THE LIFSHITZ–SLYOZOV–WAGNER EQUATION
FOR REACTION-CONTROLLED KINETICS
APOSTOLOS DAMIALIS
Abstract. We rigorously derive a weak form of the Lifshitz–Slyozov–Wagner
equation as the homogenization limit of a Stefan-type problem describing
reaction-controlled coarsening of a large number of small spherical particles.
Moreover, we deduce that the effective mean-field description holds true in the
particular limit of vanishing surface-area density of particles.
1. Introduction
The late-stage behavior of a material undergoing a first-order phase transition
(due to changes in temperature and/or pressure for example) is characterized by
thermodynamic instability resolved through phase separation and consequent coars-
ening of the emerging phase. In the case of the new phase occupying much smaller
volume fraction, and thus appearing as well-separated particles, this coarsening pro-
cess (known as Ostwald ripening) is driven by the minimization of surface energy at
the interface via diffusional mass exchange between particles while the total mass
or volume of each phase is conserved. The result of this kind of mass diffusion from
regions of high to regions of low interfacial curvature is the growth of large parti-
cles and the shrinkage and final extinction of smaller ones. For a review of some
aspects of Ostwald ripening, mainly from the physical and modeling viewpoint, see
the survey by Voorhees [21] or the book by Ratke and Voorhees [18].
In this coarsening scenario the mass-diffusion process can be controlled by two
different mechanisms: either by the diffusion of atoms away from the particles and
into the bulk, or by the reaction-rate of attachment of atoms at the phase interface.
In the former case (diffusion control), the random exchange of atoms between the
particles and the bulk is sufficiently rapid and the surrounding of each particle is in
thermal equilibrium with the atoms in it; in the latter (interface-reaction control),
detachment and attachment are slow compared to diffusion and the surrounding
bulk can be out of equilibrium with the particle interface. We refer to the physics
literature for more details, for example, Slezov and Sagalovich [19], Bartelt, Theis,
and Tromp [3]; for a related mathematical treatment see Dai and Pego [5].
The classical theory for Ostwald ripening was developed by Lifshitz and Slyozov
[9] and Wagner [22] in the case of supersaturated solid solutions in three dimensions.
The Lifshitz–Slyozov–Wagner theory statistically characterizes the evolution by the
particle-radius density n(t, R), where n(t, R) dR is defined to be the number of
particles with radii between R and dR at time t per unit volume. In the late
stages of the phase transition nucleation and coalesence of particles can be neglected
since new nuclei dissolve immediately and since particles cannot merge because of
the large distances between them. Thus, the particle-radius density satisfies the
This work was supported by the DFG through the Graduiertenkolleg RTG-1128 “Analysis,
Numerics, and Optimization of Multiphase Problems” at the Humboldt-Universität zu Berlin.
http://arxiv.org/abs/0704.0565v4
2 APOSTOLOS DAMIALIS
continuity equation (see [18, §5.1])
n(t, R) +
v(t, R)n(t, R)
where v(t, R) denotes the growth rate of particles of radius R at time t. Using a
mean-field ansatz (cf. Section 3), Lifshitz, Slyozov, and Wagner formally calculate
n(t, R) +
(Rū− 1)n(t, R)
ū(t) =
n(t, R) dR
Rn(t, R) dR,
in the diffusion-controlled case, and
n(t, R) +
n(t, R)
ū(t) =
Rn(t, R) dR
R2n(t, R) dR,
in the reaction-controlled one, both results valid in the limit of vanishing mass or
volume fraction of particles.
In [11] and [12] Niethammer rigorously derived the effective equations in the dif-
fusion-controlled case, starting from a quasi-static one-phase Stefan problem with
surface tension and kinetic undercooling,
−∆u = 0 in Ω \G,
V = ∇u · n on ∂G,
u = H + βV on ∂G,
(1.1)
and restricting it to spherical particles. The same was also done in [11] for the full
time-dependent parabolic problem but without the kinetic-drag term βV . Here, u
is a chemical potential, n is the outer normal to the particle phase G, V is the nor-
mal velocity of the phase interface ∂G, and H is its mean curvature. The domain
Ω ⊂ R3 is considered bounded and β is a parameter that comes from the nondi-
mensionalization and scales like diffusivity over mobility. The second boundary
condition is the Gibbs–Thomson law, coupling the curvature of the interface with
the chemical potential, modified by accounting for kinetic drag. Note that while
under diffusion control the parameter β is small and the kinetic drag can even
be neglected (thus yielding the well-known Mullins–Sekerka model [10]), in the
reaction-controlled case the values of β are large and, therefore, the kinetic-drag
term is necessary. For a derivation of such sharp-interface free-boundary problems
from continuum mechanics and thermodynamics see the book of Gurtin [8].
The goal in the following is to use the techniques developed in [11] and [12] to
derive the effective equations in the reaction-controlled case. This involves passing
over to a different time scale incorporating the parameter β tending to infinity
(see Section 2) and, as a result, some extra manipulations in the proofs. Except
for the scaling, in Section 2 we also give short proofs of some useful preliminaries
and discuss the validity of the mean-field description while in Section 3 we prove
pointwise estimates for approximate solutions and for the growth rates of particles.
Finally, using these estimates, in Section 4 we pass to the homogenization limit of
infinitely-many particles and obtain a weak form of the Lifshitz–Slyozov–Wagner
equation.
THE LSW EQUATION FOR REACTION-CONTROLLED KINETICS 3
In comparison with the results in the diffusion-controlled case, we make precise
that the crucial quantity that has to vanish in order to neglect direct interactions
between particles and justify the expected mean-field law is the surface-area density
of the particles in contrast to their capacity in the other case (see [11] and [12]).
This difference is of interest since the asymptotic limits of vanishing surface area
and capacity have different physical interpretations and further refine the näıve
general limit of vanishing mass or volume. For the reaction-controlled case though,
the result is in some sense to be expected since the limit of vanishing surface-
area density corresponds to the physics of the interface-reaction-controlled scenario,
where there is an obvious dependence on the area of the interface.
2. Formulation, scaling, and preliminary estimates
We start with problem (1.1) where the quasi-static approximation to the para-
bolic diffusion equation is justified by the small interfacial velocities present during
late-stage coarsening. (See the discussion in Mullins and Sekerka [10].)
We further suppose that the solid phase consists of spherical particles with cen-
ters fixed in space, a simplification that can be justified by the work of Alikakos
and Fusco [1], [2], and Velázquez [20]. Denoting these particles as Bi, where each
Bi is the closed ball B(xi, Ri(t)), the particle phase is then the union ∪Bi and its
isotropic evolution can be modeled by averaging the flux in the Stefan condition,
i.e.,
V = Ṙi(t) := −
∇u · n ,
where the average integral is defined as
for a function f on some domain D, and where the overdot denotes a derivative
with respect to time; the Gibbs–Thomson law becomes then
+ βṘi,
since in the case of spheres the mean curvature is the inverse radius.
To have many small particles in a bounded domain, for a system with size of order
O(1), say the unit cube [0, 1]3, let δ be the typical particle distance with 0 < δ ≪ 1.
For the distribution of particle centers in space, we assume, for simplicity, that they
are situated on a three-dimensional lattice of spacing δ. Then, the initial number
density of particles Ni(δ) will be bounded by 1/δ
3, and for the particles to be small
let the typical particle size be δα for α > 1. For times t ∈ [0, T ] we choose a δ small
enough so that adjacent particles of size δα will not collide during the evolution up
to a maximal time T .
Concerning the assumption on the spatial distribution of particles, a more general
assumption like infi6=j |xi−xj| > cδ, for a constant c > 0, would still be enough for
our purposes in this work. These considerations will also be used in the proof of
Lemma 3.2 where we approximate a certain sum over all particles by an integral. For
an approach using more sophisticated deterministic and stochastic assumptions on
the distribution of particles with respect to homogenization we refer to Niethammer
and Velázquez [16], [17], where also further refinements of the theory are made.
To have particle sizes of order O(1) as well, we rescale
Rδi :=
4 APOSTOLOS DAMIALIS
and motivated by the scaling invariance of problem (1.1) (cf. [5]),
uδ := δαu, tδ :=
Notice that this rescaling is another way of addressing the reaction-controlled
regime. Instead of rescaling time by β and then letting β tend to infinity, we
keep β fixed and positive, and specially rescale as above letting δ tend to zero.
Since now β plays no significant role, we will set it to unity in what follows. In
addition, one easily sees that the transformations Rδi , u
δ, and tδ preserve the form
of the equations. From hereon we also drop the superscript δ from the notation for
time and to denote the dependence on the new scale we write
Bδi := B(xi, δ
αRδi ).
Finally, note that under diffusion control the relevant scale for time would be δ3α
instead of δ2α. This difference is key to all that follows, leading to different consid-
erations on the validity of the mean-field model. (Cf. the remarks following Lemma
2.1.)
As initial data, for every particle-center xi we associate a corresponding bounded
initial radius Rδi (0) with the assumption that
supi∈NiR
i (0) ≤ R0,
uniformly for some constant R0. To consider a closed system, we impose a no-flux
Neumann boundary condition on the outer boundary of Ω, i.e.,
∇uδ · n = 0 on ∂Ω.
In case the ith particle vanishes at time ti := sup{t | Rδi (t) > 0}, for times later
than ti we define R
i to be zero, reduce the number N(t) := {j | Rδj(t) > 0} of
active particles by one, and neglect the boundary ∂Bδi in the boundary conditions.
In the following, all sums, unions, and suprema will run over the set N(t), with
N(0) ≡ Ni, and any further reference to the particle-number density will mean the
active particle-number density N unless otherwise noted.
Summarizing, the restricted and rescaled problem for the particle radii can be
considered as a nonlocal, N -dimensional system of ordinary differential equations
Ṙδi (t) =
4πδ2αRδi (t)
∇uδ · n on ∂Bδi (t), (2.1)
for times t ∈ (0, ti), ti < T , and with bounded initial data Rδi (0) for every i, while
the chemical potential is determined by
−∆uδ(t, x) = 0 in Ω \ ∪Bδi (t), (2.2)
uδ(t, x) =
Rδi (t)
+ Ṙδi (t) on ∂B
i (t), (2.3)
and the Neumann condition on the outer boundary.
Global existence and uniqueness of continuous, piecewise-smooth solutions for
a similar restricted Stefan problem was proved in [12] by an application of the
Picard–Lindelöf theorem, the only difference being the different time scale. These
solutions are not globally smooth due to the singularities arising from the extinction
of particles; however, they are smooth in the intervals between the extinction times
ti. In the following, when we mention solutions of the problem we will mean such
continuous, piecewise-smooth solutions that exist up to any given time T .
THE LSW EQUATION FOR REACTION-CONTROLLED KINETICS 5
It is easy to see that equations (2.1), (2.2), (2.3), along with the outer boundary
condition conserve the volume and decrease the interfacial area of the particle phase.
Indeed, differentiating the total volume of particles with respect to time gives
Rδi (t)
3 = 3
Rδi (t)
2Ṙδi (t) = 3
Rδi (t)
4πδ2αRδi (t)
∇uδ · n
where the last sum vanishes due to the divergence theorem, equation (2.2), and the
no-flux condition on ∂Ω. The decrease of total surface area follows from the next
a priori estimate.
Lemma 2.1. For any time t ∈ (0, T ), the solutions of the problem satisfy the
following energy equality.
(Rδi )
2|Ṙδi |2 +
Rδi (t)
4πδ2α
Ω\∪Bδ
|∇uδ|2 = 1
Rδi (0)
Proof. Multiplying −∆uδ = 0 with uδ, integrating over Ω \ ∪Bδi , and integrating
by parts gives
Ω\∪Bδ
|∇uδ|2 +
(∇uδ · n)uδ −
(∇uδ · n)uδ = 0,
where the last term vanishes due to the Neumann condition on the outer boundary.
Thus, using equations (2.3) and (2.1) we get
Ω\∪Bδ
|∇uδ|2 =
+ Ṙδi
∇uδ · n
+ Ṙδi
4πδ2α(Rδi )
2Ṙδi
= 4πδ2α
Rδi Ṙ
i + (R
2|Ṙδi |2
and after rearranging,
(Rδi )
2|Ṙδi |2 +
Rδi Ṙ
4πδ2α
Ω\∪Bδ
|∇uδ|2 = 0. (2.4)
The result follows from an integration over time. �
After normalization with respect to the initial particle-number density Ni, this
energy equality can yield useful information on the validity of the mean-field ap-
proach. In fact, we have
(Rδi )
2|Ṙδi |2+
Rδi (t)
4πNiδ2α
Ω\∪Bδ
|∇uδ|2 = 1
Rδi (0)
where the right-hand side is uniformly bounded by the assumption on the initial
radii. For the left-hand side to stay bounded as well, if the quantity Niδ
2α tends to
zero, the same must hold for |∇uδ| and it is exactly this limit of vanishing surface-
area density of particles that results in a mean field that is constant in space since,
in particular,
∇uδ → 0 in L2
0, T ;H1(Ω)
Here and in the following, to obtain global estimates that are uniform in δ we extend
uδ to the interior of particles, and thus to the whole of Ω, by its boundary values.
It is important to note that in our scaling setup, for the surface area to vanish as δ
6 APOSTOLOS DAMIALIS
tends to zero, the exponent α must be strictly larger than 3/2 since Ni is O(1/δ
These facts will be made precise in Corollary 3.3 where we give an estimate of the
mean-field effect. Note also that we do not address here the critical case α = 3/2
that corresponds to finite surface area. For that one would have to use the different
methods developed by Niethammer and Otto in [13].
Finally, note that for similar considerations under diffusion control, the corre-
sponding quantity would be the capacity Niδ
α due to the different time scale. In
three dimensions, this capacity effect fits to general homogenization results as in
the work of Cioranescu and Murat [4]; to our knowledge though, the surface-area
effect has not been explicitly discussed in the relevant literature.
3. Approximation and growth-rate estimates
As in the mean-field ansatz of Lifshitz, Slyozov, and Wagner, we suppose that
the system is dilute enough so that particles behave as if they were isolated and we
base our approximation on the solution of a single-particle problem.
Consider problem (2.1), (2.2), (2.3) for a single spherical particle centered at
the origin and with initially unscaled radius r that we rescale as rδ := r/δα, along
with the corresponding reaction-controlled rescalings for a chemical potential uδr
and time, as in Section 2. For this rescaled particle Bδr we consider the following
problem in the whole space:
ṙδ(t) =
4πδ2αrδ(t)2
∇uδr · n on ∂Bδr ,
where the chemical potential uδr(t, x) satisfies
−∆uδr(t, x) = 0 in R3 \Bδr ,
uδr(t, x) =
rδ(t)
+ ṙδ(t) for x ∈ ∂Bδr ,
and the mean-field assumption is posed as a condition at infinity, i.e.,
|x|→∞
uδr(t, x) = ū
r(t).
This problem can be explicitly solved to give
uδr(t, x) = ū
r(t) +
δαrδ(t)
1 + δαrδ(t)
1− ūδr(t)rδ(t)
ṙδ(t) =
1 + δαrδ(t)
ūδr(t)−
rδ(t)
Note that in the formal limit of δ tending to zero, the expected effective equations
take the general form
u(t, x) = ū(t) and ṙ = ū− 1
as in the reaction-controlled Lifshitz–Slyozov–Wagner theory.
Going now back to the many-particle problem, a calculation using the single-
particle growth rate above along with the requirement that the volume is conserved
gives the following expression for the mean field
ūδ =
1 + δαRδi
(Rδi )
1 + δαRδi
. (3.1)
THE LSW EQUATION FOR REACTION-CONTROLLED KINETICS 7
The effect of this mean field plus a sum of single-particle solutions will be the
monopole approximation to the solution uδ supposing that there are no direct
interactions between particles. To this end, let us define the approximate solution
ζδ(t, x) := ūδ(t) +
δαRδi (t)
1 + δαRδi (t)
1− ūδ(t)Rδi (t)
|x− xi|
(3.2)
for x ∈ Ω \ ∪Bδi (t).
Below is a maximum principle tailored to our setting that will be used to compare
the approximation and the solution in the lemma next. Its proof can be found in
[12].
Lemma 3.1. Let Ω be a Lipschitz domain and let ∪Bi ⊂ Ω be a finite collection of
disjoint closed balls. Then, a function v which is constant on each of the boundaries
∂Bi and satisfies
−∆v = 0 in Ω \ ∪Bi,
v − ci
∇v · n ≥ 0 on ∂Bi,
∇v · n ≥ 0 on ∂Ω,
where ci ≥ 0 for all i, also satisfies
v ≥ 0 in Ω \ ∪Bi.
Lemma 3.2. For any time t ∈ (0, T ) and small positive ε, the chemical potential
and its approximation satisfy
‖uδ − ζδ‖L∞(Ω\∪Bδ
)(t) ≤ Cδ2α−3−ε supRδi (t)
1 + ūδ(t) supRδi (t)
Proof. Since the difference uδ − ζδ is already harmonic in Ω \ ∪Bδi as ζδ is a su-
perposition of fundamental solutions, we would like to estimate to what extent it
satisfies the maximum principle’s boundary conditions.
For the condition on the particle boundaries, we use equations (2.1), (2.3), and
the definition of ζδ to calculate for x on the boundary ∂Bδi of the ith particle,
ζδ(t, x)−−
∇ζδ · n
uδ(t, x)−−
∇uδ · n
ζδ − 1
∇ζδ · n
ūδ − 1
δαRδj
1 + δαRδj
(1− ūδRδj)
|x− xj |
|x− xj |
and since by the divergence theorem there holds for j 6= i,
|x− xj |
· n = 0,
while for j = i,
|x− xi|
· n = − 1
δα(Rδi )
8 APOSTOLOS DAMIALIS
we continue the calculation to get
ūδ − 1
1− ūδRδi
Rδi (1 + δ
αRδi )
δαRδj
1 + δαRδj
(1− ūδRδj)
|x− xj |
j 6=i
δαRδj
1 + δαRδj
(1− ūδRδj)
|x− xj |
≤ δ2α−3 supRδj (1 + ūδ supRδj )
j 6=i
|x− xj |
≤ Cδ2α−3 supRδj(1 + ūδ supRδj). (3.3)
In the last step, keeping in mind the assumptions on the spatial distribution of
particle centers, the sum is bounded for j 6= i since it is considered as a Riemann-
sum approximation to the integral
|x− y|
which in turn is bounded using radial symmetry around the singularity and where
the factor δ3 in the sum compensates for the scaling in space.
To further fulfil the maximum principle’s outer boundary condition on ∂Ω, we
consider the comparison function ζδ+zδ, where the auxiliary function zδ solves the
problem
−∆zδ =
∇ζδ · n in Ω,
∇zδ · n = −∇ζδ · n on ∂Ω,
zδ = 0,
(3.4)
such that the comparison function ζδ + zδ has zero normal derivative on ∂Ω. To
work with the maximum principle, zδ also needs to be harmonic in Ω and for that
we need that the integral
∇ζδ · n vanishes. But,
∇ζδ · n = δα
δαRδi
1 + δαRδi
(1−Rδi ūδ)
|x− xi|
· n ,
where the last integral equals −4π, independent of i. Thus, zδ is harmonic if and
only if
ūδ =
1 + δαRδi
(Rδi )
1 + δαRδi
which is exactly the mean field (3.1) as dictated by the single-particle ansatz in
the beginning of the section. Moreover, since now zδ is harmonic, the divergence
theorem further gives
∇zδ · n = 0.
A construction as in Lemma 3 of [11] and elliptic regularity theory (see Gilbarg
and Trudinger [7]) give the estimate
‖zδ‖L∞(Ω) ≤ Cεδ2α−3−ε supRδi (1 + ūδ supRδi ),
where ε is a small positive number.
THE LSW EQUATION FOR REACTION-CONTROLLED KINETICS 9
Let us now apply the maximum principle to the function
f+ := u
δ − ζδ − zδ + Cδ2α−3−ε supRδi (1 + ūδ supRδi ).
For a large enough constant C, say 2Cε, the following hold for f+: it is harmonic,
there holds ∇f+ · n = 0 on ∂Ω by the construction of zδ, and for the constants
ci = 1/4πδ
2α(Rδi )
2, estimate (3.3) gives
uδ − ζδ − zδ + Cδ2α−3−ε supRδi (1 + ūδ supRδi )− ci
∇(uδ − ζδ − zδ) · n ≥ 0.
Thus, f+ satisfies the maximum principle’s conditions and therefore, f+ ≥ 0 in
Ω \ ∪Bδi , i.e.,
uδ − ζδ − zδ ≥ −Cδ2α−3−ε supRδi (1 + ūδ supRδi ).
Using the maximum principle with −v instead of v, the function
f− := u
δ − ζδ − zδ − Cδ2α−3−ε supRδi (1 + ūδ supRδi ),
again satisfies the corresponding conditions and, as above, yields f− ≤ 0 in Ω\∪Bδi ,
i.e.,
uδ − ζδ − zδ ≤ Cδ2α−3−ε supRδi (1 + ūδ supRδi ).
Combining the last two inequalities, we get
‖uδ − ζδ − zδ‖L∞(Ω\∪Bδ
) ≤ Cδ2α−3−ε supRδi (1 + ūδ supRδi )
and the lemma follows by the triangle inequality using the regularity of zδ. �
In the previous lemma it is clear that our approach excludes the critical case
α = 3/2. In the following we introduce, for technical reasons, a new exponent
γ > 0 with the property
δγ := max {δα, δ2α−3, δ2α−3−ε}
for each α greater than 3/2 + ε.
As a corollary to the previous lemma we can now estimate the effect of the mean
field.
Corollary 3.3. For any time t ∈ (0, T ) and γ > 0, the chemical potential and the
mean field satisfy
‖uδ − ūδ‖L∞(Ω\∪Bδ
)(t) ≤ Cδγ
1 + 2 supRδi (t)
1 + ūδ(t) supRδi (t)
Proof. By the triangle inequality and Lemma 3.2 there holds
‖uδ − ūδ‖L∞(Ω\∪Bδ
) ≤ ‖ζδ − ūδ‖L∞(Ω\∪Bδ
) + Cδ
2α−3−ε supRδj(1 + ū
δ supRδj).
To estimate ‖ζδ− ūδ‖L∞(Ω\∪Bδ
), by the definition of ζ
δ there holds for x ∈ Ω\∪Bδj ,
ζδ(t, x)− ūδ(t)
δαRδj
1 + δαRδj
(1− ūδRδj)
|x− xj |
δαRδi
1 + δαRδi
(1 + ūδRδi )
|x− xi|
j 6=i
δαRδj
1 + δαRδj
(1− ūδRδj)
|x− xj |
and since |x− xi| ≥ δαRδi in Ω \ ∪Bδj , arguing as in estimate (3.3) gives
α(1 + ūδRδi )
1 + δαRδi
+ Cδ2α−3 supRδj(1 + ū
δ supRδj)
≤ C(δα + δ2α−3 supRδj )(1 + ūδ supRδj),
10 APOSTOLOS DAMIALIS
thus,
‖ζδ − ūδ‖L∞(Ω\∪Bδ
) ≤ C(δα + δ2α−3 supRδj )(1 + ūδ supRδj),
and finally,
‖uδ − ūδ‖L∞(Ω\∪Bδ
) ≤ C(δα + δ2α−3 supRδj + δ2α−3−ε supRδj )(1 + ūδ supRδj).
Using the exponent γ, we get
‖uδ − ūδ‖L∞(Ω\∪Bδ
) ≤ Cδγ(1 + 2 supRδi )(1 + ūδ supRδi ). �
The following lemma gives an estimate for the growth rate of particles in accor-
dance with the reaction-controlled Lifshitz–Slyozov–Wagner theory.
Lemma 3.4. For any time t ∈ (0, T ) and γ > 0, for the growth rates of particles
holds
Ṙδi −
ūδ − 1
≤ Cδγ(1 + ūδ supRδi )
1 + (1 + δγ supRδi )(1 + 2 supR
Proof. Let wδi be the capacity potential of the ball B
i with respect to a larger ball
Bλδi := B(xi, λδ
αRδi ) for λ > 1, i.e., let wi solve
−∆wδi = 0 in Bλδi \Bδi ,
wδi = 0 on ∂B
wδi = 1 in B
(3.5)
An explicit calculation gives
wδi =
1− λδ
|x− xi|
(3.6)
and also
∇wδi · n =
∇wδi · n = 4π
1− λδ
αRδi . (3.7)
Using equations (2.1), (2.2), (2.3), and the Neumann boundary condition, along
with the above properties of wδi , and integrating by parts, gives
4πδ2α(Rδi )
2Ṙδi =
∇uδ · n
wδi∇uδ · n
∇wδi∇uδ
uδ∇wδi · n −
uδ∇wδi · n
+ Ṙδi
∇wδi · n −
uδ∇wδi · n
δαRδi
+ Ṙδi − ūδ
(uδ − ūδ)∇wδi · n ,
THE LSW EQUATION FOR REACTION-CONTROLLED KINETICS 11
where in the last equation we used (3.7) and added and subtracted ūδ. After
rearranging, we have
Ṙδi −
ūδ − 1
δαRδi Ṙ
4πλδαRδi
(uδ − ūδ)∇wδi · n
≤ λ− 1
δαRδi |Ṙδi |+
4πλδαRδi
(uδ − ūδ)∇wδi · n
≤ δαRδi |Ṙδi |+ ‖uδ − ūδ‖L∞(Ω\∪Bδ
), (3.8)
where in the last step we again used equation (3.7). But by using equation (2.3)
for uδ on ∂Bδi we have
Rδi |Ṙδi | ≤ 1 +Rδi |uδ| ≤ 1 +Rδi (‖uδ − ūδ‖L∞(Ω\∪Bδ
) + ū
δ). (3.9)
Substituting back in (3.8) and using Corollary 3.3 gives the final estimate. �
The next lemma ensures that the bounds in the approximation and the growth-
rate estimates are indeed uniform.
Lemma 3.5. For any time t ∈ (0, T ), the mean field and the radii of the particles
are uniformly bounded, i.e.,
ūδ(t) ≤ C and supRδi (t) ≤ C.
Proof. For the mean field (3.1) holds
ūδ =
1 + δαRδi
(Rδi )
1 + δαRδi
≤ supRδi (1 + δα supRδi )
(Rδi )
and since by Hölder’s inequality
(Rδi )
(Rδi )
δ3(Rδi )
conservation of the total volume of particles gives
ūδ ≤ C supRδi (1 + δα supRδi ).
or, using the exponent γ,
ūδ ≤ C supRδi (1 + δγ supRδi ). (3.10)
Consider now the set
t | supRδi (t) ≤
then, for times t ∈ A, plugging (3.10) in estimate (3.9) and using Corollary 3.3
gives
(Rδi )
2 ≤ C sup(Rδi )2 + C.
Integrating over the time interval (0, T ), Gronwall’s inequality implies that
supi supt∈A∩[0,T ] (R
2 ≤ C(T ),
therefore, [0, T ] ⊂ A, i.e., the radii are bounded up to time T as is the mean field
by estimate (3.10). �
Finally, the following lemma gives control over the growth rates of vanishing
particles and will prove useful for some regularity considerations in the next section.
12 APOSTOLOS DAMIALIS
Lemma 3.6. For any time t ∈ (0, T ) such that Rδi (t) ≤ 1/4 supt,δ ūδ(t) and for
sufficiently small δ, there holds
≤ Ṙδi ≤ −
and √
ti − t ≤ Rδi ≤ 2
ti − t.
Proof. For
δαRδi
1 + δαRδi
|x− xi|
it can be verified that the function uδ−g satisfies the assumptions of the maximum
principle in Lemma 3.1 for the constants ci = 1/4πδ
2α(Rδi )
2, thus yielding uδ ≥ g
in Ω \ ∪Bδi . But since uδ = g on the boundary ∂Bδi , monotonicity implies that
∇uδ · n ≥ ∇g · n on ∂Bδi and taking the average integrals over ∂Bδi we have
Ṙδi ≥ −
Rδi (1 + δ
αRδi )
≥ − 2
Moreover, Lemma 3.4 gives
Ṙδi ≤ ūδ −
+ Cδγ(1 + ūδ supRδi )
1 + (1 + δγ supRδi )(1 + 2 supR
Using now the assumption that Rδi ≤ 1/4 supt,δ ūδ and since from Lemma 3.5 it
follows that for sufficiently small δ the O(δγ) term is uniformly bounded by 1/4Rδi ,
we get
Ṙδi ≤
≤ − 1
Let now
y1 :=
ti − t, y2 := 2
ti − t
be sub- and supersolutions that respectively solve
ẏ1 = −
, ẏ2 = −
By comparison, we get the lemma’s second assertion, i.e., y1 ≤ Rδi ≤ y2. �
4. Homogenization
In order to pass to the homogenization limit of infinitely-many particles, we need
first describe the particle-radius density in the limit. To that end, define at any
time t ∈ (0, T ) the empirical measure νδt as
〈φ, νδt 〉 =
t, Rδi (t)
dνδt :=
t, Rδi (t)
for φ ∈ Cc,
i.e., for functions φ(t, R) continuous and compactly supported in the radius variable.
Using now the estimates from the previous section, we can prove the following
Lemma 4.1. For a subsequence δ → 0 and for a function ū ∈ W 1, p(0, T ), for
p < 2, holds
ūδ → ū in L2(0, T ),
uδ → ū in L2
0, T ;H1(Ω)
THE LSW EQUATION FOR REACTION-CONTROLLED KINETICS 13
Furthermore, the measures νδt converge to a family νt of probability measures such
φdνδt →
φa(t) dνt uniformly in t,
where a(t) denotes the percentage of active particles in the limit.
Proof. As a consequence of Lemma 3.6, we have
sup ‖Ṙδi ‖Lp(0,T ) ≤ C(p) for p < 2,
thus, conservation of volume and boundedness of the radii give for p < 2,
Lp(0,T )
≤ C sup ‖Ṙδi ‖Lp(0,T ) ≤ C.
Therefore, ūδ ∈ W 1, p(0, T ), for p < 2, and the compactness following from the
Rellich–Kondrachov theorem gives that ūδ converges to a limit ū in L2.
Taking into consideration that uδ is extended to the whole of Ω and using the
lemmas in the previous section, ζδ converges to ū in L2(Ω) and uδ − ζδ converges
uniformly to 0 as δ → 0, therefore, uδ converges to ū in L2(Ω). By the energy
equality in Lemma 2.1 we have further control over ‖∇uδ‖L2 and thus, we have
strong convergence in L2(0, T ;H1(Ω)).
For the measures νδt holds
‖νδt ‖ := sup‖φ‖Cc≤1|〈φ, ν
t 〉| ≤ 1
in the norm of (Cc)
∗, so for a subsequence δ → 0 there holds that νδt converges
weakly-* to νt. Furthermore, for positive functions φ the limit measure νt is non-
negative and from this it follows that νt becomes zero if there are no particles left
in the system.
Choosing now a function ψ(t) that depends only on time, we calculate
ψ(t) dνδt =
ψ(t).
The ratio N/Ni is the percentage of active particles at time t. This ratio is bounded
by 1 and decreasing, therefore it is uniformly bounded in the space BV (0, T ) and
by the compact embedding of BV (0, T )∩L∞(0, T ) in L2(0, T ), it converges in L2,
for a subsequence δ → 0, to a limit a ∈ BV (0, T ).
If we project now the measure νt to the interval [0, T ], we get that the projection
satisfies
proj[0, T ] νt = a(t) dt
and according to [6, Ch. 1, Thm. 10], the decomposition and convergence to νt
follow from the slicing of measures. �
We conclude with the following theorem which states that the limit measure νt
satisfies the Lifshitz–Slyozov–Wagner equation in a weak sense. Note that in the
theorem’s statement, the initial condition is defined as
t, Rδi (t)
dνδ0 :=
0, Rδi (0)
Theorem 4.2. The measure νt satisfies the Lifshitz–Slyozov–Wagner equation in
the sense that
φ(t, R) +
ū− 1
φ(t, R)
a(t) dνt +
φ(0, R) dν0 = 0, (4.1)
14 APOSTOLOS DAMIALIS
for all smooth and compactly supported functions φ ∈ C∞c ([0, T )× R+), where the
mean field ū is given by
R dνt
R2 dνt.
Proof. We begin by computing the mean-field limit ū(t). For a continuous function
φ(t) there holds, by the definition of ūδ,
1 + δαR
dνδt =
∑ R2i
1 + δαRi
1 + δαRi
1 + δαR
dνδt .
Taking the limit δ → 0 on both sides, Lemma 4.1 gives
R dνt
R2 dνt.
Consider now a smooth and compactly supported function φ as in the theorem’s
statement. Then, the fundamental theorem of calculus and Lemma 3.4 give
t, Rδi (t)
dνδt +
0, Rδi (0)
t, Rδi (t)
+ Ṙδi (t)
t, Rδi (t)
dνδt +
0, Rδi (0)
t, Rδi (t)
ūδ − 1
t, Rδi (t)
dνδt +O(δ
0, Rδi (0)
dνδ0 .
The result follows by taking the limit for a subsequence δ → 0 and using the strong
convergence of ūδ. �
As a concluding remark, we note that the well-posedness (existence, uniqueness,
and continuous dependence on initial data) of the weak formulation (4.1) can be
treated by the methods developed by Niethammer and Pego in [14] and [15].
Acknowledgments
Thanks are due to Barbara Niethammer for her substantial help and to Nick
Alikakos and Bob Pego for helpful discussions. Thanks are also due to the anony-
mous referee for a careful reading of the manuscript.
References
[1] N. D. Alikakos and G. Fusco. The equations of Ostwald ripening for dilute systems. J. Stat.
Phys. 95 No. 5/6 (1999), pp. 851–866.
[2] N. D. Alikakos and G. Fusco. Ostwald ripening for dilute systems under quasistationary
dynamics. Comm. Math. Phys. 238 No. 3 (2003), pp. 429–479.
[3] N. C. Bartelt, W. Theis, and R. M. Tromp. Ostwald ripening of two-dimensional islands on
Si(001). Phys. Rev. B 54 (1996), pp. 11741–11751.
[4] D. Cioranescu and F. Murat. A strange term coming from nowhere. In Topics in the math-
ematical modelling of composite materials, A. Cherkaev, R. Kohn eds. Birkhäuser, Boston,
MA, 1997, pp. 45–94.
THE LSW EQUATION FOR REACTION-CONTROLLED KINETICS 15
[5] S. Dai and R. L. Pego. Universal bounds on coarsening rates for mean-field models of phase
transitions. SIAM J. Math. Anal. 37 No. 2 (2005), pp. 347–371.
[6] L. C. Evans. Weak convergence methods for nonlinear partial differential equations. American
Mathematical Society, Providence, RI, 1990.
[7] D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order.
Springer-Verlag, Berlin, second edition, 1983.
[8] M. E. Gurtin. Thermomechanics of evolving phase boundaries in the plane. The Clarendon
Press, Oxford, 1993.
[9] I. M. Lifshitz and V. V. Slyozov. The kinetics of precipitation from supersaturated solid
solutions. J. Phys. Chem. Solids 19 (1961), pp. 35–50.
[10] W. W. Mullins and R. F. Sekerka. Morphological stability of a particle growing by diffusion
or heat flow. J. Appl. Phys. 34 No. 2 (1963), pp. 323–329.
[11] B. Niethammer. Derivation of the LSW-theory for Ostwald ripening by homogenization meth-
ods. Arch. Rat. Mech. Anal. 147 (1999), pp. 119–178.
[12] B. Niethammer. The LSW model for Ostwald ripening with kinetic undercooling. Proc. R.
Soc. Edinburgh 130A No. 6 (2000), pp. 1337–1361.
[13] B. Niethammer and F. Otto. Ostwald ripening: The screening length revisited. Calc. Var.
13 No. 1 (2001), pp. 33–68.
[14] B. Niethammer and R. L. Pego. On the initial-value problem in the Lifshitz–Slyozov–Wagner
theory of Ostwald ripening. SIAM J. Math. Anal. 31 No. 3 (2000), pp. 467–485.
[15] B. Niethammer and R. L. Pego. Well-posedness for measure transport in a family of nonlocal
domain coarsening models. Indiana Univ. Math. J. 54 No. 2 (2005), pp. 499–530.
[16] B. Niethammer and J. J. L. Velázquez. Homogenization in coarsening systems I: Deterministic
case. Math. Mod. Meth. Appl. Sci. 14 No. 8 (2004), pp. 1211–1233.
[17] B. Niethammer and J. J. L. Velázquez. Homogenization in coarsening systems II: Stochastic
case. Math. Mod. Meth. Appl. Sci. 14 No. 9 (2004), pp. 1–24.
[18] L. Ratke and P. W. Voorhees. Growth and coarsening: Ostwald ripening in material process-
ing. Springer-Verlag, Berlin, 2002.
[19] V. V. Slezov and V. V. Sagalovich. Diffusive decomposition of solid solutions. Sov. Phys.
Usp. 30 No. 1 (1987), pp. 23–45.
[20] J. J. L. Velázquez. On the effect of stochastic fluctuations in the dynamics of the Lifshitz–
Slyozov–Wagner model. J. Stat. Phys. 99 No. 1/2 (2000), pp. 231–252.
[21] P. W. Voorhees. The theory of Ostwald ripening. J. Stat. Phys. 38 No. 1/2 (1985), pp. 231–
[22] C. Wagner. Theorie der Alterung von Niederschlägen durch Umlösen. Z. Elektrochem. 65
No. 7/8 (1961), pp. 581–591.
Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099
Berlin, Germany
E-mail address: damialis@mathematik.hu-berlin.de
Current address: Department of Mathematics, University of Athens, Panepistemiopolis, 15784
Athens, Greece
mailto:damialis@mathematik.hu-berlin.de
	1. Introduction
	2. Formulation, scaling, and preliminary estimates
	3. Approximation and growth-rate estimates
	4. Homogenization
	Acknowledgments
	References

  We rigorously derive a weak form of the Lifshitz-Slyozov-Wagner equation as
the homogenization limit of a Stefan-type problem describing
reaction-controlled coarsening of a large number of small spherical particles.
Moreover, we deduce that the effective mean-field description holds true in the
particular limit of vanishing surface-area density of particles.

Introduction
The late-stage behavior of a material undergoing a first-order phase transition
(due to changes in temperature and/or pressure for example) is characterized by
thermodynamic instability resolved through phase separation and consequent coars-
ening of the emerging phase. In the case of the new phase occupying much smaller
volume fraction, and thus appearing as well-separated particles, this coarsening pro-
cess (known as Ostwald ripening) is driven by the minimization of surface energy at
the interface via diffusional mass exchange between particles while the total mass
or volume of each phase is conserved. The result of this kind of mass diffusion from
regions of high to regions of low interfacial curvature is the growth of large parti-
cles and the shrinkage and final extinction of smaller ones. For a review of some
aspects of Ostwald ripening, mainly from the physical and modeling viewpoint, see
the survey by Voorhees [21] or the book by Ratke and Voorhees [18].
In this coarsening scenario the mass-diffusion process can be controlled by two
different mechanisms: either by the diffusion of atoms away from the particles and
into the bulk, or by the reaction-rate of attachment of atoms at the phase interface.
In the former case (diffusion control), the random exchange of atoms between the
particles and the bulk is sufficiently rapid and the surrounding of each particle is in
thermal equilibrium with the atoms in it; in the latter (interface-reaction control),
detachment and attachment are slow compared to diffusion and the surrounding
bulk can be out of equilibrium with the particle interface. We refer to the physics
literature for more details, for example, Slezov and Sagalovich [19], Bartelt, Theis,
and Tromp [3]; for a related mathematical treatment see Dai and Pego [5].
The classical theory for Ostwald ripening was developed by Lifshitz and Slyozov
[9] and Wagner [22] in the case of supersaturated solid solutions in three dimensions.
The Lifshitz–Slyozov–Wagner theory statistically characterizes the evolution by the
particle-radius density n(t, R), where n(t, R) dR is defined to be the number of
particles with radii between R and dR at time t per unit volume. In the late
stages of the phase transition nucleation and coalesence of particles can be neglected
since new nuclei dissolve immediately and since particles cannot merge because of
the large distances between them. Thus, the particle-radius density satisfies the
This work was supported by the DFG through the Graduiertenkolleg RTG-1128 “Analysis,
Numerics, and Optimization of Multiphase Problems” at the Humboldt-Universität zu Berlin.
http://arxiv.org/abs/0704.0565v4
2 APOSTOLOS DAMIALIS
continuity equation (see [18, §5.1])
n(t, R) +
v(t, R)n(t, R)
where v(t, R) denotes the growth rate of particles of radius R at time t. Using a
mean-field ansatz (cf. Section 3), Lifshitz, Slyozov, and Wagner formally calculate
n(t, R) +
(Rū− 1)n(t, R)
ū(t) =
n(t, R) dR
Rn(t, R) dR,
in the diffusion-controlled case, and
n(t, R) +
n(t, R)
ū(t) =
Rn(t, R) dR
R2n(t, R) dR,
in the reaction-controlled one, both results valid in the limit of vanishing mass or
volume fraction of particles.
In [11] and [12] Niethammer rigorously derived the effective equations in the dif-
fusion-controlled case, starting from a quasi-static one-phase Stefan problem with
surface tension and kinetic undercooling,
−∆u = 0 in Ω \G,
V = ∇u · n on ∂G,
u = H + βV on ∂G,
(1.1)
and restricting it to spherical particles. The same was also done in [11] for the full
time-dependent parabolic problem but without the kinetic-drag term βV . Here, u
is a chemical potential, n is the outer normal to the particle phase G, V is the nor-
mal velocity of the phase interface ∂G, and H is its mean curvature. The domain
Ω ⊂ R3 is considered bounded and β is a parameter that comes from the nondi-
mensionalization and scales like diffusivity over mobility. The second boundary
condition is the Gibbs–Thomson law, coupling the curvature of the interface with
the chemical potential, modified by accounting for kinetic drag. Note that while
under diffusion control the parameter β is small and the kinetic drag can even
be neglected (thus yielding the well-known Mullins–Sekerka model [10]), in the
reaction-controlled case the values of β are large and, therefore, the kinetic-drag
term is necessary. For a derivation of such sharp-interface free-boundary problems
from continuum mechanics and thermodynamics see the book of Gurtin [8].
The goal in the following is to use the techniques developed in [11] and [12] to
derive the effective equations in the reaction-controlled case. This involves passing
over to a different time scale incorporating the parameter β tending to infinity
(see Section 2) and, as a result, some extra manipulations in the proofs. Except
for the scaling, in Section 2 we also give short proofs of some useful preliminaries
and discuss the validity of the mean-field description while in Section 3 we prove
pointwise estimates for approximate solutions and for the growth rates of particles.
Finally, using these estimates, in Section 4 we pass to the homogenization limit of
infinitely-many particles and obtain a weak form of the Lifshitz–Slyozov–Wagner
equation.
THE LSW EQUATION FOR REACTION-CONTROLLED KINETICS 3
In comparison with the results in the diffusion-controlled case, we make precise
that the crucial quantity that has to vanish in order to neglect direct interactions
between particles and justify the expected mean-field law is the surface-area density
of the particles in contrast to their capacity in the other case (see [11] and [12]).
This difference is of interest since the asymptotic limits of vanishing surface area
and capacity have different physical interpretations and further refine the näıve
general limit of vanishing mass or volume. For the reaction-controlled case though,
the result is in some sense to be expected since the limit of vanishing surface-
area density corresponds to the physics of the interface-reaction-controlled scenario,
where there is an obvious dependence on the area of the interface.
2. Formulation, scaling, and preliminary estimates
We start with problem (1.1) where the quasi-static approximation to the para-
bolic diffusion equation is justified by the small interfacial velocities present during
late-stage coarsening. (See the discussion in Mullins and Sekerka [10].)
We further suppose that the solid phase consists of spherical particles with cen-
ters fixed in space, a simplification that can be justified by the work of Alikakos
and Fusco [1], [2], and Velázquez [20]. Denoting these particles as Bi, where each
Bi is the closed ball B(xi, Ri(t)), the particle phase is then the union ∪Bi and its
isotropic evolution can be modeled by averaging the flux in the Stefan condition,
i.e.,
V = Ṙi(t) := −
∇u · n ,
where the average integral is defined as
for a function f on some domain D, and where the overdot denotes a derivative
with respect to time; the Gibbs–Thomson law becomes then
+ βṘi,
since in the case of spheres the mean curvature is the inverse radius.
To have many small particles in a bounded domain, for a system with size of order
O(1), say the unit cube [0, 1]3, let δ be the typical particle distance with 0 < δ ≪ 1.
For the distribution of particle centers in space, we assume, for simplicity, that they
are situated on a three-dimensional lattice of spacing δ. Then, the initial number
density of particles Ni(δ) will be bounded by 1/δ
3, and for the particles to be small
let the typical particle size be δα for α > 1. For times t ∈ [0, T ] we choose a δ small
enough so that adjacent particles of size δα will not collide during the evolution up
to a maximal time T .
Concerning the assumption on the spatial distribution of particles, a more general
assumption like infi6=j |xi−xj| > cδ, for a constant c > 0, would still be enough for
our purposes in this work. These considerations will also be used in the proof of
Lemma 3.2 where we approximate a certain sum over all particles by an integral. For
an approach using more sophisticated deterministic and stochastic assumptions on
the distribution of particles with respect to homogenization we refer to Niethammer
and Velázquez [16], [17], where also further refinements of the theory are made.
To have particle sizes of order O(1) as well, we rescale
Rδi :=
4 APOSTOLOS DAMIALIS
and motivated by the scaling invariance of problem (1.1) (cf. [5]),
uδ := δαu, tδ :=
Notice that this rescaling is another way of addressing the reaction-controlled
regime. Instead of rescaling time by β and then letting β tend to infinity, we
keep β fixed and positive, and specially rescale as above letting δ tend to zero.
Since now β plays no significant role, we will set it to unity in what follows. In
addition, one easily sees that the transformations Rδi , u
δ, and tδ preserve the form
of the equations. From hereon we also drop the superscript δ from the notation for
time and to denote the dependence on the new scale we write
Bδi := B(xi, δ
αRδi ).
Finally, note that under diffusion control the relevant scale for time would be δ3α
instead of δ2α. This difference is key to all that follows, leading to different consid-
erations on the validity of the mean-field model. (Cf. the remarks following Lemma
2.1.)
As initial data, for every particle-center xi we associate a corresponding bounded
initial radius Rδi (0) with the assumption that
supi∈NiR
i (0) ≤ R0,
uniformly for some constant R0. To consider a closed system, we impose a no-flux
Neumann boundary condition on the outer boundary of Ω, i.e.,
∇uδ · n = 0 on ∂Ω.
In case the ith particle vanishes at time ti := sup{t | Rδi (t) > 0}, for times later
than ti we define R
i to be zero, reduce the number N(t) := {j | Rδj(t) > 0} of
active particles by one, and neglect the boundary ∂Bδi in the boundary conditions.
In the following, all sums, unions, and suprema will run over the set N(t), with
N(0) ≡ Ni, and any further reference to the particle-number density will mean the
active particle-number density N unless otherwise noted.
Summarizing, the restricted and rescaled problem for the particle radii can be
considered as a nonlocal, N -dimensional system of ordinary differential equations
Ṙδi (t) =
4πδ2αRδi (t)
∇uδ · n on ∂Bδi (t), (2.1)
for times t ∈ (0, ti), ti < T , and with bounded initial data Rδi (0) for every i, while
the chemical potential is determined by
−∆uδ(t, x) = 0 in Ω \ ∪Bδi (t), (2.2)
uδ(t, x) =
Rδi (t)
+ Ṙδi (t) on ∂B
i (t), (2.3)
and the Neumann condition on the outer boundary.
Global existence and uniqueness of continuous, piecewise-smooth solutions for
a similar restricted Stefan problem was proved in [12] by an application of the
Picard–Lindelöf theorem, the only difference being the different time scale. These
solutions are not globally smooth due to the singularities arising from the extinction
of particles; however, they are smooth in the intervals between the extinction times
ti. In the following, when we mention solutions of the problem we will mean such
continuous, piecewise-smooth solutions that exist up to any given time T .
THE LSW EQUATION FOR REACTION-CONTROLLED KINETICS 5
It is easy to see that equations (2.1), (2.2), (2.3), along with the outer boundary
condition conserve the volume and decrease the interfacial area of the particle phase.
Indeed, differentiating the total volume of particles with respect to time gives
Rδi (t)
3 = 3
Rδi (t)
2Ṙδi (t) = 3
Rδi (t)
4πδ2αRδi (t)
∇uδ · n
where the last sum vanishes due to the divergence theorem, equation (2.2), and the
no-flux condition on ∂Ω. The decrease of total surface area follows from the next
a priori estimate.
Lemma 2.1. For any time t ∈ (0, T ), the solutions of the problem satisfy the
following energy equality.
(Rδi )
2|Ṙδi |2 +
Rδi (t)
4πδ2α
Ω\∪Bδ
|∇uδ|2 = 1
Rδi (0)
Proof. Multiplying −∆uδ = 0 with uδ, integrating over Ω \ ∪Bδi , and integrating
by parts gives
Ω\∪Bδ
|∇uδ|2 +
(∇uδ · n)uδ −
(∇uδ · n)uδ = 0,
where the last term vanishes due to the Neumann condition on the outer boundary.
Thus, using equations (2.3) and (2.1) we get
Ω\∪Bδ
|∇uδ|2 =
+ Ṙδi
∇uδ · n
+ Ṙδi
4πδ2α(Rδi )
2Ṙδi
= 4πδ2α
Rδi Ṙ
i + (R
2|Ṙδi |2
and after rearranging,
(Rδi )
2|Ṙδi |2 +
Rδi Ṙ
4πδ2α
Ω\∪Bδ
|∇uδ|2 = 0. (2.4)
The result follows from an integration over time. �
After normalization with respect to the initial particle-number density Ni, this
energy equality can yield useful information on the validity of the mean-field ap-
proach. In fact, we have
(Rδi )
2|Ṙδi |2+
Rδi (t)
4πNiδ2α
Ω\∪Bδ
|∇uδ|2 = 1
Rδi (0)
where the right-hand side is uniformly bounded by the assumption on the initial
radii. For the left-hand side to stay bounded as well, if the quantity Niδ
2α tends to
zero, the same must hold for |∇uδ| and it is exactly this limit of vanishing surface-
area density of particles that results in a mean field that is constant in space since,
in particular,
∇uδ → 0 in L2
0, T ;H1(Ω)
Here and in the following, to obtain global estimates that are uniform in δ we extend
uδ to the interior of particles, and thus to the whole of Ω, by its boundary values.
It is important to note that in our scaling setup, for the surface area to vanish as δ
6 APOSTOLOS DAMIALIS
tends to zero, the exponent α must be strictly larger than 3/2 since Ni is O(1/δ
These facts will be made precise in Corollary 3.3 where we give an estimate of the
mean-field effect. Note also that we do not address here the critical case α = 3/2
that corresponds to finite surface area. For that one would have to use the different
methods developed by Niethammer and Otto in [13].
Finally, note that for similar considerations under diffusion control, the corre-
sponding quantity would be the capacity Niδ
α due to the different time scale. In
three dimensions, this capacity effect fits to general homogenization results as in
the work of Cioranescu and Murat [4]; to our knowledge though, the surface-area
effect has not been explicitly discussed in the relevant literature.
3. Approximation and growth-rate estimates
As in the mean-field ansatz of Lifshitz, Slyozov, and Wagner, we suppose that
the system is dilute enough so that particles behave as if they were isolated and we
base our approximation on the solution of a single-particle problem.
Consider problem (2.1), (2.2), (2.3) for a single spherical particle centered at
the origin and with initially unscaled radius r that we rescale as rδ := r/δα, along
with the corresponding reaction-controlled rescalings for a chemical potential uδr
and time, as in Section 2. For this rescaled particle Bδr we consider the following
problem in the whole space:
ṙδ(t) =
4πδ2αrδ(t)2
∇uδr · n on ∂Bδr ,
where the chemical potential uδr(t, x) satisfies
−∆uδr(t, x) = 0 in R3 \Bδr ,
uδr(t, x) =
rδ(t)
+ ṙδ(t) for x ∈ ∂Bδr ,
and the mean-field assumption is posed as a condition at infinity, i.e.,
|x|→∞
uδr(t, x) = ū
r(t).
This problem can be explicitly solved to give
uδr(t, x) = ū
r(t) +
δαrδ(t)
1 + δαrδ(t)
1− ūδr(t)rδ(t)
ṙδ(t) =
1 + δαrδ(t)
ūδr(t)−
rδ(t)
Note that in the formal limit of δ tending to zero, the expected effective equations
take the general form
u(t, x) = ū(t) and ṙ = ū− 1
as in the reaction-controlled Lifshitz–Slyozov–Wagner theory.
Going now back to the many-particle problem, a calculation using the single-
particle growth rate above along with the requirement that the volume is conserved
gives the following expression for the mean field
ūδ =
1 + δαRδi
(Rδi )
1 + δαRδi
. (3.1)
THE LSW EQUATION FOR REACTION-CONTROLLED KINETICS 7
The effect of this mean field plus a sum of single-particle solutions will be the
monopole approximation to the solution uδ supposing that there are no direct
interactions between particles. To this end, let us define the approximate solution
ζδ(t, x) := ūδ(t) +
δαRδi (t)
1 + δαRδi (t)
1− ūδ(t)Rδi (t)
|x− xi|
(3.2)
for x ∈ Ω \ ∪Bδi (t).
Below is a maximum principle tailored to our setting that will be used to compare
the approximation and the solution in the lemma next. Its proof can be found in
[12].
Lemma 3.1. Let Ω be a Lipschitz domain and let ∪Bi ⊂ Ω be a finite collection of
disjoint closed balls. Then, a function v which is constant on each of the boundaries
∂Bi and satisfies
−∆v = 0 in Ω \ ∪Bi,
v − ci
∇v · n ≥ 0 on ∂Bi,
∇v · n ≥ 0 on ∂Ω,
where ci ≥ 0 for all i, also satisfies
v ≥ 0 in Ω \ ∪Bi.
Lemma 3.2. For any time t ∈ (0, T ) and small positive ε, the chemical potential
and its approximation satisfy
‖uδ − ζδ‖L∞(Ω\∪Bδ
)(t) ≤ Cδ2α−3−ε supRδi (t)
1 + ūδ(t) supRδi (t)
Proof. Since the difference uδ − ζδ is already harmonic in Ω \ ∪Bδi as ζδ is a su-
perposition of fundamental solutions, we would like to estimate to what extent it
satisfies the maximum principle’s boundary conditions.
For the condition on the particle boundaries, we use equations (2.1), (2.3), and
the definition of ζδ to calculate for x on the boundary ∂Bδi of the ith particle,
ζδ(t, x)−−
∇ζδ · n
uδ(t, x)−−
∇uδ · n
ζδ − 1
∇ζδ · n
ūδ − 1
δαRδj
1 + δαRδj
(1− ūδRδj)
|x− xj |
|x− xj |
and since by the divergence theorem there holds for j 6= i,
|x− xj |
· n = 0,
while for j = i,
|x− xi|
· n = − 1
δα(Rδi )
8 APOSTOLOS DAMIALIS
we continue the calculation to get
ūδ − 1
1− ūδRδi
Rδi (1 + δ
αRδi )
δαRδj
1 + δαRδj
(1− ūδRδj)
|x− xj |
j 6=i
δαRδj
1 + δαRδj
(1− ūδRδj)
|x− xj |
≤ δ2α−3 supRδj (1 + ūδ supRδj )
j 6=i
|x− xj |
≤ Cδ2α−3 supRδj(1 + ūδ supRδj). (3.3)
In the last step, keeping in mind the assumptions on the spatial distribution of
particle centers, the sum is bounded for j 6= i since it is considered as a Riemann-
sum approximation to the integral
|x− y|
which in turn is bounded using radial symmetry around the singularity and where
the factor δ3 in the sum compensates for the scaling in space.
To further fulfil the maximum principle’s outer boundary condition on ∂Ω, we
consider the comparison function ζδ+zδ, where the auxiliary function zδ solves the
problem
−∆zδ =
∇ζδ · n in Ω,
∇zδ · n = −∇ζδ · n on ∂Ω,
zδ = 0,
(3.4)
such that the comparison function ζδ + zδ has zero normal derivative on ∂Ω. To
work with the maximum principle, zδ also needs to be harmonic in Ω and for that
we need that the integral
∇ζδ · n vanishes. But,
∇ζδ · n = δα
δαRδi
1 + δαRδi
(1−Rδi ūδ)
|x− xi|
· n ,
where the last integral equals −4π, independent of i. Thus, zδ is harmonic if and
only if
ūδ =
1 + δαRδi
(Rδi )
1 + δαRδi
which is exactly the mean field (3.1) as dictated by the single-particle ansatz in
the beginning of the section. Moreover, since now zδ is harmonic, the divergence
theorem further gives
∇zδ · n = 0.
A construction as in Lemma 3 of [11] and elliptic regularity theory (see Gilbarg
and Trudinger [7]) give the estimate
‖zδ‖L∞(Ω) ≤ Cεδ2α−3−ε supRδi (1 + ūδ supRδi ),
where ε is a small positive number.
THE LSW EQUATION FOR REACTION-CONTROLLED KINETICS 9
Let us now apply the maximum principle to the function
f+ := u
δ − ζδ − zδ + Cδ2α−3−ε supRδi (1 + ūδ supRδi ).
For a large enough constant C, say 2Cε, the following hold for f+: it is harmonic,
there holds ∇f+ · n = 0 on ∂Ω by the construction of zδ, and for the constants
ci = 1/4πδ
2α(Rδi )
2, estimate (3.3) gives
uδ − ζδ − zδ + Cδ2α−3−ε supRδi (1 + ūδ supRδi )− ci
∇(uδ − ζδ − zδ) · n ≥ 0.
Thus, f+ satisfies the maximum principle’s conditions and therefore, f+ ≥ 0 in
Ω \ ∪Bδi , i.e.,
uδ − ζδ − zδ ≥ −Cδ2α−3−ε supRδi (1 + ūδ supRδi ).
Using the maximum principle with −v instead of v, the function
f− := u
δ − ζδ − zδ − Cδ2α−3−ε supRδi (1 + ūδ supRδi ),
again satisfies the corresponding conditions and, as above, yields f− ≤ 0 in Ω\∪Bδi ,
i.e.,
uδ − ζδ − zδ ≤ Cδ2α−3−ε supRδi (1 + ūδ supRδi ).
Combining the last two inequalities, we get
‖uδ − ζδ − zδ‖L∞(Ω\∪Bδ
) ≤ Cδ2α−3−ε supRδi (1 + ūδ supRδi )
and the lemma follows by the triangle inequality using the regularity of zδ. �
In the previous lemma it is clear that our approach excludes the critical case
α = 3/2. In the following we introduce, for technical reasons, a new exponent
γ > 0 with the property
δγ := max {δα, δ2α−3, δ2α−3−ε}
for each α greater than 3/2 + ε.
As a corollary to the previous lemma we can now estimate the effect of the mean
field.
Corollary 3.3. For any time t ∈ (0, T ) and γ > 0, the chemical potential and the
mean field satisfy
‖uδ − ūδ‖L∞(Ω\∪Bδ
)(t) ≤ Cδγ
1 + 2 supRδi (t)
1 + ūδ(t) supRδi (t)
Proof. By the triangle inequality and Lemma 3.2 there holds
‖uδ − ūδ‖L∞(Ω\∪Bδ
) ≤ ‖ζδ − ūδ‖L∞(Ω\∪Bδ
) + Cδ
2α−3−ε supRδj(1 + ū
δ supRδj).
To estimate ‖ζδ− ūδ‖L∞(Ω\∪Bδ
), by the definition of ζ
δ there holds for x ∈ Ω\∪Bδj ,
ζδ(t, x)− ūδ(t)
δαRδj
1 + δαRδj
(1− ūδRδj)
|x− xj |
δαRδi
1 + δαRδi
(1 + ūδRδi )
|x− xi|
j 6=i
δαRδj
1 + δαRδj
(1− ūδRδj)
|x− xj |
and since |x− xi| ≥ δαRδi in Ω \ ∪Bδj , arguing as in estimate (3.3) gives
α(1 + ūδRδi )
1 + δαRδi
+ Cδ2α−3 supRδj(1 + ū
δ supRδj)
≤ C(δα + δ2α−3 supRδj )(1 + ūδ supRδj),
10 APOSTOLOS DAMIALIS
thus,
‖ζδ − ūδ‖L∞(Ω\∪Bδ
) ≤ C(δα + δ2α−3 supRδj )(1 + ūδ supRδj),
and finally,
‖uδ − ūδ‖L∞(Ω\∪Bδ
) ≤ C(δα + δ2α−3 supRδj + δ2α−3−ε supRδj )(1 + ūδ supRδj).
Using the exponent γ, we get
‖uδ − ūδ‖L∞(Ω\∪Bδ
) ≤ Cδγ(1 + 2 supRδi )(1 + ūδ supRδi ). �
The following lemma gives an estimate for the growth rate of particles in accor-
dance with the reaction-controlled Lifshitz–Slyozov–Wagner theory.
Lemma 3.4. For any time t ∈ (0, T ) and γ > 0, for the growth rates of particles
holds
Ṙδi −
ūδ − 1
≤ Cδγ(1 + ūδ supRδi )
1 + (1 + δγ supRδi )(1 + 2 supR
Proof. Let wδi be the capacity potential of the ball B
i with respect to a larger ball
Bλδi := B(xi, λδ
αRδi ) for λ > 1, i.e., let wi solve
−∆wδi = 0 in Bλδi \Bδi ,
wδi = 0 on ∂B
wδi = 1 in B
(3.5)
An explicit calculation gives
wδi =
1− λδ
|x− xi|
(3.6)
and also
∇wδi · n =
∇wδi · n = 4π
1− λδ
αRδi . (3.7)
Using equations (2.1), (2.2), (2.3), and the Neumann boundary condition, along
with the above properties of wδi , and integrating by parts, gives
4πδ2α(Rδi )
2Ṙδi =
∇uδ · n
wδi∇uδ · n
∇wδi∇uδ
uδ∇wδi · n −
uδ∇wδi · n
+ Ṙδi
∇wδi · n −
uδ∇wδi · n
δαRδi
+ Ṙδi − ūδ
(uδ − ūδ)∇wδi · n ,
THE LSW EQUATION FOR REACTION-CONTROLLED KINETICS 11
where in the last equation we used (3.7) and added and subtracted ūδ. After
rearranging, we have
Ṙδi −
ūδ − 1
δαRδi Ṙ
4πλδαRδi
(uδ − ūδ)∇wδi · n
≤ λ− 1
δαRδi |Ṙδi |+
4πλδαRδi
(uδ − ūδ)∇wδi · n
≤ δαRδi |Ṙδi |+ ‖uδ − ūδ‖L∞(Ω\∪Bδ
), (3.8)
where in the last step we again used equation (3.7). But by using equation (2.3)
for uδ on ∂Bδi we have
Rδi |Ṙδi | ≤ 1 +Rδi |uδ| ≤ 1 +Rδi (‖uδ − ūδ‖L∞(Ω\∪Bδ
) + ū
δ). (3.9)
Substituting back in (3.8) and using Corollary 3.3 gives the final estimate. �
The next lemma ensures that the bounds in the approximation and the growth-
rate estimates are indeed uniform.
Lemma 3.5. For any time t ∈ (0, T ), the mean field and the radii of the particles
are uniformly bounded, i.e.,
ūδ(t) ≤ C and supRδi (t) ≤ C.
Proof. For the mean field (3.1) holds
ūδ =
1 + δαRδi
(Rδi )
1 + δαRδi
≤ supRδi (1 + δα supRδi )
(Rδi )
and since by Hölder’s inequality
(Rδi )
(Rδi )
δ3(Rδi )
conservation of the total volume of particles gives
ūδ ≤ C supRδi (1 + δα supRδi ).
or, using the exponent γ,
ūδ ≤ C supRδi (1 + δγ supRδi ). (3.10)
Consider now the set
t | supRδi (t) ≤
then, for times t ∈ A, plugging (3.10) in estimate (3.9) and using Corollary 3.3
gives
(Rδi )
2 ≤ C sup(Rδi )2 + C.
Integrating over the time interval (0, T ), Gronwall’s inequality implies that
supi supt∈A∩[0,T ] (R
2 ≤ C(T ),
therefore, [0, T ] ⊂ A, i.e., the radii are bounded up to time T as is the mean field
by estimate (3.10). �
Finally, the following lemma gives control over the growth rates of vanishing
particles and will prove useful for some regularity considerations in the next section.
12 APOSTOLOS DAMIALIS
Lemma 3.6. For any time t ∈ (0, T ) such that Rδi (t) ≤ 1/4 supt,δ ūδ(t) and for
sufficiently small δ, there holds
≤ Ṙδi ≤ −
and √
ti − t ≤ Rδi ≤ 2
ti − t.
Proof. For
δαRδi
1 + δαRδi
|x− xi|
it can be verified that the function uδ−g satisfies the assumptions of the maximum
principle in Lemma 3.1 for the constants ci = 1/4πδ
2α(Rδi )
2, thus yielding uδ ≥ g
in Ω \ ∪Bδi . But since uδ = g on the boundary ∂Bδi , monotonicity implies that
∇uδ · n ≥ ∇g · n on ∂Bδi and taking the average integrals over ∂Bδi we have
Ṙδi ≥ −
Rδi (1 + δ
αRδi )
≥ − 2
Moreover, Lemma 3.4 gives
Ṙδi ≤ ūδ −
+ Cδγ(1 + ūδ supRδi )
1 + (1 + δγ supRδi )(1 + 2 supR
Using now the assumption that Rδi ≤ 1/4 supt,δ ūδ and since from Lemma 3.5 it
follows that for sufficiently small δ the O(δγ) term is uniformly bounded by 1/4Rδi ,
we get
Ṙδi ≤
≤ − 1
Let now
y1 :=
ti − t, y2 := 2
ti − t
be sub- and supersolutions that respectively solve
ẏ1 = −
, ẏ2 = −
By comparison, we get the lemma’s second assertion, i.e., y1 ≤ Rδi ≤ y2. �
4. Homogenization
In order to pass to the homogenization limit of infinitely-many particles, we need
first describe the particle-radius density in the limit. To that end, define at any
time t ∈ (0, T ) the empirical measure νδt as
〈φ, νδt 〉 =
t, Rδi (t)
dνδt :=
t, Rδi (t)
for φ ∈ Cc,
i.e., for functions φ(t, R) continuous and compactly supported in the radius variable.
Using now the estimates from the previous section, we can prove the following
Lemma 4.1. For a subsequence δ → 0 and for a function ū ∈ W 1, p(0, T ), for
p < 2, holds
ūδ → ū in L2(0, T ),
uδ → ū in L2
0, T ;H1(Ω)
THE LSW EQUATION FOR REACTION-CONTROLLED KINETICS 13
Furthermore, the measures νδt converge to a family νt of probability measures such
φdνδt →
φa(t) dνt uniformly in t,
where a(t) denotes the percentage of active particles in the limit.
Proof. As a consequence of Lemma 3.6, we have
sup ‖Ṙδi ‖Lp(0,T ) ≤ C(p) for p < 2,
thus, conservation of volume and boundedness of the radii give for p < 2,
Lp(0,T )
≤ C sup ‖Ṙδi ‖Lp(0,T ) ≤ C.
Therefore, ūδ ∈ W 1, p(0, T ), for p < 2, and the compactness following from the
Rellich–Kondrachov theorem gives that ūδ converges to a limit ū in L2.
Taking into consideration that uδ is extended to the whole of Ω and using the
lemmas in the previous section, ζδ converges to ū in L2(Ω) and uδ − ζδ converges
uniformly to 0 as δ → 0, therefore, uδ converges to ū in L2(Ω). By the energy
equality in Lemma 2.1 we have further control over ‖∇uδ‖L2 and thus, we have
strong convergence in L2(0, T ;H1(Ω)).
For the measures νδt holds
‖νδt ‖ := sup‖φ‖Cc≤1|〈φ, ν
t 〉| ≤ 1
in the norm of (Cc)
∗, so for a subsequence δ → 0 there holds that νδt converges
weakly-* to νt. Furthermore, for positive functions φ the limit measure νt is non-
negative and from this it follows that νt becomes zero if there are no particles left
in the system.
Choosing now a function ψ(t) that depends only on time, we calculate
ψ(t) dνδt =
ψ(t).
The ratio N/Ni is the percentage of active particles at time t. This ratio is bounded
by 1 and decreasing, therefore it is uniformly bounded in the space BV (0, T ) and
by the compact embedding of BV (0, T )∩L∞(0, T ) in L2(0, T ), it converges in L2,
for a subsequence δ → 0, to a limit a ∈ BV (0, T ).
If we project now the measure νt to the interval [0, T ], we get that the projection
satisfies
proj[0, T ] νt = a(t) dt
and according to [6, Ch. 1, Thm. 10], the decomposition and convergence to νt
follow from the slicing of measures. �
We conclude with the following theorem which states that the limit measure νt
satisfies the Lifshitz–Slyozov–Wagner equation in a weak sense. Note that in the
theorem’s statement, the initial condition is defined as
t, Rδi (t)
dνδ0 :=
0, Rδi (0)
Theorem 4.2. The measure νt satisfies the Lifshitz–Slyozov–Wagner equation in
the sense that
φ(t, R) +
ū− 1
φ(t, R)
a(t) dνt +
φ(0, R) dν0 = 0, (4.1)
14 APOSTOLOS DAMIALIS
for all smooth and compactly supported functions φ ∈ C∞c ([0, T )× R+), where the
mean field ū is given by
R dνt
R2 dνt.
Proof. We begin by computing the mean-field limit ū(t). For a continuous function
φ(t) there holds, by the definition of ūδ,
1 + δαR
dνδt =
∑ R2i
1 + δαRi
1 + δαRi
1 + δαR
dνδt .
Taking the limit δ → 0 on both sides, Lemma 4.1 gives
R dνt
R2 dνt.
Consider now a smooth and compactly supported function φ as in the theorem’s
statement. Then, the fundamental theorem of calculus and Lemma 3.4 give
t, Rδi (t)
dνδt +
0, Rδi (0)
t, Rδi (t)
+ Ṙδi (t)
t, Rδi (t)
dνδt +
0, Rδi (0)
t, Rδi (t)
ūδ − 1
t, Rδi (t)
dνδt +O(δ
0, Rδi (0)
dνδ0 .
The result follows by taking the limit for a subsequence δ → 0 and using the strong
convergence of ūδ. �
As a concluding remark, we note that the well-posedness (existence, uniqueness,
and continuous dependence on initial data) of the weak formulation (4.1) can be
treated by the methods developed by Niethammer and Pego in [14] and [15].
Acknowledgments
Thanks are due to Barbara Niethammer for her substantial help and to Nick
Alikakos and Bob Pego for helpful discussions. Thanks are also due to the anony-
mous referee for a careful reading of the manuscript.
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Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099
Berlin, Germany
E-mail address: damialis@mathematik.hu-berlin.de
Current address: Department of Mathematics, University of Athens, Panepistemiopolis, 15784
Athens, Greece
mailto:damialis@mathematik.hu-berlin.de
	1. Introduction
	2. Formulation, scaling, and preliminary estimates
	3. Approximation and growth-rate estimates
	4. Homogenization
	Acknowledgments
	References
