Uniform moment propagation for the Becker-D\"oring equation

TL;DR

This paper proves that algebraic and stretched exponential moments remain uniformly bounded over time for the Becker-Döring equations, using maximum principle techniques and convergence rates.

Contribution

It introduces a novel approach to establish uniform moment propagation for the Becker-Döring equations combining maximum principle and convergence analysis.

Findings

Uniform-in-time propagation of algebraic moments

Uniform-in-time propagation of stretched exponential moments

Utilization of maximum principle and convergence rates

Abstract

We show uniform-in-time propagation of algebraic and stretched exponential moments for the Becker-D\"oring equations. Our proof is based upon a suitable use of the maximum principle together with known rates of convergence to equilibrium.

Equations

$$\frac{\mathrm{d}}{\mathrm{d}t}{c}_i(t)$$

$$\frac{\mathrm{d}}{\mathrm{d}t}{c}_1(t)$$

$$W_i(t):=a_i{c}_1(t)c_i(t)-b_{i+1}{c}_{i+1}(t)i\in \mathbb{N}.$$

$$\{i\}+\{1\}\underset{b_{i+1}}{\stackrel{a_i}{\leftrightharpoons }}\{i+1\},$$

$$\frac{\mathrm{d}}{\mathrm{d}t}\sum _{i=1}^{\infty }{c}_i(t){\varphi }_i=\sum _{i=1}^{\infty }{W}_i(t)({\varphi }_{i+1}-{\varphi }_i-{\varphi }_1),$$

$$\varrho :=\sum _{i=1}^{\infty }i{c}_i(0)=\sum _{i=1}^{\infty }i{c}_i(t),$$

$$Q_1=1,Q_{i+1}=\frac{a_i}{b_{i+1}}{Q}_{i}i\in \mathbb{N}$$

$$\sum _{i=1}^{\infty }i{Q}_i{z}^i<+\infty \text{ for all 0⩽z<zs}$$

$${\varrho }_s:=\sum _{i=1}^{\infty }i{Q}_i{z}_s^i\in [0,+\infty ].$$

$${\mathcal{Q}}_i=Q_i{\overline{z}}^i,i⩾1,$$

$$H(\mathbf{c}(t)|\mathcal{Q}):=\sum _{i=1}^{\infty }\left(c_i(t)\log \frac{c_i(t)}{{\mathcal{Q}}_i}-c_i(t)+{\mathcal{Q}}_i\right),t⩾0.$$

$$\sum _{i=1}^{\infty }\exp \left(\eta i\right)c_i(0)<+\infty ⟹\underset{t⩾0}{\sup }\sum _{i=1}^{\infty }\exp \left({\eta }^{\prime }i\right)c_i(t)<\infty$$

$$0<a_i⩽\overline{a}{i}^{\gamma }$$

$$C_{1}i⩽a_i⩽C_{2}i$$

$$0<b_i⩽\overline{b}{a}_i,$$

$$\underset{i\to +\infty }{\lim }\frac{Q_{i+1}}{Q_i}=\frac{1}{z_s}\text{for some 0<zs<+∞.}$$

$$\{Q_i{z}_s^i{\}}_i$$

$$a_i=i^{\gamma },b_i=a_i\left(z_s+\frac{q}{i^{1-\mu }}\right),$$

$$a_i=i^{\gamma },b_i=z_s{\left(i-1\right)}^{\gamma }{e}^{\sigma i^{\mu }-\sigma (i-1{)}^{\mu }},$$

$$\sum _{i=1}^{\infty }{i}^r{c}_i^0<+\infty \text{for r=max{2−γ,1+γ}.}$$

$$M_k(0):=\sum _{i=1}^{\infty }{i}^k{c}_i(0)<\infty .$$

$$M_k(t):=\sum _{i=1}^{\infty }{i}^k{c}_i(t)⩽C\text{for all t⩾0.}$$

$${\mathcal{E}}_{\mu }(0):=\sum _{i=1}^{\infty }\exp \left(\alpha i^{\mu }\right)c_i(0)<\infty .$$

$${\mathcal{E}}_{\mu }(t):=\sum _{i=1}^{\infty }\exp \left(\alpha i^{\mu }\right)c_i(t)⩽C\text{for all t⩾0.}$$

$$\frac{\mathrm{d}}{\mathrm{d}t}{M}_k(t)=\sum _{i=1}^{\infty }\left(a_i{c}_1(t)c_i(t)-b_{i+1}{c}_{i+1}(t)\right)\left((i+1{)}^k-i^k-1\right),$$

$$G_j(t)=\sum _{i=j}^{\infty }{c}_i(t),j⩾1.$$

$$\frac{\mathrm{d}}{\mathrm{d}t}{G}_j(t)=a_{j-1}{c}_1(t)\left(G_{j-1}(t)-G_j(t)\right)+b_j\left(G_{j+1}(t)-G_j(t)\right)j⩾2.$$

$$G_j=\sum _{i=j}^{\infty }{c}_i,j\in \mathbb{N}.$$

$$\frac{M_{k+1}\left(\mathbf{c}\right)}{k+1}⩽M_k\left(\mathcal{G}\right)⩽M_{k+1}\left(\mathbf{c}\right).$$

$${\mathcal{E}}_{\mu }(\mathbf{c})=\sum _{i⩾1}\exp \left(\alpha i^{\mu }\right)c_i.$$

$${\eta }_1\sum _{j=1}^{\infty }{\psi }_j{G}_j⩽{\mathcal{E}}_{\mu }(\mathbf{c})⩽{\eta }_2\sum _{j=1}^{\infty }{\psi }_j{G}_j$$

Full text

1. 1 Introduction

2. 1.1 A quick summary on asymptotic behaviour

3. 1.2 Main results

4. 1.3 Method of proof

5. 1.4 Organization of the paper

6. 2 Tail density and the maximum principle

7. 3 On the propagation of moments

1. Introduction

In this note we consider the Becker-Döring equations

[TABLE]

where

[TABLE]

The unknowns here are the functions $\bm{c}(t)=\left(c_{i}(t)\right)_{i\geqslant 1}$ which depend on time $t\geqslant 0$ and where, for each $i\in\mathbb{N}$, $c_{i}(t)$ represents the density of clusters of size $i$ at time $t\geqslant 0$ (that is, clusters composed of exactly $i$ individual particles). The non-negative numbers $a_{i},b_{i}$ denote respectively the coagulation and fragmentation coefficients. These equations are a model for the dynamics of cluster growth in which clusters can only gain or shed one particle; that is, the only reactions taking place are

[TABLE]

where $\{i\}$ represents the concentration of clusters of size $i$. The quantity $W_{i}$ then represents the net rate of this reaction, obtained by standard mass-action dynamics. It is a well accepted model for the kinetics of first order phase transitions, applicable to a wide variety of phenomena such as crystallisation, vapor condensation, aggregation of lipids or phase separation in alloys. The model is traced back to [5], and the basis of its mathematical theory was set in [3, 4]. There have been a number of works on the long-time behaviour of solutions, which is especially interesting since it exhibits phase-change phenomena, metastability, and fast relaxation to equilibrium depending on the regime one is considering. We mention here the works by [9, 10, 11, 13, 14, 15, 16, 17, 18, 20], leaving out many relevant ones. We direct the reader to the references in the aforementioned works for a more complete picture, and to the survey paper [19].

Despite the amount of works devoted to the model, it seems to us that the question of propagation of moments has not been fully answered, and it is our purpose to fill that gap in this paper. The basic question we address is the following: if $\sum_{i=1}^{\infty}i^{k}c_{i}(0)<+\infty$ is finite for some $k>1$, is it true that $\sum_{i=1}^{\infty}i^{k}c_{i}(t)\leqslant C$ for some $C>0$ and all $t\geqslant 0$? We show an affirmative answer for subcritical solutions, which is the natural case in which one expects it to hold.

Before describing our results with more detail we need to set some notation and give some background on the asymptotic behaviour of equation (1.1).

1.1. A quick summary on asymptotic behaviour

Equation (1.1) can be written, in weak form, as

[TABLE]

for all slowly growing sequences $(\phi_{i})_{i\geqslant 1}$. In particular, taking $\phi_{i}=i$, one sees that the density of the solution, defined by

[TABLE]

is formally conserved under time evolution. Defining the detailed balance coefficients $Q_{i}$ recursively by

[TABLE]

one can see that any sequence of the form $(Q_{i}z^{i})_{i\geqslant 1}$ is formally an equilibrium of (1.1). However, such a sequence may not have a finite density. The largest $z_{s}\geqslant 0$ (possibly $z_{s}=+\infty$) for which

[TABLE]

is called the critical monomer density, or sometimes the monomer saturation density (alternatively, $z_{s}$ is the radius of convergence of the power series with coefficients $iQ_{i}$). The critical density (or, again, saturation density) is then defined by

[TABLE]

This critical density plays a fundamental role in the long-time behaviour of solutions to (1.1): it was proved in [4] and [3] that any solution with density $\varrho>\varrho_{s}$ will converge (in a weak sense) to the only equilibrium with density $\varrho_{s}$, with the excess mass $\varrho-\varrho_{s}$ becoming concentrated in larger and larger clusters as time passes. In contrast, any solution with initial density $\varrho\leqslant\varrho_{s}$ will converge (strongly) as $t\to\infty$ to an equilibrium solution with its same density $\varrho$. We focus here on the so-called subcritical solutions for which $\varrho<\varrho_{s}$, which converge to the equilibrium $\bm{\mathcal{Q}}:=\left(\mathcal{Q}_{i}\right)_{i\geqslant 1}$ given by

[TABLE]

where $\overline{z}\in[0,z_{s})$ is the unique number such that $\varrho=\sum_{i=1}^{\infty}Q_{i}\overline{z}^{i}.$ The rate of convergence to this equilibrium in exponentially weighted $\ell_{1}(\mathbb{N})$ norms was studied in [11] and subsequently improved in [9]. Convergence for solutions with finite algebraic moments (which applies to a wider range of initial conditions) has been studied in [10, 13, 14].

The approach in [11] is based on the entropy-entropy production method and has been recently revisited by the authors of the present paper in [10]. It consists in estimating in a careful way the evolution of the relative free energy

[TABLE]

We observe that $H(\bm{c}(t)|\bm{\mathcal{Q}})$ is finite whenever the solution $\bm{c}(t)=(c_{i}(t))_{i\geqslant 1}$ is nonnegative and has finite density (see for example Lemma 7.1 and 7.2 in [8]). We refer to [10] for more details on the entropy-entropy production method in the context of the Becker-Döring equations.

1.2. Main results

A fundamental tool in the application of the entropy method is a uniform control of suitable moments of the solution $\bm{c}(t)$ to (1.1), i.e. the control of suitable weighted-$\ell_{1}(\mathbb{N})$ estimates. For instance, the analysis of [11] deals with subcritical solutions with finite exponential moments and is based on the property that

[TABLE]

for $\eta>0$ and some $0<\eta^{\prime}<\eta$. This was proved in [11], and is to our knowledge the only available result on uniform propagation of moments.

We would like to have a similar information for algebraic moments $\sum_{i\geqslant 1}i^{k}c_{i}(t)$ or stretched exponential moments of the form $\sum_{i\geqslant 1}\exp\left(\alpha i^{\mu}\right)c_{i}$, for some $\alpha>0$ and $0<\mu<1$. Propagation of these moments on a finite time interval is known to hold from the results of [4] (see Lemma 3.3 hereafter), but the estimate on the time interval $[0,T]$ deteriorates as $T$ increases. We intend to fill this blank with the uniform in time propagation results in the next two theorems. For our results we assume that

[TABLE]

If the second option holds we call $\gamma=1$ for consistency. We also assume that

[TABLE]

for all $i\geqslant 1$ and some $\overline{b}>0$, and that

[TABLE]

(Note that $z_{s}$ is indeed the critical monomer density, in agreement with (1.6).) Additionally we may assume that the critical equilibrium is non-increasing:

[TABLE]

though this is not a fundamental requirement and small changes can be made to adapt the proofs if the sequence $\{Q_{i}z_{s}^{i}\}_{i\geqslant i_{0}}$ is non-increasing only for some fixed $i_{0}\in\mathbb{N}$.

Assumptions (1.9)-(1.12) are natural and satisfied in most physically relevant situations. It is worth mentioning that some of the assumptions can follow from others, with additional conditions. For instance, if one assumes that $\frac{b_{i+1}}{b_{i}}$ is bounded from below then the definition of $Q_{i}$ and assumption (1.11) imply that (1.10) is satisfied. Common coefficients that appear in the theory of density conserving phase transitions [4, 15] are

[TABLE]

for some $0<\gamma\leqslant 1$, $q>0$, and $0<\mu<1$. A different modelling assumption yields the coefficients

[TABLE]

for some $0<\gamma\leqslant 1,\sigma>0$ and $0<\mu<1$. It is easy to verify that these type of coefficients satisfy all our assumptions. Regarding the initial datum $\bm{c}^{0}=(c_{i}^{0})_{i\geqslant 1}$, we assume it is non-negative and has some finite moments:

[TABLE]

With this at hand, we can now state our first result:

Theorem 1.1** (Uniform propagation of moments).**

Assume (1.9)–(1.12), and let $\bm{c}(t)=(c_{i}(t))_{i\geqslant 1}$ be a solution to the Becker-Döring equations (1.1) with non-negative, subcritical initial datum $\bm{c}(0)$ and density $\varrho<\varrho_{s}$. Let $k\geqslant\max\{2-\gamma,1+\gamma\}$ be such that

[TABLE]

There exists a constant $C>0$ depending only on $k$, $M_{k}(0)$, the density $\varrho$ and the coefficients $(a_{i})_{i\geqslant 1}$, $(b_{i})_{i\geqslant 1}$ such that

[TABLE]

The constant $C$ can be estimated explicitly from the proof. Our second result deals with the uniform propagation of stretched exponential moments in a similar way:

Theorem 1.2** **(**Uniform propagation of stretched

exponential moments**).

Assume (1.9)–(1.12) hold, with the first option in (1.9) being true (for some $0\leqslant\gamma<1$). Let $\bm{c}(t)=(c_{i}(t))_{i\geqslant 1}$ be a solution to the Becker-Döring equations (1.1) with non-negative, subcritical initial datum $\bm{c}(0)$ and density $\varrho$. Let $0<\mu\leqslant 1-\gamma$ and $\alpha>0$ be such that

[TABLE]

There exists a constant $C>0$ depending only on $\mu$, $\alpha$, $\mathcal{E}_{\mu}(0)$, the density $\varrho$ and the coefficients $(a_{i})_{i\geqslant 1}$, $(b_{i})_{i\geqslant 1}$ such that

[TABLE]

Notice that these two results prove uniform propagation of the considered moments whereas the result of [11] recalled in (1.8) is of a slightly different nature because of the deterioration of the constant $\eta$ which measures the strength of the exponential. We do not know whether uniform propagation is true for exponential moments (that is, we do not know whether one can take $\eta^{\prime}=\eta$ in (1.8)); our method does not immediately apply in this case since the short-time propagation in Lemma 3.3 does not apply to exponential moments.

We mention here that, besides its own interest, Theorem 1.1 plays a crucial role in the determination of the convergence rate to equilibrium for solutions to (1.1) recently established in [10].

1.3. Method of proof

A natural attempt to prove the above results would be to directly compute the evolution of $M_{k}(t)$ or $\mathcal{E}_{\mu}(t)$. Namely, picking $\phi_{i}=i^{k}$ in the weak form (1.3), we get the evolution of $M_{k}(t)$

[TABLE]

and one may try to obtain a suitable differential inequality for $M_{k}$ in the spirit of similar results for kinetic equations (see [1] for an example on the Boltzmann equation). This method is rather efficient to obtain local in time bounds on $M_{k}(t)$ (or $\mathcal{E}_{\mu}(t)$) but seems difficult to apply to get uniform bounds on $[0,\infty)$. The difficulty stems from the fact that the “loss term” $b_{i+1}c_{i+1}(t)$ appearing in the evolution does not always compensate the “gain term” $a_{i}c_{1}(t)c_{i}(t)$. A deeper reason for this is that boundedness of moments must depend on the mass of the solution (since moments are never uniformly bounded for supercritical solutions), so any estimate that gives uniform bounds must somehow involve the mass of the solution. In practice, it is the value of $c_{1}(t)$ that appears when one tries to bound the time evolution of moments, and any uniform estimate seems to require some a priori knowledge on the behaviour of $c_{1}(t)$. This is in contrast with the situation for the Boltzmann equation (with hard potential interactions) where the optimal Povzner’s inequality allows us to control the contribution of gain part of the collision operator by that of its loss part (see for example [6]). It should be remarked that the behaviour of moments for the Boltzmann equation does not depend on the mass of the initial datum, but only on which moments are initially finite, which is a fundamental difference with the present case. Another important difference is the fact that there is no creation of moments (of any kind) for the Becker-Döring equations (see [4]).

We adopt here a different approach relying on a maximum principle. A crucial role in our study will be played by the tail density $\mathcal{G}(t)=\left(G_{j}(t)\right)_{j\geqslant 1}$ given by

[TABLE]

The main properties of $\mathcal{G}$ which are relevant for us are established in Lemmas 2.2 and 2.3. Tail density was already introduced in [12] in order to establish uniqueness of the solution to (1.1) and a variant of it was used in [7] to show strong convergence to equilibrium for a generalised discrete coagulation–fragmentation model.

It is important to notice that moments of $\bm{c}(t)$ can be estimated by suitable moments of $\mathcal{G}(t)$, so that Theorems 1.1–1.2 can be stated in terms of moments of $\mathcal{G}(t)$ (the rough idea being that the $k$-th moment of $\bm{c}$ is equivalent to the $(k-1)$-th moment of $\mathcal{G}$; see Lemma 2.2). Of course, the main interest is that the equation solved by $\mathcal{G}(t)$ is somewhat simpler than (1.1): one has

[TABLE]

The evolution equation for $\mathcal{G}(t)$ depends on $c_{1}(t)$, and the entire nonlinear structure of the interaction between clusters is driven by it (assuming $c_{1}(t)$ to be known in (1.1) would yield a linear system of ODEs). Since the coefficient of $c_{1}(t)$ is non-negative in the above equation, if one is able to control $c_{1}(t)$ from above on some given interval, then one can bound the above evolution of $\mathcal{G}(t)$ by a suitable infinite system of differential inequalities, represented by an infinite matrix whose off–diagonal entries are non-negative. This is the key ingredient that yields a maximum principle for the evolution of $\mathcal{G}(t)$ (see Lemmas 2.3 and 2.4). The proof then consists in establishing the existence of suitable supersolutions to the Becker-Döring equations whose moments are strongly related to the moments of $\mathcal{G}(t)$ in order to apply the maximum principle. As already said, this will be possible once a suitable bound on $c_{1}(t)$ has been established. To prove an a priori bound for $c_{1}(t)$ we resort to general results of [8] (when $\gamma<1$) and [10] (when $a_{i}\sim i$) where the rate of convergence to equilibrium for solutions to (1.1) has been established under mild assumptions on the initial data. Notice that the rate obtained in [8] is far from being optimal but applies to a wide range of initial data, and ensures at least the existence of some explicit time $T>0$ such that $c_{1}(t)<z_{s}$ for $t\geqslant T$. This is enough to apply the method we just described.

1.4. Organization of the paper

In the next section we introduce the main tools for the proof of both Theorems 1.1 and 1.2, namely the introduction of the tail density $\mathcal{G}(t)$ and the maximum principle. The proofs of Theorems 1.1 and 1.2 are then given in Section 3 after recalling the result on convergence to equilibrium in [8].

2. Tail density and the maximum principle

A key idea for showing our main theorems is to find a quantity, which we will call the tail density, that obeys a maximum principle for the equation and whose moments are intimately connected to the moments of $\bm{c}(t)$, the solution to the Becker-Döring equations.

Definition 2.1**.**

Let $\bm{c}=\left\{c_{i}\right\}_{i\in\mathbb{N}}$ be a non-negative, summable sequence. We define the tail density of $\bm{c}$ as the sequence $\mathcal{G}=\left\{G_{j}\right\}_{j\in\mathbb{N}}$ given by

[TABLE]

The tail density enjoys the following properties:

Lemma 2.2**.**

Let $\bm{c}=\left\{c_{i}\right\}_{i\in\mathbb{N}}$ be a non-negative, summable sequence. Then

1. (i)

the tail density $\mathcal{G}=\left\{G_{j}\right\}_{j\in\mathbb{N}}$ is a non-negative, non-increasing sequence. 2. (ii)

For any $k\geqslant 0$

[TABLE] 3. (iii)

Given $\gamma\in[0,1)$, let $\alpha>0$ and $\mu\in(0,1-\gamma).$ Introduce

[TABLE]

Then, there exist $\eta_{1},\eta_{2}>0$, depending only on $\mu$ and $\alpha$, such that

[TABLE]

where $\psi_{j}:=j^{\mu-1}\exp\left(\alpha j^{\mu}\right),$ for all $j\geqslant 1.$

Proof.

Point $(i)$ is clear from the definition of the tail density. To show $(ii)$ we notice that

[TABLE]

Since

[TABLE]

where we were allowed to change summation due to the non-negativity of the elements, the proof of $(ii)$ complete.

To prove point $(iii)$ we write $c_{i}=G_{i}-G_{i+1}$ to obtain

[TABLE]

Since

[TABLE]

we have

[TABLE]

In addition,

[TABLE]

Thus, using the fact that $\exp\left(\alpha j^{\mu}-\alpha(j-1)^{\mu}\right)\underset{j\rightarrow\infty}{\longrightarrow}1$ when $0<\mu<1$, from (2.4) we conclude that

[TABLE]

where $C=\alpha\mu\,\inf_{j\geqslant 2}\exp\left(\alpha(j-1)^{\mu}-\alpha j^{\mu}\right)$. This proves the result. ∎

We have also the following whenever $\bm{c}(t)$ is a solution to (1.1):

Lemma 2.3**.**

Let $\bm{c}(t)$ be a solution to the Becker-Döring equation with non-negative, finite density initial datum. Assume (1.9) and (1.10) to hold. Then its associated tail density $\mathcal{G}(t)$ is continuously differentiable, and satisfies

[TABLE]

In particular, if there exist $t_{0}>0$ and $\omega>0$ such that

[TABLE]

then

[TABLE]

Proof.

We notice that for any $k,N\in\mathbb{N}$ with $1<k\leqslant N$

[TABLE]

[TABLE]

where we used (1.9) and (1.10). Recalling now that $\sum_{i=1}^{\infty}ic_{i}(t)$ converges uniformly on any interval thanks to Proposition 3.1 in [4], we deduce that $\sum_{i=1}^{\infty}\frac{\,\mathrm{d}}{\,\mathrm{d}t}c_{i}(t)$ converges uniformly on any interval for any $\gamma\in[0,1]$. Thus, since $\bm{c}(t)$ is continuously differentiable, so it $\mathcal{G}(t)$, and

[TABLE]

completing the proof. The second assertion follows immediately from (2.7) and the non-negativity of all the elements involved. ∎

Looking at inequality (2.8) we notice that the infinite system of differential inequalities for tail densities can be represented by an infinite constant matrix with entries only in the diagonal, and above and below it. Moreover, the off–diagonal entries are non-negative. Unsurprisingly, this will entail a maximum principle to the system.

For any given vectors $\bm{u},\bm{v}\in\mathbb{R}^{n}$, we denote by $\bm{u}\leqslant\bm{v}$ the case where $u_{i}\leqslant v_{i}$ for all $i=1,\dots,n$. Given $z\in\mathbb{R}$, we also denote $z_{+}=\max(z,0)$ and, if $\bm{u}=(u^{1},\ldots,u^{n})\in\mathbb{R}^{n}$, we set $\bm{u}_{+}=(u^{1}_{+},\ldots,u^{n}_{+}).$

Lemma 2.4** (Maximum principle for linear ODE systems).**

Let $T>0$ and consider the vector of continuously differentiable functions $\bm{u}=(u_{1},\dots,u_{n})\colon[0,T)\to\mathbb{R}^{n}$. Assume that

[TABLE]

where $A$ is a constant $n\times n$ matrix whose off-diagonal entries are non-negative. Then, if $\bm{u}(0)\leqslant 0$ we have that $\bm{u}(t)\leqslant 0$ for all $t\in[0,T)$.

Proof.

Let $0\leqslant t<T$ be given. Since $\bm{u}$ is differentiable at $t$ we have that for $0<s<T-t$

[TABLE]

Set $A=(a_{i,j})_{i,j=1,\ldots,n}$ and call $s_{0}:=\inf_{i=1,\dots,n}|a_{i,i}|^{-1}\in(0,+\infty]$. As the off-diagonal entries of $A$ are non-negative we find that for $0<s<s_{*}:=\min\{s_{0},T-t\}$, all the entries of the matrix $I+sA$ are non-negative. Thus,

[TABLE]

for all $0<s<s_{*}$. Denoting by $y(t)$ the $\ell_{1}$-norm of $[\bm{u}(t)]_{+}$, i.e. $y(t):=\sum_{j=1}^{n}u^{j}_{+}(t)$, we see that

[TABLE]

where

[TABLE]

Dividing by $s$ and taking the limit as $s\to 0$ we see that

[TABLE]

A generalised version of Gronwall’s lemma, following from a generalised comparison theorem that can be found in Lemma 16.4, p. 215 of [2], implies that

[TABLE]

Since $y(0)=0$ we conclude the proof. ∎

Remark 2.5*.*

The above proof is a simple version of the invariance of the cone of points with non-positive coordinates using the so-called sub-tangent condition (as given for example in Theorem 16.5, p. 215 of [2]). A matrix with non-negative off–diagonal entries is known as a Metzler matrix, and its sign-preserving properties are well-known. We have given a full proof for the sake of completeness, and since we need the result when we deal with an inequality (and not an equality).

In order to use this maximum principle we define the notion of supersolution for the Becker-Döring equations:

Definition 2.6** (Supersolution).**

Let $0<\varrho$ and $0<\omega$ be given. We say that a non-negative sequence $(r_{j})_{j\geqslant 1}$ is a $(\omega,\varrho)$-supersolution to Becker-Döring equations if

1. (1)

$r_{1}\geqslant\varrho$ 2. (2)

For all $j\geqslant 2$ it holds that

[TABLE]

Remark 2.7*.*

Notice that, strictly speaking, a sequence $(r_{j})_{j\geqslant 1}$ with the above properties is not a supersolution to (1.1) (in the classical ODEs sense) but rather a supersolution of the system:

[TABLE]

with $x_{i}(t)\leqslant\varrho$ for all $i\geqslant 1,$ $t\geqslant 0$. Notice also that, whenever (2.8) holds true, $\mathcal{G}(t)$ is a subsolution of (2.11) on $[t_{0},\infty)$.

The values of $\omega$ and $\varrho$ that are helpful to obtain a maximum principle are connected to those of $c_{1}(t)$ and the mass of $\bm{c}$ in the following way:

Proposition 2.8** (Maximum principle).**

Let $\bm{c}(t)=(c_{i}(t))_{i\geqslant 1}$ be a solution to the Becker-Döring equations with non-negative initial condition $\bm{c}(0)$. Assume that (1.9) and (1.10) hold and that the density of $\bm{c}$ is $0<\varrho<\varrho_{s}$. Let $\mathcal{G}(t)$ denote the tail density of $\bm{c}(t).$ Take $\omega>0$ and $0\leqslant t_{0}<t_{1}$, and denote $I:=[t_{0},t_{1}]$. Assume that

[TABLE]

Let $(r_{j})_{j\geqslant 1}$ be a $(\omega,\varrho)$-supersolution to the associated Becker-Döring equations. Then if

[TABLE]

we find that

[TABLE]

Proof.

Since $[t_{0},t_{1}]$ is compact and the sequence $\mathcal{G}(t)=(G_{j}(t))_{j\geqslant 1}$ is a non-increasing sequence of continuous functions that converge pointwise to zero, we conclude from Dini’s Theorem that

[TABLE]

Given $\varepsilon>0$, set

[TABLE]

There exists an $M\geqslant 1$, independent in $t$, such that

[TABLE]

In addition, as $H_{1}(t)=\varrho-r_{1}-\varepsilon$, the condition $\varrho\leqslant r_{1}$ implies that $H_{1}(t)<0$.

Lemma 2.3 and condition (2.10) for the supersolution sequence imply that

[TABLE]

on $I$. Due to (2.12) and the fact that $H_{1}(t)<0$ we can consider the system (2.13) for $j=2,\ldots,M$ only. This system can be rewritten as

[TABLE]

where $\alpha_{j}=a_{j-1}\omega+b_{j}$. As all the off–diagonal entries of the above matrix are non-negative and since our initial conditions imply

[TABLE]

we find that due to our maximum principle (Lemma 2.4)

[TABLE]

Together with the bounds on $H_{1}$ and $(H_{j})_{j\geqslant M+1}$ we conclude that on $I$

[TABLE]

As $\varepsilon$ was arbitrary, we get our desired result. ∎

As we can see, if $\varrho$ is the density of $\bm{c}(t)$, the associated $(\omega,\varrho)$-supersolution will control $\mathcal{G}(t)$, with appropriate initial conditions. The question remains as to which $\omega$ one may choose. This choice will be crucial to the existence of a supersolution that bounds $\mathcal{G}(t)$ at a suitable time. Since in the subcritical case $c_{1}(t)$ converges to $\overline{z}<z_{s}$, it seems natural to choose $\omega$ close to, but larger than, $\overline{z}$. This is indeed the required ingredient to construct a supersolution. The following lemma, which not only gives us the existence of a supersolution but also gives us moment connections between it and $\mathcal{G}$, is reminiscent to Lemma 3.4 in [7].

Lemma 2.9**.**

Assume that conditions (1.9) to (1.11) hold. Let $0<\varrho<\varrho_{s}$ and $0<\omega<z_{s}$ be given. Consider a non-negative, non-increasing sequence $({g}_{j})_{j\geqslant 1}$ that tends to [math] as $j$ goes to infinity and such that $g_{1}\leqslant\varrho$. Then, there exists a $(\omega,\varrho)$-supersolution $(r_{j})_{j\geqslant 1}$ to the associated Becker-Döring equations which tends to [math] as $j$ goes to infinity, and satisfies

[TABLE]

Moreover, $(r_{j})_{j\geqslant 1}$ can be chosen so that for any $1\leqslant\delta<z_{s}/\omega$ and any positive, eventually non-decreasing sequence $(\phi_{j})_{j\geqslant 1}$ satisfying

[TABLE]

we have that

[TABLE]

where $C>0$ is a fixed constant that depends only on $(\phi_{j})_{j\geqslant 1}$, $\varrho$, $\omega$, $\delta$ and the coefficients $(a_{i})_{i\geqslant 1}$, $(b_{i})_{i\geqslant 1}$.

Proof.

According to (1.5) and (1.11), one notices that $\lim_{j\to\infty}b_{j}/a_{j-1}=z_{s}$, we can find $1<\lambda\in(\delta,\frac{z_{s}}{\omega})$ and $N\geqslant 1$ such that

[TABLE]

For $j\geqslant N$, we set

[TABLE]

and define

[TABLE]

We will now show that this sequence is well defined and is bounded. Indeed, using the fact that

[TABLE]

and the fact that $\frac{\varrho}{\lambda\omega}\leqslant s_{N}\leqslant\varrho\left(1+\frac{1}{\lambda\omega}\right)$ and $1\leqslant\delta<\lambda$, we can use a simple induction to show that

[TABLE]

Moreover, as $s_{j+1}\leqslant\frac{s_{j}}{\lambda}+h_{j+1}$ for all $j\geqslant N$, we have that for any $p\geqslant N$

[TABLE]

Due to the non-negativity of $g_{j}$ and the fact that $1<\lambda$ we conclude that

[TABLE]

As $p$ is arbitrary, this shows that the sum converges and thus that $(r_{j})_{j\geqslant N}$ is well defined with $\lim_{j\to\infty}r_{j}=0.$ Moreover, using that $g_{N+1}\leqslant\varrho$ we see from the previous inequality that

[TABLE]

From its definition, $(r_{j})_{j\geqslant N}$ is clearly non-negative and non-increasing. In addition

[TABLE]

where we used the fact that $(g_{j})_{j\geqslant 1}$ goes to zero as $j$ goes to infinity. Due to the choice of $N$, we have that for all $j\geqslant N+1$

[TABLE]

All of the above show that we have managed to construct a supersolution to the associated Becker-Döring equation from the point $j=N+1$. We are left with defining it for $j<N$ and to check the supersolution condition for $j\leqslant N$. We set for any $j<N$

[TABLE]

Clearly, by its definition

[TABLE]

which also shows that $\varrho\leqslant r_{1}$. In addition, one checks that

[TABLE]

Indeed, the last inequality is valid since, due to the choice of $N$,

[TABLE]

Thus $(1)$ and $(2)$ from Definition 2.6, are satisfied up to $j=N$. Together with our definition for $j\geqslant N$ we conclude that $(r_{j})_{j\geqslant 1}$ is an $(\omega,\varrho)$–supersolution to the associated Becker-Döring equation. Moreover,

[TABLE]

We turn our attention now to the second part of the proof. Due to the conditions on $(\phi_{j})_{j\geqslant 1}$ we can find $\delta<\delta_{*}<\lambda$ and $M\geqslant N\geqslant 1$ such that for all $j\geqslant M$

[TABLE]

[TABLE]

Consider the sum $\sum_{j=1}^{\infty}r_{j}\phi_{j}$. Using again that $s_{j+1}\leqslant\frac{s_{j}}{\lambda}+h_{j+1}$ for any $j\geqslant M\geqslant N$, we have that

[TABLE]

Thus

[TABLE]

From the above we can estimate that

[TABLE]

which implies that

[TABLE]

[TABLE]

Thus, as

[TABLE]

we conclude that

[TABLE]

where

[TABLE]

This completes the proof. ∎

Remark 2.10*.*

It is important to note that the sequences $(\phi_{j})_{j\geqslant 1}$ given by

[TABLE]

both satisfy condition (2.14) with $\delta=1$. This means that we can build a supersolution with comparable moments, and stretched exponential moments, to those of $\mathcal{G}(t)$. This will be a crucial element in the proof of our main theorems. Note that $\phi_{j}=\exp\left(\eta\,j\right)$ $(j\geqslant 1)$ is also allowed, as long as $\eta<\log\left(\frac{z_{s}}{\omega}\right)$.

With these tools at hand, we have the main ingredient to prove our main theorem.

3. On the propagation of moments

From the previous section we know that as long as $c_{1}(t)<z_{s}$ in a certain time interval, we are able to construct a supersolution to the associated Becker-Döring equations whose moments are strongly related to the moments of $\bm{c}(t)$, thus allowing us to take advantage of the maximum principle in Lemma 2.8 to obtain a uniform bound. However, the condition on $c_{1}(t)$ is not necessarily valid at all times. Before we prove our main theorems, we show that one can find an explicit time, $T_{0}\geqslant 0$, such that for all $T>T_{0}$, $c_{1}(t)<z_{s}$. Before that time the moments, and stretched exponential moments, grow at most exponentially in time (which was already noted in previous works).

The fact that $c_{1}(t)<z_{s}$ after a certain time $T_{0}$ is a consequence of a stronger statement about the convergence to equilibrium of the solution to the Becker-Döring equations. The quantitative version we state here uses the relative free energy mentioned in the introduction and can be easily deduced from results in [8] and [10]:

Theorem 3.1**.**

Consider the Becker-Döring equations with coagulation and fragmentation coefficients $(a_{i})_{i\geqslant 1},(b_{i})_{i\geqslant 1}$ such that conditions (1.9) to (1.12) hold. Assume that $\bm{c}(t)=(c_{i}(t))_{i\geqslant 1}$ is a solution to the Becker-Döring equations with non-negative, subcritical initial datum $\bm{c}(0)$ satisfying (1.13). Then, there exists a constant $C>0$, depending only on the coagulation and fragmentation coefficients, the density $\varrho$, the initial moment of $\bm{c}$ of order $\max\{2-\gamma,1+\gamma\}$ and the initial relative free energy, such that

[TABLE]

Proof.

The result in [8] is valid under (1.9)–(1.13) for $\gamma\in[0,1)$ (and holds true for more general discrete coagulation models). For $\gamma=1$ (i.e., the second option in (1.9)), the rate is actually exponential thanks to a recent result by the authors: see Theorem 1.3 in [10]. Notice that for the case $\gamma=1$ no assumptions on the propagation of moments are needed in [10]. ∎

As a consequence we have:

Corollary 3.2**.**

Assume the conditions of Theorem 3.1. Then for any $\delta>0$ one has

[TABLE]

where $T_{\delta}=\max\left(1,\exp\left(\frac{C^{2}}{\delta^{2}}-1\right)\right)$, with $C>0$ is the explicit constant from Theorem 3.1.

Proof.

Since

[TABLE]

the result follows immediately. ∎

The last issue that we need to deal with before being able to tackle our main theorem is the issue of the possible growth of our moments, and stretched exponential moments, in the time until $c_{1}(t)$ is in the right range to use our machinery from the Section 2. This has actually been shown in [4]:

Lemma 3.3**.**

Consider the Becker-Döring equations with coagulation and fragmentation coefficients $(a_{i})_{i\geqslant 1},(b_{i})_{i\geqslant 1}$ such that (1.9) and (1.10) hold true. Let $(\phi_{i})_{i\geqslant 1}$ be a non-negative sequence such that

[TABLE]

for some $\varepsilon>0$. Then, if $\bm{c}(t)$ is the solution to the Becker-Döring equations with non-negative initial datum $\bm{c}(0)$ and density $\varrho$ such that

[TABLE]

there exists a positive constant $C_{\phi}$ depending on $\mathcal{A}_{\phi},\varepsilon$ and $\varrho$ such that

[TABLE]

Proof.

A detailed proof of the result is given in [4]. For the sake of completeness we provide a formal proof here, from which a fully rigorous one can be obtained by standard approximation arguments. Using (1.3) with the sequence $(\phi_{i})_{i\geqslant 1}$ we get

[TABLE]

where we used that $(\phi_{i})_{i\geqslant 1}$ is non-negative. The first sum is estimated with (3.2):

[TABLE]

while using (3.2) and (1.10) we find that

[TABLE]

The result then follows with $C_{\phi}=\left(\varrho+\varepsilon^{-1}\overline{b}\right)\mathcal{A}_{\phi}.$ ∎

Remark 3.4*.*

The two main types of moments we consider, namely

[TABLE]

satisfy the assumptions of Lemma 3.3. On the contrary, the previous lemma cannot be applied to exponential moments (with weight $e^{\mu i}$) if $a_{i}$ diverges to $+\infty$ with $i$.

We are now ready to prove our main theorems.

Proof of Theorem 1.1.

Since we are dealing with a subcritical solution, using Corollary 3.2 we can find an explicit time $T_{0}>0$ such that

[TABLE]

Due to Lemma 3.3 we can find an explicit constant $C_{k}>0$ such that for all $t\leqslant T_{0}$

[TABLE]

Considering the tail density sequence $\mathcal{G}(t)$, we use Lemma 2.9 to find a supersolution to the associated Becker-Döring equation for $t\geqslant T_{0}$ such that

[TABLE]

and

[TABLE]

where we have used Lemma 2.2 and 3.3. According to the maximum principle, Proposition 2.8, and the fact that $c_{1}(t)<\omega$ for $t>T_{0}$ we find that

[TABLE]

and thus, using Lemma 2.2 again, and (3.3), we have that for all $t\geqslant T_{0}$

[TABLE]

This concludes he proof. ∎

Proof of Theorem 1.2.

We set

[TABLE]

The link between $\mathcal{E}_{\mu}(t)$ and the tail density $\mathcal{G}(t)$ is given by (2.3), namely

[TABLE]

for some positive constants $\eta_{1},\eta_{2}>0$ depending only on $\alpha,\mu$ and $(\psi_{j})_{j\geqslant 1}:=\left(\alpha\mu j^{\mu-1}\exp\left(\alpha j^{\mu}\right)\right)_{j\geqslant 1}$. At this point we just mimic the proof of Theorem 1.1: We find an explicit time $T_{0}>0$ such that for any $t>T_{0}$ we have that $c_{1}(t)<\omega<z_{s}.$ Then, using Lemma 2.9, and noting that our $\psi_{j}$ satisfies its conditions, we find a supersolution to the associated Becker-Döring equation, $(r_{j})_{j\geqslant 1}$ such that $G_{j}(T_{0})\leqslant r_{j}$ $(j\geqslant 1)$ and

[TABLE]

Invoking Proposition 2.8, we get $G_{j}(t)\leqslant r_{j}$ for all $t\geqslant T_{0}$ and all $j\geqslant 1.$ Using again (3.4), we have then

[TABLE]

completing the proof by using Lemma 3.3 with $\phi_{i}=\exp\left(\alpha i^{\mu}\right)$ $(i\geqslant 1)$. ∎

Acknowledgement

JAC was supported by project MTM2014-52056-P, funded by the Spanish government and the European Regional Development Fund. AE was supported by the Austrian Science Fund (FWF) grant M 2104-N32. We would like to thank André Schlichting for useful comments on our previous work, which led to us realising that no uniform bounds for moments were available.