{+}). Then the system (1.1) has a unique solution defined for all positive time.

Remark 1 In general, if the kinetic coefficients are not Lipschitz functions of x, a global solution may fail to exist. We give below (Section 6) a counterexample in which there is no global solution.

3. Existence of solutions for system (1.1)

For convenience we rewrite the problem as

\{ \begin{matrix}∂_{t}f + ∂_{x}(af) = 0 & for & t ∈ (0, T), x > 0, \\ f_{|t=0} = f_{0} & for & x ≥ 0, \end{matrix} (3.1)

where a is defined by

a(t, x) = k(x)c(t) − q(x) (3.2)

and c is related to f by

c(t) + ∫_{0}^{t} xf(t, x)dx = ρ. (3.3)

We will first prove existence for this problem on the time interval [0, T], for any fixed positive T. This will be done by finding a fixed point of the following map:

T : c ∈ C^{0}([0, T]) ⟼ c̃(t) = [ρ − ∫_{0}^{∞} xf(t, x)dx]_{+} , (3.4)

where f denotes the solution of (3.1), associated to c by (3.2). Here [·]_{+} stands for the positive part ([s]_{+} = max(s, 0)). The fixed point will be obtained by applying Schauder’s theorem to the set

B = {c ∈ C^{0}([0, T]), 0 ≤ c(t) ≤ ρ}, (3.5)

viewed as a (closed convex) subset of the space of continuous functions defined on the interval [0, T]. To obtain the necessary continuity and compactness properties of T we start by analyzing the linear problem (3.1-3.2) with c given.

3.1. Characteristic curves

In what follows we will repeatedly make use of the following inequalities, which are immediate consequences of (2.1)-(2.3), and hold true provided c ∈ B:

a(t, x) ≤ Ax for x ≥ 0, t > 0,  (3.6)
|a(t, x)| ≤ A(1 + x), for x ≥ 0, t > 0,  (3.7)
|a(t, x) − a(t, y)| ≤ A|x − y|, for x, y ≥ 0, t > 0,  (3.8)

where we have set A = K(1 + ρ). In particular, property (3.8) allows us to consider the characteristic curves associated to a. Namely, for all t ≥ 0, x ≥ 0, there exists a unique curve s ∈ R ⟼ X(s; t, x), solution of the following problem

\{ \begin{matrix}\frac{d}{ds}X(s; t, x) = a(s, X(s; t, x)), \\ X(t; t, x) = x. \end{matrix} (3.9)

Let us now derive a few bounds on X(s; t, x). In the statement of all the estimates given below, it is always implicitly assumed that s, t, x are such that X(s; t, x) > 0, that is, we consider characteristic curves before their exit time. Indeed, all the formulae used further below for deriving existence and properties of solutions never involve characteristics beyond their exit time.

Lemma 1 Let c be in the set B. Assume that (2.1)-(2.3) hold true. Then for x ≥ 0, t > 0, as long as it does not exit the domain X > 0, the characteristic curve X(s; t, x) defined by (3.9) satisfies

for s_{1} ≤ s_{2} ≤ t : 0 ≤ X(s_{2}; t, x) ≤ X(s_{1}; t, x)e^{A(s_{2}−s_{1})},  (3.10)
if x_{n} → ∞, then for all t > 0 : X(0; t, x_{n}) → ∞,  (3.11)
for s ≥ t : X(s; t, x) ≤ xe^{A(s−t)},  (3.12)
for all t ≥ 0 : X(t; 0, y) ≤ ye^{At},  (3.13)
for all s, t : |a(s, X(t; 0, y))| ≤ A(1 + ye^{At}),  (3.14)
for s_{2} ≤ s_{1} ≤ t : X(s_{2}; t, x) ≤ (X(s_{1}; t, x) + 1)e^{A(s_{1}−s_{2})} − 1.  (3.15)

Proof. We first remark that in view of (2.3), if x > 0, then for all s ≤ t we have X(s; t, x) > 0. More generally X(s; t, x) ≥ 0 implies X(σ; t, x) ≥ 0 for all 0 ≤ σ ≤ s: non negative points may be reached only from non negative earlier positions. Let us fix s_{1} ≤ s ≤ s_{2} ≤ t. From the definition of the characteristic curves and inequality (3.6) we get

\begin{matrix}0 ≤ X(s; t, x) & = & X(s_{1}; t, x) + ∫_{s_{1}}^{s} a(σ, X(σ; t, x)) dσ \\ & ≤ & X(s_{1}; t, x) + A ∫_{s_{1}}^{s} X(σ; t, x) dσ. \end{matrix} (3.16)

Applying Gronwall’s lemma on the interval [s_{1}, s_{2}], we are led to (3.10). The limit (3.11) now follows upon taking s_{2} = t, s_{1} = 0. Inequality (3.12) is obtained by exactly the same method as (3.10). (3.13) is a particular case of (3.12). Combining (3.13) and (3.7) yields (3.14). Finally, (3.15) is obtained by applying Gronwall’s lemma to the quantity y(s) = X(t − s; t, x) and making use of inequality (3.7).

In order to obtain various continuity properties we will need to estimate the distance between the characteristic curves associated to two differents fields.

Lemma 2 Consider the characteristic curves X_{1}, X_{2} associated to given functions c_{1}, c_{2} and let z(s; t, x) = X_{2}(s; t, x) − X_{1}(s; t, x). Then the following inequalities

\begin{matrix}for t ≤ t_{1} ≤ t_{2} : & |z(t_{2}; t, x)| ≤ |z(t_{1}; t, x)|e^{A_{2}(t_{2}−t_{1})} \\ & +Ke^{A_{2}t_{2}} ∫_{t_{1}}^{t_{2}}(1 + xe^{A_{1}(s−t)})|c_{2}(s) − c_{1}(s)|e^{−A_{2}s} ds, \end{matrix} (3.17)

\begin{matrix}for t_{2} ≤ t_{1} ≤ t : & |z(t_{2}; t, x)| ≤ |z(t_{1}; t, x)|e^{A_{2}(t_{1}−t_{2})} \\ & +K(x + 1)e^{(A_{1}t−A_{2}t_{2})} ∫_{t_{2}}^{t_{1}} |c_{2}(s) − c_{1}(s)|e^{(A_{2}−A_{1})s} ds \end{matrix} (3.18)

hold true. Here A_{1,2} denotes any constant greater than K(1 + sup_{t∈(0,T)} |c_{1,2}|).

Proof. We have

z(s; t, x) = z(t_{1}; t, x) + ∫_{t_{1}}^{s}{a_{2}(σ, X_{2}(σ; t, x)) − a_{1}(σ, X_{1}(σ; t, x))} dσ. (3.19)

The integrand may be estimated as follows

|a_{2}(σ, X_{2}(σ; t, x)) − a_{1}(σ, X_{1}(σ; t, x))| ≤ |a_{2}(σ, X_{2}(σ; t, x)) − a_{2}(σ, X_{1}(σ; t, x))|
+ |a_{2}(σ, X_{1}(σ; t, x)) − a_{1}(σ, X_{1}(σ; t, x))|
≤ A_{2}|z(σ; t, x)|
+ K(1 + X_{1}(σ; t, x))|c_{2}(σ) − c_{1}(σ)|.

Coming back to (3.19) we obtain a differential inequality for |z|, and Gronwall’s lemma gives

|z(t_{2}; t, x)| ≤ |z(t_{1}; t, x)|e^{A_{2}(t_{2}−t_{1})} + K ∫_{t_{1}}^{t_{2}}(1 + X_{1}(s; t, x))|c_{2}(s) − c_{1}(s)|e^{A_{2}(t_{2}−s)} ds.

Inserting (3.12) into this inequality we are led to (3.17). As for the case of (3.15), the same method applied to the quantity y(s) = z(t − s; t, x) yields the inequality (3.18).

3.2. Mild solution

Let c be given in the set B. We define a mild solution of (3.1-3.2) by integrating the equation along the characteristic curves. Let us write (3.1) in the following non conservative form

∂_{t}f + a∂_{x}f − bf = 0 (3.20)

where in view of (2.1), b(t, x) = −∂_{x}a(t, x) = q′(x)−c(t)k′(x) belongs to L^{∞}((0, T)×R^{+}). The mild solution f is defined for almost every (t, x) ∈ (0, T) × R^{+} by the relation

f(t, x) = f_{0}(X(0; t, x)) exp (∫_{0}^{t} b(σ, X(σ; t, x)) dσ) . (3.21)

(see Appendix A for details). Let us now deduce from (3.21) some integrability properties of the mild solution.

Lemma 3 Let c ∈ B, f_{0} ≥ 0 be fixed, and let θ be a fixed nonnegative number. Assume that (2.1)-(2.3) hold. Let f be defined by (3.21). If x^{θ}f_{0}(x) ∈ L^{1}(R^{+}), then for all t ∈ (0, T), x^{θ}f(t, x) ∈ L^{1}(R^{+}). If the zeroth and first moments are initially finite, then first moment M_{1}(t) of f is a C^{1} function of time, and its time derivative is given by

\frac{dM_{1}(t)}{dt} = ∫_{X(0;t,0)}^{∞} a(t, X(t; 0, y))f_{0}(y) dy = ∫_{0}^{∞} a(t, x)f(t, x) dx. (3.22)

Proof. Let us introduce the natural change of variable y = X(0; t, x). It follows from (A.2) that this change of variables is a diffeomorphism and x = X(t; 0, y). Moreover, x ≥ 0 implies y ≥ 0 by (3.10) while x is non negative provided y ≥ X(0; t, 0) ≥ 0. One deduces from (3.21) and (A.3) that

\begin{matrix}∫_{x_{1}}^{x_{2}} x^{θ}f(t, x) dx & = & ∫_{x_{1}}^{x_{2}} x^{θ}f_{0}(X(0; t, x)) exp (∫_{0}^{t} b(σ, X(σ; t, x)) dσ) dx \\ & = & ∫_{X(0;t,x_{1})}^{X(0;t,x_{2})}(X(t; 0, y))^{θ}f_{0}(y) dy. \end{matrix} (3.23)

Therefore, from (3.10) and (3.13) we obtain

0 ≤ ∫_{x_{1}}^{x_{2}} x^{θ}f(t, x) dx ≤ exp(θAt) ∫_{X(0;t,x_{1})}^{X(0;t,x_{2})} y^{θ}f_{0}(y) dy
≤ exp(θAt) ∫_{x_{1} exp (−At)}^{∞} y^{θ}f_{0}(y) dy.

This shows that the moments which are initially finite remain so for all time t ∈ (0, T).

We now investigate differentiability of the first moment M_{1}(t). To this end we compute the difference quotient

\frac{M_{1}(t + h) − M_{1}(t)}{h} ; = \frac{1}{h} (∫_{X(0;t+h,0)}^{∞} X(t + h; 0, y)f_{0}(y) dy − ∫_{X(0;t,0)}^{∞} X(t; 0, y)f_{0}(y) dy) . (3.24)

It is relevant to write this quotient as sum of integrals where the domains of integration are such that the quantities X(·; 0, y) involved in these integrals are non negative. Then, we remark that

\begin{matrix}X(0; t + h, 0) − X(0; t, 0) & = & ∫_{t}^{t+h} \frac{∂X}{∂τ} (0; τ, 0) dτ \\ & = & − ∫_{t}^{t+h} a(τ, 0) exp (∫_{0}^{τ} b(σ; τ, x)dσ) dτ \end{matrix} (3.25)

by (A.4). Since a(τ, 0) = k(0)c(τ) − q(0) ≤ k(0)ρ − q(0) ≤ 0, X(0; t + h, 0) − X(0; t, 0) has the sign of h and, furthermore

|X(0; t + h, 0) − X(0; t, 0)| ≤ Ae^{AT} h. (3.26)

In other words, if h > 0 (resp. h < 0), then X(0; t + h, 0) decreases to X(0; t, 0) (resp. increases) with a speed of order h. We thus write

\begin{matrix}\frac{M_{1}(t + h) − M_{1}(t)}{h} & = & \frac{1}{h} ∫_{X(0;t,0)}^{∞}(X(t + h; 0, y) − X(t; 0, y))f_{0}(y) dy \\ & + & \frac{1}{h} ∫_{X(0;t+h,0)}^{X(0;t,0)} X(t + h; 0, y)f_{0}(y) dy if h < 0, \end{matrix} (3.27)

and

\begin{matrix}\frac{M_{1}(t + h) − M_{1}(t)}{h} & = & \frac{1}{h} ∫_{X(0;t+h,0)}^{∞}(X(t + h; 0, y) − X(t; 0, y))f_{0}(y) dy \\ & + & \frac{1}{h} ∫_{X(0;t,0)}^{X(0;t+h,0)} X(t; 0, y)f_{0}(y) dy  if h > 0. \end{matrix}  (3.28)

We only detail the case h < 0. The first integral in (3.27) may be rewritten as:

∫_{X(0;t,0)}^{∞}(\frac{1}{h} ∫_{t}^{t+h} a(s, X(s; 0, y)) ds f_{0}(y)) dy.

By the Lebesgue differentiation theorem, as h goes to 0, the integrand in this last integral converges pointwise to a(t, X(t; 0, y))f_{0}(y) for every t and almost every y. On the other hand from (3.14) we obtain

\begin{aligned}|\frac{f_{0}(y)}{h} ∫_{t}^{t+h} a(s, X(s; 0, y)) ds| & ≤ & \frac{f_{0}(y)}{|h|} ∫_{t}^{t+h} A(1 + y exp(As)) ds \\ & ≤ & A(1 + y exp(AT))f_{0}(y). \end{aligned}

Thus we may conclude by the Lebesgue dominated convergence theorem that the first integral in (3.27) converges to the right hand side of (3.22).

Let us rewrite the second integral in (3.27) as follows

\begin{matrix}\frac{1}{h} ∫_{X(0;t+h,0)}^{X(0;t,0)} X(t + h; 0, y)f_{0}(y) dy & = & \frac{1}{h} ∫_{X(0;t+h,0)}^{X(0;t,0)} X(t; 0, y)f_{0}(y) dy \\ & + & \frac{1}{h} ∫_{X(0;t+h,0)}^{X(0;t,0)} ∫_{t}^{t+h} a(s, X(s; 0, y)) dsf_{0}(y) dy. \end{matrix} (3.29)

The second integral in the right hand side of (3.29) may be shown to converge to zero by using exactly the same argument, thanks to inequality (3.14). Thus it only remains to show that

for a.e. t ∈ [0, T] : lim_{h→0} \frac{1}{h} ∫_{X(0;t+h,0)}^{X(0;t,0)} X(0; t, y)f_{0}(y) dy = 0. (3.30)

The proof of this fact is technical, and therefore will be given in Appendix B. Thus admitting this fact, (3.22) is established. The derivative of M_{1} may then be shown to be a continuous function of time by proceeding analogously, i.e. using the continuity properties of a and X, and dominated convergence. This completes the proof of the lemma.

Remark 2 A similar result may be obtained by duality as in [10] where relations of the form ∫_{0}^{∞} f(t, x)ψ(x)dx = ∫_{0}^{∞} f_{0}(x)ψ(X(t; 0, x))dx are derived, where ψ is a test function.

3.3. Existence : fixed point step

This section is devoted to the proof of the following Lemma 4 Assume that (2.1)-(2.3) hold true. Let f_{0} ∈ L^{1}(R^{+}, xdx) ∩ L^{1}(R^{+}, dx) be nonegative almost everywhere. Then, the map T defined in (3.4) maps the set B into itself, is continuous, and the image of B is a precompact subset of C^{0}([0, T]).

Here, continuity and compactness are understood for the uniform norm on [0, T]. Proof. The result discussed in the previous section allows us to consider the map T defined in (3.4). By construction, it is obvious that T (B) ∈ B. Moreover, writing c̃(t) = T (c)(t), it follows from Stampacchia’s theorem that

\frac{d}{dt}c̃(t) = \{ \begin{matrix}0 & if & ∫_{0}^{∞} xf(t, x)dx > ρ, \\ −\frac{d}{dt} (∫_{0}^{∞} xf(t, x)dx) & & otherwise. \end{matrix}

Thus Lemma 3 implies that c̃ belongs to W^{1,∞}(0, T) and Ascoli’s theorem gives the compactness statement. It remains to show the continuity of T . Let c_{n} be a sequence of elements in B which converges to c ∈ B. We need to show that

lim_{n→∞} sup_{0≤t≤T} |∫_{0}^{∞} xf_{n}(t, x)dx − ∫_{0}^{∞} xf(t, x)dx| = 0. (3.31)

Using (3.17) with A = A_{1} = A_{2} we have

|z_{n}(t; 0, y)| := |X_{n}(t; 0, y)−X(t; 0, y)| ≤ (1+y)KT e^{AT} ∥c_{n} −c∥_{C^{0}(0,T)}(3.32)

since X_{n}(0; 0, y) = y = X(0; 0, y). Thus, we obtain

|∫_{0}^{∞} xf_{n}(t, x)dx − ∫_{0}^{∞} xf(t, x)dx|

= |∫_{X_{n}(0;t,0)}^{∞} X_{n}(t; 0, y)f_{0}(y)dy − ∫_{X(0;t,0)}^{∞} X(t; 0, y)f_{0}(y)dy|

≤ ∫_{X(0;t,0)}^{∞} |z_{n}(t; 0, y)|f_{0}(y) dy + |∫_{X_{n}(0;t,0)}^{X(0;t,0)} X_{n}(t; 0, y)f_{0}(y)dy|

≤ KT e^{AT} ∥c_{n} − c∥_{C^{0}(0,T)} ∫_{0}^{∞} (1 + y)f_{0}(y)dy + |∫_{X_{n}(0;t,0)}^{X(0;t,0)} X_{n}(t; 0, y)f_{0}(y)dy| .

It remains to treat the last integral. To this end, we write

\begin{matrix}|X_{n}(t; 0, y)| & = & |X_{n}(t; 0, y) − X_{n}(t; 0, X_{n}(0; t, 0))| \\ & = & |∫_{y}^{X_{n}(0;t,0)} \frac{∂X_{n}}{∂x} (t; 0, ξ) dξ| \\ & = & |∫_{y}^{X_{n}(0;t,0)} exp (− ∫_{0}^{t} b(σ, X_{n}(σ; 0, ξ)) dσ) dξ| \\ & ≤ & e^{At}|y − X_{n}(0; t, 0)|. \end{matrix} (3.33)

Here, we used successively (A.3) and (2.1). However, as y lies between X(0; t, 0) and X_{n}(0; t, 0), we can estimate this last quantity by e^{At}|X(0; t, 0) − X_{n}(0; t, 0)| = e^{At}|z_{n}(0; t, 0)| ≤ Kte^{2At}∥c_{n} − c∥_{C^{0}(0,T)} by using (3.18). Let us denote by χ_{n} the characteristic function of the segment defined by X_{n}(0; t, 0) and X(0; t, 0). Therefore, X_{n}(t; 0, y)χ_{n}(y) converges to 0 uniformly in t and y as n → 0. This suffices to end the proof of the required convergence (3.31).

We are now ready to derive the existence of solutions for bounded times:

Lemma 5 Under the assumptions of Theorem 1, for any T > 0, the problem (1.1) has at least one solution on the time interval [0, T].

Proof. By Schauder’s theorem, we obtain a fixed point c ∈ B of the map T as a consequence of Lemma 4. To ensure that the fixed point actually is a solution of the Lifshitz-Slyozov system (1.1), it remains to show that c satisfies (3.3) instead of the relation given by (3.4) which involves the truncation [·]_{+}. Suppose there exists a time t_{∗} ∈ [0, T] such that

∫_{0}^{∞} xf(t_{∗}, x)dx = ρ and ∫_{0}^{∞} xf(t, x)dx < ρ for t < t_{∗}, (3.34)

f being the solution of (3.1-3.2) associated to c. Note that (3.34) makes sense in view of the regularity result in Lemma 3. Thus, T (c) = c implies c(t_{∗}) = 0 and it follows that

\frac{d}{dt} (∫_{0}^{∞} xf(·, x)dx) (t_{∗}) = ∫_{0}^{∞} a(t_{∗}, x)f(t_{∗}, x)dx = − ∫_{0}^{∞} q(x)f(t_{∗}, x)dx (3.35)

is non positive since q(x)f(t_{∗}, x) ≥ 0 a.e. in R^{+}. Moreover, the quantity (3.35) vanishes only if q(x)f(t_{∗}, x) = 0 a.e. in R^{+}, which would contradict (2.2) and (3.34). Thus, we actually have

\frac{d}{dt} (∫_{0}^{∞} xf(·, x)dx) (t_{∗}) < 0. (3.36)

If t_{∗} = 0, one deduces that ∫_{0}^{∞} xf(t, x)dx < ρ in a neighborhood of the initial time. If

t_{∗} > 0, then (3.36) implies that t ⟼ ∫_{0}^{∞} xf(t, x)dx is non increasing for t < t_{∗}, t close

to t_{∗} which contradicts (3.34). Finally, one obtains ∫_{0}^{∞} xf(t, x)dx < ρ for all t > 0, and

(3.3) holds. This completes the proof of the lemma.

4. Uniqueness

This section is devoted to the uniqueness proof for (1.1). More precisely, we show the following continuity result.

Lemma 6 Assume that (2.1) and (2.3) hold true. For i ∈ {1, 2}, let f_{i}, c_{i} be a solution of (3.1-3.3) with initial data f_{0,i} ∈ L^{1}(R^{+}, (1 + x) dx), and total mass ρ_{i}. Then, there exists a constant C, depending only on K, ρ_{i}, f_{0,i} and T, such that

∥c_{2} − c_{1}∥_{C^{0}(0,T)} ≤ C (|ρ_{2} − ρ_{1}| + ∥f_{0,2} − f_{0,1}∥_{L^{1}(R^{+},xdx)}) . (4.1)

The uniqueness result immediately follows. Indeed if ρ_{2} = ρ_{1} and f_{0,2} = f_{0,1} we obtain c_{2} = c_{1}, which yields the equality of the characteristic curves, thus f_{2} = f_{1}. This uniqueness result and Lemma 5 immediately yield Theorem 1. Proof. Using (3.3) and (3.23), we get

|c_{2}(t) − c_{1}(t)| ≤ |ρ_{2} − ρ_{1}| + I_{1} + I_{2} + I_{3}, (4.2)

with

\{ \begin{matrix}I_{1} = ∫_{X_{1}(0;t,0)}^{∞} |z(t; 0, y)|f_{0,2}(y) dy, \\ I_{2} = ∫_{X_{1}(0;t,0)}^{∞} X_{1}(t; 0, y)|f_{0,2}(y) − f_{0,1}(y)| dy, \\ I_{3} = |∫_{X_{2}(0;t,0)}^{X_{1}(0;t,0)} X_{2}(t; 0, y)f_{0,2}(y) dy| , \end{matrix} (4.3)

where z still denotes X_{2} − X_{1}. We now treat these three integrals. From (3.17) we obtain

|z(t; 0, y)| ≤ K(1 + y)e^{A_{2}t} ∫_{0}^{t} |c_{2}(s) − c_{1}(s)| exp((A_{1} − A_{2})s) ds,

thus

I_{1} ≤ K exp((A_{1} + A_{2})T)(M_{1}(f_{0,2}) + M_{0}(f_{0,2})) ∫_{0}^{t} |c_{2}(s) − c_{1}(s)| ds.

Similarly (3.13) leads to

I_{2} ≤ e^{A_{1}T} ∫_{0}^{∞} y|f_{0,2}(y) − f_{0,1}(y)| dy.

Finally, we treat I_{3} by using a method similar to that used for (3.33). Indeed, we have

I_{3} = |∫_{X_{2}(0;t,0)}^{X_{1}(0;t,0)}(X_{2}(t; 0, y) − X_{2}(t; 0, X_{2}(0; t, 0))) f_{0,2}(y) dy|
≤ |∫_{X_{2}(0;t,0)}^{X_{1}(0;t,0)} (∫_{X_{2}(0;t,0)}^{y} \frac{∂X_{2}}{∂x} (t; 0, ξ) dξ) f_{0,2}(y) dy|
≤ |∫_{X_{2}(0;t,0)}^{X_{1}(0;t,0)} e^{A_{2}t}|X_{1}(0; t, 0) − X_{2}(0; t, 0)| f_{0,2}(y) dy|
≤ e^{A_{2}t}|z(0; t, 0)| ∫_{0}^{∞} f_{0,2}(y) dy.

Hence, (3.18) yields

\begin{aligned}I_{3} & ≤ & e^{A_{2}t}Ke^{A_{1}t} ∫_{0}^{t} |c_{2}(s) − c_{1}(s)|e^{(A_{2}−A_{1})s} ds∥f_{0,2}∥_{L^{1}(R,dx)} \\ & ≤ & k(T) ∫_{0}^{t} |c_{2}(s) − c_{1}(s)|e^{(A_{2}−A_{1})s} ds \end{aligned}

Collecting these three inequalities and coming back to (4.2) we obtain the following estimate

\begin{matrix}|c_{2}(t) − c_{1}(t)| ≤ |ρ_{2} − ρ_{1}| \\ +k(T) ∫_{0}^{t} |c_{2}(s) − c_{1}(s)| ds + k(T)M_{1}(|f_{0,2} − f_{0,1}|). \end{matrix}

Applying Gronwall’s lemma again gives the asserted result.

5. Measure-valued solutions

In this section, we extend the result obtained previously to the context of measurevalued initial data. Here we use the classical notion of measure-valued solution, which for the reader’s convenience, is recalled in Appendix A. In that setting, our existence result is the following.

Theorem 3 Assume that the kinetic coefficients satisfy (2.1)-(2.3). Let the initial data μ_{0} be a nonnegative measure on R^{+} satisfying

⟨μ_{0}, 1⟩ < ∞, ⟨μ_{0}, x⟩ ≤ ρ.

Then the system (1.1) has a unique solution defined for all positive time. If the kinetic coefficients satisfy k(0) = 0 = q(0) and (2.1)-(2.2), it suffices to assume that the condition on the first moment is fulfilled.

One important case is the one where the initial datum is a Dirac measure, which means that clusters are perfectly localized in size: the system is then said to be monodisperse (see [4]). Let us indicate some modifications in the proof which allow us to deal with this less regular case.

Assume that the initial data μ_{0} lies in M_{+}^{1}, the space of positive bounded Borel measures on R^{+}. Then, we define a family of measures on R^{+}, parametrized by t, by using the following natural generalization of formulation (3.21)

μ_{t}(B) = μ_{0}(X(0; t, B)) (5.1)

for any Borel set B in R^{+}, or equivalently

⟨μ_{t}(x), ϕ(x)⟩ = ⟨μ_{0}(y), ϕ(X(t; 0, y))χ_{[X(0;t,0),∞[}(y)⟩, (5.2)

with

χ_{[X(0;t,0),∞[}(y) = \{ \begin{matrix}1 if y ≥ X(0; t, 0), \\ 0 otherwise. \end{matrix} (5.3)

Relation (5.2) makes sense for all test function ϕ ∈ C_{0}^{0}(R^{+}), continuous and vanishing at infinity, since [X(0; t, 0), ∞[ is a Borel set and y ⟼ ϕ(X(t; 0, y)) is bounded on this set.

As in the integrable case, (5.2) clearly preserves positivity and moment properties of μ_{0}. Crucial to our approach is the discussion of the derivability property of the first moment.

Lemma 7 Let c be in the set B and μ_{0} ∈ M_{+}^{1}. Assume that (2.1)-(2.3) hold. Let μ be defined by (5.2). If the zeroth and first moments are initially finite, then the first moment M_{1}(t) = ⟨μ_{t}, x⟩ of μ_{t} is a C^{0} function of time and for all time t, M_{1} has bounded left and right derivatives on t. Precisely, t is a point of discontinuity of \frac{dM_{1}}{dt} if X(0; t, 0) is an atomic point of μ_{0} (see formula (5.9) and (5.11) below). In particular \frac{dM_{1}(t)}{dt} belongs to W^{1,∞}((0, T)).

Proof. From now on, we adopt the notation χ_{t}(y) to designate the characteristic function defined by (5.3). Similarly, for T ≥ s ≥ t ≥ 0, we recall that X(0; s, 0) ≥ X(0; t, 0) ≥ 0 and we set

χ_{t}^{s}(y) = χ_{[X(0;t,0),X(0;s,0)]}(y) = \{ \begin{matrix}1 if y ∈ [X(0; t, 0), X(0; s, 0)], \\ 0 otherwise. \end{matrix}

In order to discuss continuity-differentiability property of M_{1} in the measure-valued case, it is necessary to treat separately the approximation from the right and from the left, i. e. accroding to the sign of h, of the derivative. Thus, we rewrite the quantity

M_{1}(t + h) − M_{1}(t) = ⟨μ_{0}(y), X(t + h; 0, y)χ_{t+h}(y)⟩ − ⟨μ_{0}(y), X(t; 0, y)χ_{t}(y)⟩ as follows

M_{1}(t + h) − M_{1}(t) \{ \begin{matrix}= ⟨μ_{0}(y), (X(t + h; 0, y) − X(t; 0, y))χ_{t+h}(y)⟩ \\ −⟨μ_{0}(y), X(t; 0, y)χ_{t}^{t+h}(y)⟩ \\ = I_{h}^{r} + J_{h}^{r},  for h > 0, \\ = ⟨μ_{0}(y), (X(t + h; 0, y) − X(t; 0, y))χ_{t}(y)⟩ \\ +⟨μ_{0}(y), X(t; 0, y)χ_{t+h}^{t}(y)⟩ \\ = I_{h}^{l} + J_{h}^{l},  for h < 0. \end{matrix} (5.4)

We denote I_{h}^{r} = ⟨μ_{0}(y), Φ_{h}^{r}(y)⟩, J_{h}^{r} = ⟨μ_{0}(y), Ψ_{h}^{r}(y)⟩ and similar notation with l replacing r. We easily check that Φ_{h}^{r,l}(y) and Φ_{h}^{r,l}(y) are dominated by μ_{0}−integrable functions (for instance 2e^{AT} (1 + y), by (3.13)). Moreover, dividing by h yields

\begin{matrix}|1/h Φ_{h}^{r,l}(y)| & = & \frac{1}{|h|} |∫_{t}^{t+h} \frac{∂X}{∂s} (s; 0, y) ds| = \frac{1}{|h|} |∫_{t}^{t+h} a(s, X(s; 0, y)) ds| \\ & ≤ & \frac{1}{|h|} |∫_{t}^{t+h} A(1 + X(s; 0, y)) ds| ≤ A(1 + e^{AT} y), \end{matrix} (5.5)

by using (3.7) and (3.13). On the other hand, we also have (for the sake of clarity we drop the χ)

\begin{matrix}|1/h Ψ_{h}^{r,l}(y)| = \frac{1}{|h|}|X(t; 0, y) − X(t; 0, X(0; t, 0))| = \frac{1}{|h|} |∫_{X(0;t,0)}^{y} \frac{∂X}{∂ξ} (t; 0, ξ) dξ| \\ = \frac{1}{|h|} |∫_{X(0;t,0)}^{y} exp (∫_{t}^{0} b(σ, X(σ; 0, ξ)) dσ) dξ| ≤ \frac{e^{AT}}{|h|} |y − X(0; t, 0)| \\ ≤ \frac{e^{AT}}{|h|} |X(0; t + h, 0) − X(0; t, 0)|, \end{matrix} (5.6)

by using (A.3) and taking into account only the y between X(0; t, 0) and X(0; t + h, 0). Therefore, we obtain

|1/h Ψ_{h}^{r,l}(y)| ≤ Ae^{2AT} . (5.7)

Hence, it will be possible to pass to the limit h → 0 by using the Lebesgue dominated convergence theorem in the relations (5.4) and 1/h(5.4) if we are able to pass to the limit pointwise (everywhere, and thus μ_{0}-a.e.) in Φ_{h}^{r,l}(y), Ψ_{h}^{r,l}(y), and 1/h Φ_{h}^{r,l}(y), 1/h Ψ_{h}^{r,l}(y).

As in Appendix B, we can distinguish three cases depending on the local behaviour of c. First, if

c(τ) = ρ (5.8)

in a neighborhood of t, then X(0; t + h, 0) = X(0; t, 0) for |h| sufficiently small, and the corresponding J_{h} vanishes. Second, if (5.8) holds for t + h_{0} ≤ τ < t , then we have

• for y > X(0; t, 0), Φ_{h}^{l}(y) = X(t + h; 0, y) − X(t; 0, y) → 0 and

\frac{Φ_{h}^{l}(y)}{h} = \frac{X(t + h; 0, y) − X(t; 0, y)}{h} → a(t, X(t; 0, y)),

• for y < X(0; t, 0), Φ_{h}^{l}(y) = 0, since χ_{t}(y) = 0,

• for y = X(0; t, 0), Φ_{h}^{l}(y) = X(t + h; 0, y) − 0 → X(t; 0, X(0; t, 0)) = 0 and

\frac{Φ_{h}^{l}(y)}{h} → a(t, X(t; 0, y)) = a(t, 0).

This argument may be reproduced in the case where (5.8) holds for t ≤ τ ≤ t + h_{0}. Third, and finally, we treat the case where c(τ) < ρ, and thus a(τ, 0) < 0, for τ in a neighborhood of t.

First, we consider the left approximation, i.e. h < 0:

• for y < X(0; t, 0), Φ_{h}^{l}(y) = 0, since χ_{t}(y) = 0,

• for y ≥ X(0; t, 0), Φ_{h}^{l}(y) = X(t + h; 0, y) − X(t; 0, y) → 0 and

\frac{Φ_{h}^{l}(y)}{h} = \frac{X(t + h; 0, y) − X(t; 0, y)}{h} → a(t, X(t; 0, y))

Next, we are concerned with Ψ_{h}^{l}. The sign condition on a implies that X(0; t + h, 0) increases to X(0; t, 0) as h < 0 goes to 0. Let y < X(0; t, 0). There exists h_{y} < 0 such that for all h_{y} < h < 0, one has y < X(0; t + h, 0) < X(0; t, 0) which forces χ_{t+h}^{t}(y) = 0 and thus Ψ_{h}^{l}(y) = 0. For y = X(0; t, 0), one has Ψ_{h}^{l}(y) = X(t; 0, X(0; t, 0)) = 0. This allows us to conclude that

lim_{h→0^{−}} J_{h}^{l} = 0 = lim_{h→0^{−}} 1/h J_{h}^{l}

while we showed that lim_{h→0^{−}} I_{h}^{l} = 0 and

lim_{h→0^{−}} 1/h I_{h}^{l} = ⟨μ_{0}(y), a(t, X(t; 0, y))χ_{t}(y)⟩.

We now turn to the right approximation, i.e. h > 0. Reasoning as above, we show that Ψ_{h}^{r}(y) vanishes for h > 0 small enough, and thus lim_{h→0^{+}} J_{h}^{r} = 0 = lim_{h→0^{+}} 1/h J_{h}^{r}. The discussion for I_{h}^{r} is slightly different. We have to consider three situations:

• if y < X(0; t, 0) < X(0; t + h, 0), then χ_{t+h}(y) = 0 and Φ_{h}^{r}(y) = 0,

• if y > X(0; t, 0), then by the sign condition on a there exists h_{y} > 0 such that X(0; t, 0) ≤ X(0; t + h, 0) < y for all 0 < h < h_{y} which leads to

\{ \begin{matrix}Φ_{h}^{r}(y) = X(t + h; 0, y) − X(t; 0, y) → 0, \\ 1/h Φ_{h}^{r}(y) → a(t, X(t; 0, y)), \end{matrix}

as h goes to 0.

• if y = X(0; t, 0), then for all h > 0, X(0; t + h, 0) > X(0; t, 0) = y which forces Φ_{h}^{r}(y) = 0 since χ_{t+h}(y) = 0. Denoting χ_{t}^{∗}(y) = χ_{]X(0;t,0),∞[}(y) (1 for y ∈]X(0; t, 0), ∞[, 0 otherwise), we conclude that

\{ \begin{matrix}lim_{h→0^{+}} I_{h}^{r} = 0, \\ lim_{h→0^{+}} 1/h I_{h}^{r} = ⟨μ_{0}(y), a(t, X(t; 0, y))χ_{t}^{∗}(y)⟩. \end{matrix}

Finally we can summarize the previous discussion by defining left and right derivative of M_{1} as follows

\{ \begin{matrix}D^{l}M_{1}(t) & = & lim_{h→0^{−}} \frac{M_{1}(t + h) − M_{1}(t)}{h} = ⟨μ_{0}(y), a(t, X(t; 0, y))χ_{t}(y)⟩, \\ D^{r}M_{1}(t) & = & lim_{h→0^{+}} \frac{M_{1}(t + h) − M_{1}(t)}{h} = ⟨μ_{0}(y), a(t, X(t; 0, y))χ_{t}^{∗}(y)⟩ \\ & = & ⟨μ_{0}(y), a(t, X(t; 0, y))χ_{t}(y)⟩ − a(t, 0)μ_{0}(\{X(0; t, 0)\}). \end{matrix} (5.9)

We note that D^{r}M_{1}(t) ≥ D^{l}M_{1}(t) and actually \frac{d}{dt}M_{1}(t) is continuous on t provided

either a(t, 0) = 0 or \{X(0; t, 0)\} is μ_{0}−negligible, while \frac{d}{dt}M_{1}(t) has a jump when X(0; t, 0) is an atomic point of μ_{0}.

Equation (5.9) is maybe clearer if we split μ_{0} into its atomic and its diffuse part as follows

μ_{0} = ∑_{n∈N} c_{n}δ_{y_{n}} + ν_{0} (5.10)

where ν_{0}({y}) = 0 for all y ≥ 0 and in view of the assumptions on μ_{0}, we have c_{n} ≥ 0, ν_{0} ≥ 0, ⟨ν_{0}, 1⟩ + ∑_{n∈N} c_{n} < ∞ and ⟨ν_{0}(y), y⟩ + ∑_{n∈N} y_{n}c_{n} = ρ. Now, formula (5.9) reads as

follows

\{ \begin{matrix}if X(0; t, 0) ≠ y_{k}, \\ \frac{d}{dt}M_{1}(t) = ∑_{n∈N} c_{n}a(t, X(t; 0, y_{n}))χ_{t}(y_{n}) + ⟨ν_{0}(y), a(t, X(t; 0, y))χ_{t}(y)⟩, \\ if X(0; t, 0) = y_{k}, \\ \frac{d}{dt}M_{1}(t) has left and right value \\ \{ & \begin{matrix}D^{l}M_{1}(t) = ∑_{n∈N} c_{n}a(t, X(t; 0, y_{n}))χ_{t}(y_{n}) + ⟨ν_{0}(y), a(t, X(t; 0, y))χ_{t}(y)⟩, \\ D^{r}M_{1}(t) = D^{l}M_{1}(t) − c_{k}a(t, 0). \end{matrix} \end{matrix} (5.11)

Actually, this formula means that clusters which are initially at position y_{n} = X(0; t_{n}, 0) are dissolved at time t_{n} and then fall out of consideration. We conclude by remarking that, in view of the estimates on a and X(t; 0, y), the right hand sides in (5.11) are bounded on (0, T).

Next, we turn to the fixed point step, and we modify the definition of T as follows

T : c ∈ B ⟼ c̃(t) = [ρ − ⟨μ_{t}(x), x⟩]_{+} ,

where μ is associated to c by (5.2). Clearly T (B) ⊂ B while, as in Section 3, Lemma 7 provides the W^{1,∞}(0, T) regularity of the image of B. Let c_{n} be a sequence of elements in B which converges to c ∈ B and prove the following measure-valued analogue to (3.31)

lim_{n→∞} sup_{0≤t≤T} |⟨(μ_{n})_{t}(x), x⟩ − ⟨μ_{t}(x), x⟩| = 0. (5.12)

Indeed, we have

|⟨(μ_{n})_{t}(x), x⟩ − ⟨μ_{t}(x), x⟩|
≤ ⟨μ_{0}(y), |z_{n}(t; 0, y)| χ_{[X(0;t,0),∞[}(y)⟩ + |⟨μ_{0}(y), X_{n}(t; 0, y)χ_{n}(y)⟩|
≤ KT e^{AT} ∥c_{n} − c∥_{C^{0}(0,T)}⟨μ_{0}(y), (1 + y)⟩ + |⟨μ_{0}(y), X_{n}(t; 0, y)χ_{n}(y)⟩|

where χ_{n}(y) still stands for the characteristic function of the nonnegative reals y which belong to the segment defined by X_{n}(0; t, 0) and X(0; t, 0). In Section 3.3 we showed that X_{n}(t; 0, y)χ_{n}(y) converges to 0 uniformly in t and y as n → 0, which is exactly what we need to pass to the limit n → ∞ in the last term and proves that T is continuous. Then, Schauder’s theorem allows to claim the existence of a fixed point T (c) = c ∈ B. It remains to show that the truncation [·]_{+} is irrelevant for the fixed point. Suppose there exists a time t_{∗} satisfying

⟨μ_{t_{∗}}(x), x⟩ = ρ and ⟨μ_{t}(x), x⟩ < ρ for t < t_{∗}, (5.13)

μ_{t} being associated to c by (5.2). Then, c(t_{∗}) = 0 and (5.9) becomes

\{ \begin{matrix}D^{l}M_{1}(t_{∗}) = −⟨μ_{0}(y), q(X(t_{∗}; 0, y))χ_{[X(0;t_{∗},0),∞[}(y)⟩, \\ D^{r}M_{1}(t_{∗}) = −⟨μ_{0}(y), q(X(t_{∗}; 0, y))χ_{]X(0;t_{∗},0),∞[}(y)⟩. \end{matrix} (5.14)

By (2.2) the right hand side in (5.14) are negative. Thus, if t_{∗} = 0, this simply means that M_{1}(t) < ρ in a neighborhood of the initial time, while for t_{∗} > 0, this leads to a contradiction with (5.13). Therefore, we conclude M_{1}(t) < ρ for all t > 0, and ρ = c(t) + M_{1}(t) holds. This ends the proof of the existence part.

Uniqueness is established by following the proof in Section 4. Indeed, keeping the same notation, we write

|c_{2}(t) − c_{1}(t)| ≤ |ρ_{2} − ρ_{1}| + I_{1} + I_{2} + I_{3},

where, here, we denote

\{ \begin{matrix}I_{1} = ⟨μ_{0,2}(y), |z(t; 0, y)|χ_{[X_{1}(0;t,0),∞[}(y)⟩, \\ I_{2} = ⟨|μ_{0,2} − μ_{0,1}|(y), X_{1}(t; 0, y)χ_{[X_{1}(0;t,0),∞[}(y)⟩, \\ I_{3} = |⟨μ_{0,2}(y), X_{2}(t; 0, y)χ_{1,2}(y)⟩|, \end{matrix}

where z = X_{2} −X_{1} and χ_{1,2} stands for the characteristic function of the segment defined by X_{1}(0; t, 0) and X_{2}(0; t, 0). The arguments used in Section 4 still work and we evaluate these terms as follows

I_{1} ≤ K exp((A_{1} + A_{2})T) ⟨μ_{0,2}(y), (1 + y)⟩ ∫_{0}^{t} |c_{2}(s) − c_{1}(s)| ds,
I_{2} ≤ e^{A_{1}T} ⟨|μ_{0,2} − μ_{0,1}|(y), y⟩,
I_{3} ≤ |⟨μ_{0,2}(y), (∫_{X_{2}(0;t,0)}^{y} \frac{∂X_{2}}{∂x} (t; 0, ξ) dξ) χ_{1,2}(y)⟩|
≤ ⟨μ_{0,2}(y), e^{A_{2}t}|z(0; t, 0)|⟩
≤ e^{A_{2}t}Ke^{A_{1}t} ∫_{0}^{t} |c_{2}(s) − c_{1}(s)|e^{(A_{2}−A_{1})s} ds ⟨μ_{0,2}, 1⟩.

Hence, we actually have

\begin{matrix}|c_{2}(t) − c_{1}(t)| ≤ |ρ_{2} − ρ_{1}| \\ +k(T) ∫_{0}^{t} |c_{2}(s) − c_{1}(s)| ds + k(T)⟨|μ_{0,2} − μ_{0,1}|(y), y⟩, \end{matrix}

and Gronwall’s lemma again leads to the analogue of Lemma 6.

6. Explicit solutions

In this section we investigate the very simple case where the kinetic coefficients depend on the size variable as powers

k(x) = kx^{r}, q(x) = qx^{r}, (6.1)

where r is an integer. This case is amenable to a special treatment because certain computations may be explicitly written. Since we assume that the same power arises in the definition of k and q, in this case there is no critical radius: at any given time, all grains shrink or expand, according to the sign of the quantity kc(t) − q.

In the special case (6.1), we can remark that the Laplace transform of the density function (with respect to x) satisfies an equation of order r. The idea of using the Laplace transform in this setting is not new, for instance for discrete analogs of (1.1) it is equivalent (modulo a change of variables) to the use of the generating function (see [11]). For p ∈ C with sufficiently large real part, define F (t, p) by

F (t, p) = ∫_{0}^{∞} e^{−px}f(t, x) dx.

Equation (1.1) gives

\frac{∂F}{∂t} + (−1)^{r}p(kc − q)\frac{∂^{r}F}{∂p^{r}} = 0. (6.2)

In the case of constant coefficients, r = 0, this is an ordinary differential equation which may be integrated to obtain an explicit expression for F. In the case of first-order coefficients, r = 1, (6.2) is a transport equation which may be solved by the method of characteristics. The inversion formula for the Laplace transform may then be used to recover the density function. After some manipulations we obtain

f(t, x) = f_{0}(x − γ(t)) for r = 0,  (6.3)
f(t, x) = β(t)f_{0}(xβ(t)) for r = 1,  (6.4)

where we have set

γ(t) = ∫_{0}^{t}(kc(s) − q) ds, β(t) = exp(γ(t)).

This is an alternative approach to formula (3.21). Let us precisely focus our interest on these special cases r = 0, r = 1.

6.1. Explicit expression of the solution in the case of constant or first-order coefficients

The specificity of these two cases is best understood by considering the moments of the density

M_{s}(t) = ∫_{0}^{∞} x^{s}f(t, x) dx, s ≥ 1.

An elementary computation leads to

\frac{dM_{s}}{dt} = [k(ρ − M_{1}) − q]sM_{s−1+r}. (6.5)

This relation shows that M_{1} (and therefore any M_{s}) satisfies a closed equation only in the case where r = 1. Indeed, the equation (6.5) may be integrated to obtain M_{1}(t) in an explicit form; coming back to the conservation equation we get after some elementary manipulations

if kρ − q ≠ 0 : c(t) = \frac{ρ(kc_{0} − q) exp((q − kρ)t) + q(ρ − c_{0})}{(kc_{0} − q) exp((q − kρ)t) + k(ρ − c_{0})},  (6.6)
if kρ − q = 0 : c(t) = \frac{c_{0} + ρ(ρ − c_{0})kt}{1 + (ρ − c_{0})kt}.  (6.7)

The expression of f may now be obtained by either solving equation (1.1) by the methods of characteristics, or using (6.4) and computing β(t).

Next, let us examine the case r = 0. We obtain a closed equation for M_{1} if we can ensure that M_{0} remains constant. This will be the case if, on the one hand, we assume that kc(0) − q ≥ 0, which means that the characteristics at x = 0 are directed towards the interior of the domain, and, on the other hand, we add the boundary condition

f(t, 0) = 0, for t ≥ 0

to the problem (1.1). Then, the equation for M_{1} may be solved explicitly. Coming back to c(t) = ρ − M_{1} gives

c(t) = \frac{q}{k} + (c_{0} − \frac{q}{k}) exp(−M_{0}kt). (6.8)

Finally, by using (6.3) we obtain the expression of f.

Note that (6.8) shows that c(t) ≥ \frac{q}{k}, i.e. the characteristics always keep the same orientation at x = 0. Physically this means that the system remains supersaturated at all time. Thus monomers are continuously being added onto clusters, and the critical radius is zero. Moreover, we also note that (6.3), (6.8) still give a solution of (1.1), without additional condition, when kc(0) − q < 0 (outgoing characteristics) and r = 0, provided the support of the initial data remains at a sufficiently large distance from the origin in order to ensure the conservation of the zeroth moment. Let us summarize the various formulae we obtained.

Proposition 1 Let the kinetic coefficients be given by (6.1).

• If r = 0, assume that f_{0}(0) = 0 and extend f_{0} by the value zero for negative numbers. Assume that kc(0) − q ≥ 0 (or kc(0) − q < 0 with supp(f_{0}) ⊂ (R, ∞), where R is large enough). Then, the solution to (1.1) is given by (6.3), (6.8) with

γ(t) = \frac{kc(0) − q}{kM_{0}} (1 − exp(−M_{0}kt)).

Here M_{0} denotes the zeroth moment of the density function which remains constant. • If r = 1, then the solution to (1.1) is given by (6.6) or (6.7), and (6.4), with

if kρ − q ≠ 0 : β(t) = \frac{(kc_{0} − q) exp((q − kρ)t) + k(ρ − c_{0})}{(kρ − q)},  (6.9)
if kρ − q = 0 : β(t) = 1 + k(ρ − c_{0})t.  (6.10)

6.2. Asymptotic behaviour

For general coefficients, the asymptotic behaviour of the solution is essentially an open problem. In particular if there exists a critical size x_{c}, then it has not yet been rigorously established that x_{c} increases monotonically in time, as argued in [6]. We describe some results in that direction in [3].

For r = 0, 1 the asymptotic behaviour may be directly read from the explicit expression of the solution given above. In the case of a nonlinear power law, even though the previous method does not yield an explicit expression of the solution, the equation for the moments may still be used to investigate the asymptotic behaviour of the solutions. It can be described as follows.

For general r ≠ 1, let us now derive formally some formulae. The differential equation (3.9) may be integrated to yield the explicit form of the characteristic curves

X(s; t, x) = [x^{1−r} + (r − 1) ∫_{s}^{t}(kc(σ) − q) dσ]^{\frac{1}{1−r}} . (6.11)

Coming back to the definition of a mild solution we obtain an expression for the density

f(t, x) = β^{r}(t, x)f_{0}(xβ(t, x))

where

β(t, x) = [1 + (r − 1)γ(t)x^{r−1}]^{\frac{1}{1−r}} , (6.12)

and γ is defined as before. Note that (6.3) and (6.4) are particular cases of this formula, obtained by letting r approach 0 or 1 respectively. Finally, the total density may be computed by using (3.23)

∫_{0}^{∞} xf(t, x) dx = ∫_{X(0;t,0)}^{X(0;t,∞)}[1 − (r − 1)γ(t)y^{r−1}]^{\frac{1}{1−r}} yf_{0}(y) dy. (6.13)

The endpoints involved in this last integral are given by (6.11). For instance if r < 1 and γ(t) ≤ 0 we obtain

X(0; t, 0) = [(r − 1)γ(t)]^{\frac{1}{1−r}} , X(0; t, ∞) = ∞.

If kc(t) − q ≥ 0, then the characteristics are directed towards the interior of the domain and the problem has to be supplemented with a boundary condition. Then, as before when treating the case r = 0, we choose the boundary condition f(t, 0) = 0 which leads to the conservation of the zeroth moment. Therefore, the integration in (6.13) is taken as starting from 0. This also means that the solution has been obtained by extending the initial data f_{0}(x) by 0 for negative x. Correspondingly, if β(x) < 0 for some value of x > 0, t > 0, then f(t, x) = 0.

Before we describe the behaviour as time goes to ∞ in the relations written above, let us note that these explicit computations can be used to obtain an example of nonexistence. It is clear that the case r > 1 is not covered by our existence result in Theorem 1, because the characteristic curves may blow up in finite time. In fact it is easy to see that in this case a global solution may fail to exist. For instance, let us assume that the coefficients are given by (6.1) with r = 2, and that kc_{0} − q > 0. If one takes an initial data f_{0} which never vanishes, it is easy to check that the first moment instantaneously blows up.

Then, as 0 ≤ r ≤ 1, the asymptotic behaviour of the solutions can be described as follows.

Proposition 2 Assume the kinetic coefficients are given by (6.1) with 0 ≤ r ≤ 1, that the initial data f_{0} has finite zeroth and first moments, and that the system has a unique global solution. Then as t → ∞ the behaviour of the solution is as follows.

• If ρ ≤ \frac{q}{k}, then M_{1}(f) → 0, and c → ρ. (6.14)

• If ρ ≥ \frac{q}{k}, then

f → f_{∞} pointwise, and c → \frac{q}{k} , (6.15)

where f_{∞} is given by

f_{∞}(x) = β_{∞}(x)^{r}f_{0}(xβ_{∞}(x)), β_{∞}(x) = [1 + (r − 1)γ_{∞}x^{r−1}]^{\frac{1}{1−r}} ,

and the number γ_{∞} is uniquely determined by the following condition

\frac{q}{k} + ∫_{[(r−1)γ_{∞}]^{\frac{1}{1−r}}}^{∞} [1 + (1 − r)γ_{∞}y^{r−1}]^{\frac{1}{1−r}} yf_{0}(y) dy = ρ if c_{0} < \frac{q}{k} , (6.16)
\frac{q}{k} + ∫_{0}^{∞}[1 + (1 − r)γ_{∞}y^{r−1}]^{\frac{1}{1−r}} yf_{0}(y) dy = ρ if c_{0} > \frac{q}{k} .  (6.17)

Remark: Since assumption (6.1) implies there is no critical radius, (6.14) and (6.15) correspond respectively to the undersaturated (all grains shrink) and supersaturated case (all grains expand). Thus the asymptotic behaviour is obvious on physical grounds: in the undersaturated case, only monomers remain at infinite times, whereas in the supersaturated case the monomer concentration converges to the equilibrium value \frac{q}{k}. Proof. We will use the relations (6.12), (6.13) together with equation (6.5) for the r−th moment, and the relation

\frac{dc}{dt} = −(kc − q)M_{r}. (6.18)

This last relation shows that c is a monotonous function of time. Indeed since \frac{q}{k} ≠ ρ and M_{r} ≠ 0 for c(t) ≠ ρ, the time derivative of c is non zero in a neighborhood of \frac{q}{k} (except precisely at the equilibrium value \frac{q}{k}). Thus the quantity kc − q maintains a constant sign, and γ also is monotonous. Let us start with the undersaturated case. From (6.18) we obtain that c increases with time, therefore has a limit c_{∞} ≤ ρ < \frac{q}{k}. Thus the time derivative of γ converges to the strictly negative value kc_{∞} − q. Thus γ is negative for all times, and γ → −∞. From (6.13) we obtain:

\begin{matrix}∫_{0}^{∞} xf(t, x) dx & = & ∫_{[(r−1)γ(t)]^{\frac{1}{1−r}}}^{∞} [1 − (r − 1)y^{r−1}γ(t)]^{\frac{1}{1−r}} yf_{0}(y) dy \\ & ≤ & ∫_{[(r−1)γ(t)]^{\frac{1}{1−r}}}^{∞} yf_{0}(y) dy. \end{matrix} (6.19)

The last integral goes to zero as t goes to infinity, thus the first moment of f converges to zero. Passing to the limit in the mass conservation equation we obtain c_{∞} = ρ.

Let us now treat the supersaturated case. If c_{0} < \frac{q}{k} < ρ, then again c(t) increases to its limit value c_{∞} ≤ \frac{q}{k}. As before if c_{∞} < \frac{q}{k} we have γ → −∞, thus M_{1} → 0. Passing to the limit in the mass conservation equation we obtain c_{∞} = ρ, which contradicts the assumption. Therefore we have c_{∞} = \frac{q}{k} and γ decreases to some finite negative value γ_{∞}. Passing to the limit in the mass conservation equation we obtain the relation \frac{q}{k} + I(γ_{∞}) = ρ, with

I(γ_{∞}) := ∫_{[(r−1)γ_{∞}]^{\frac{1}{1−r}}}^{∞} [1 + (1 − r)γ_{∞}y^{r−1}]^{\frac{1}{1−r}} yf_{0}(y) dy.

It is easy to see that the quantity I(γ_{∞}) is a strictly increasing function of γ_{∞}, and satisfies:

I(0) = ∫_{0}^{∞} yf_{0}(y) dy > ρ − \frac{q}{k} , I(−∞) = 0,

so that the condition (6.16) determines the limit value γ_{∞} in a unique way. If \frac{q}{k} < c_{0} < ρ, then by the same argument as before, the quantity γ increases with time. The first moment of f may be estimated from below:

\begin{matrix}∫_{0}^{∞} xf(t, x) dx & = & ∫_{0}^{∞}[1 + (1 − r)γ(t)y^{r−1}]^{\frac{1}{1−r}} yf_{0}(y) dy \\ & ≥ & [(1 − r)γ]^{\frac{1}{1−r}} ∫_{0}^{∞} f_{0}(y) dy. \end{matrix} (6.20)

Note that, as mentioned above, integration starts at 0. The lower bound for M_{1}(f) goes to infinity if γ → +∞. Thus we conclude that γ has a finite limit γ_{∞}, which in turn implies that c converges to the equilibrium value \frac{q}{k}. Proceeding as before we obtain that γ_{∞} is uniquely determined by the relation (6.17), and this completes the proof of the proposition.

Appendix A. Some elementary facts about linear transport equations

In this appendix we collect (without proof) some very well-known facts about characteristic curves, as well as the notions of mild and measure-valued solution for linear transport equations. Let a = a(t, x) be a given function, which is continuous with respect to both t, x variables, and for all time t, has an x−derivative, which also is continuous with respect to both variables. The characteristics associated to a are the curves defined by:

\{ \begin{matrix}\frac{∂}{∂s}X(s; t, x) = a(s, X(s; t, x)), \\ X(t; t, x) = x. \end{matrix} (A.1)

Given any t ∈ [0, ∞[, x ∈ [0, ∞[, the curve s ⟼ X(s; t, x) is uniquely defined for all s ≥ 0. In the paper we repeatedly make use of the well-known formula:

X(t; s, X(s; t, x)) = x,  (A.2)
J(s; t, x) := \frac{∂X}{∂x} (s; t, x) = exp (∫_{s}^{t} b(σ, X(σ; t, x)) dσ) ,  (A.3)
\frac{∂X}{∂t} (s; t, x) = −a(t, x) exp (∫_{s}^{t} b(σ, X(σ; t, x)) dσ) ,  (A.4)

in which we have set b = −\frac{∂a}{∂x}. If a function f = f(t, x) is a smooth solution of the Cauchy problem

\{ \begin{matrix}\frac{∂}{∂t}f + \frac{∂}{∂x}(af) = 0, t > 0, x ∈ [0, ∞[, \\ f(0, x) = f_{0}(x) x ∈ [0, ∞[, \end{matrix} (A.5)

and f_{0} is also smooth, then a straightforward computation shows that

f(t, x) = f_{0}(X(0; t, x))J(0; t, x). (A.6)

If we now take f_{0}(x) to be defined for any x outside a null set O, we claim that the right hand side of this last relation is still defined for almost every (t, x) ∈ [0, T] × [0, ∞[. From the relation (A.2) we see that for any t, we have X(0; t, x) ∈ O iff x ∈ X(t; 0, O), thus we see from the definition of X that the set

O_{t} := {x ∈ [0, ∞[: X(0; t, x) ∈ O}

has measure zero. The map which to (t, x) associates X(0; t, x) is clearly continuous, thus the inverse image of O is clearly measurable; it then follows from Fubini’s theorem that it has measure zero, and this proves the claim. Motivated by this trivial remark, in the case where f_{0}(x) is defined for almost every x, we will call mild solution to the Cauchy problem the unique function f defined for almost every (t, x) by (A.6). Let us now recall the notion of measure-valued solution to the Cauchy problem. If the initial data μ_{0} is in M_{+}^{1}, the measure valued solution to the Cauchy problem is the map

μ : [0, T] → M_{+}^{1}

uniquely defined by the prescription μ_{t}(B) = μ_{0}(X(0; t, B)), for any t ∈ [0, T] and any Borel set B. Note that by the previous argument, this definition makes sense since X(0; t, B) is indeed a Borel set for any t, and it is clear that the map μ is continuous on the interval [0, T], with values in M_{+}^{1} endowed with the weak-star topology. A result in [10] states that it is the only distributional solution of the Cauchy problem with this time regularity. This yields the obvious equivalence between the mild solution and the measure-valued solution. Theorem 4 Assume f_{0} ∈ L^{1}([0, ∞[) is a nonnegative function. For any t ∈ [0, T], define the measure μ_{t} = f(t, ·)dx, where dx denotes the one-dimensional Lebesgue measure, and f denotes the mild solution to the Cauchy problem. Then μ_{t} is the measurevalued solution to the Cauchy problem.

Note that the measure-valued μ_{t} is always absolutely continuous with respect to dx in the case where μ_{0} also is. Thus the mild solution is just the Radon-Nykodim derivative of μ_{t} with respect to dx. Although is just an immediate consequence of the result in [10], we can obtain it at once from the relation

\begin{aligned}μ_{t}(B) & = & ∫_{B} f(t, x) dx = ∫_{B} f_{0}(X(0; t, x))J(0; t, x) dx \\ & = & ∫_{X(0;t,B)} f_{0}(y) dy = μ_{0}(X(0; t, B)). \end{aligned}

Appendix B. Proof of (3.30)

With the notations of Lemma 3, for any t ∈]0, T [ and any |h| sufficiently small we define:

Φ_{h}(t) = \frac{1}{h} ∫_{X(0;t,0)}^{X(0;t+h,0)} X(0; t, y)f_{0}(y) dy.

We need to show that for almost every t ∈]0, T [, the quantity Φ_{h}(t) converges to zero with h. Let us first remark that setting α(t) = X(0; t, 0), we have

α′(t) = (q(0) − k(0)c(t)) exp (∫_{0}^{t} b(σ, X(σ; t, 0)) dσ) ≥ 0. (B.1)

Let us first treat the case q(0) − k(0)ρ > 0. Then we have α′(t) > 0 for any t, and we may write:

\begin{aligned}Φ_{h}(t) & = & [\frac{X(0;t+h,0)−X(0;t,0)}{h}] \\ & & [\frac{1}{X(0;t+h,0)−X(0;t,0)} ∫_{X(0;t,0)}^{X(0;t+h,0)} X(0; t, y)f_{0}(y) dy] . \end{aligned}

From (B.1) we obtain that for any t ∈]0, T [, the first bracket in this product has a finite limit when h goes to zero; by the Lebesgue differentiation theorem we see that the second bracket will converge to:

X(t; 0, X(0; t, 0))f_{0}(X(0; t, 0)) = 0,

provided X(0; t, 0) is a Lebesgue point of the integrand. Since the map y ⟼ X(t; 0, y) is of class C^{1}, this will be true if X(0; t, 0) is a Lebesgue point of f_{0}. Let us denote by X the complement in [0, ∞[ of the set of Lebesgue points of f_{0}. Defining

I = {t ∈]0, T [: α(t) ∈ X }, we need to show that I is negligible (for the Lebesgue measure dt). Since X has measure zero, we have

|I| = |α^{−1}(X)| = ∫_{X} \frac{dx}{|α′(α^{−1}(x))|} = 0,

and this completes the proof in the case q(0) − k(0)ρ > 0.

Let us now treat the case where ρ = q(0)/k(0). We will again prove that the set A of times t ∈ [0, T] such that (3.30) has full measure in [0, T]. For convenience we denote the complement of A by Ā. Let us fix t ∈]0, T [ arbitrarily. The discussion depends on the value of c(t), and can be splitted into three different cases:

• case 1: if c(t) ≠ ρ, then from the continuity of c we have c(τ) < ρ for all τ in some open neighborhood V (t) of t. Then we may use the same argument as in the case q(0) − k(0)ρ > 0 to obtain that the set V (t) ∩ Ā is negligible.

• case 2: if now c(τ) = ρ for all τ in some open neighborhood V (t) of t, then we have Φ_{h}(τ) = 0 for all sufficiently small h, i.e. V (t) ⊂ A.

• case 3: finally if t is such that there is a number ϵ > 0 with c(τ) = ρ for any τ ∈ [t − ϵ, t] and c(τ) < ρ for any τ ∈]t, t + ϵ], we can combine the arguments given in the two previous cases. More precisely for τ ∈]t − ϵ, t[ we are essentially in case 2, and we obtain that ]t − ϵ, t[⊂ A. For τ ∈]t, t + ϵ] we may apply the argument used in case 1 to obtain a neighborhood V (τ) of τ such that |V (τ)∩Ā| = 0. By the Borel Lebesgue property, any compact interval [t_{1}, t_{2}] ⊂]t, t + ϵ] can be covered by a finite number of such neighborhoods, which implies that |[t_{1}, t_{2}] ∩ Ā| = 0. Thus we may conclude:

|]t, t + ϵ] ∩ Ā| = |(⋃_{n=1}^{∞}[t + \frac{ϵ}{n}, t + ϵ]) ∩ Ā| ≤ ∑_{n=1}^{∞} |[t + \frac{ϵ}{n}, t + ϵ] ∩ Ā| = 0

Recalling that ]t − ϵ, t[⊂ A, we obtain that |]t − ϵ, t + ϵ[∩Ā| = 0. The case where c(τ) = ρ for any τ ∈ [t, t + ϵ] and c(τ) < ρ for any τ ∈]t − ϵ, t[ is of course totally analogous to case 3.

Summarizing, we have shown that for any t ∈]0, T [, there exists a neighborhood V (t) such that |V (t)∩Ā| = 0. By the same device as in case 3 we obtain that for any compact interval [t_{1}, t_{2}] ⊂]0, T [ we have |[t_{1}, t_{2}] ∩ Ā| = 0, and therefore |Ā| = 0, which completes the proof of the lemma.

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