Smoluchowski's coagulation equation: uniqueness, non-uniqueness and a hydrodynamic limit for the stochastic coalescent

32 Pages·0.28 MB·English

by  James R. Norris

SMOLUCHOWSKI’S COAGULATION EQUATION: UNIQUENESS,
NON-UNIQUENESS AND A HYDRODYNAMIC LIMIT FOR THE
STOCHASTIC COALESCENT
8
9
9 JAMES R. NORRIS
1

n
Abstract. Sufficient conditions are given for existence and uniqueness in Smolu-
a
J chowski’s coagulation equation, for a wide class of coagulation kernels and initial

9 mass distributions. An example of non-uniqueness is constructed. The stochastic
coalescentis shownto convergeweaklytothe solutionofSmoluchowski’sequation.

]
R
P
.
h
t
a 1. Introduction
m
[ Coagulationofparticles, inpairsandovertime,withinalargesystemofparticles, is

a phenomenon widely observed and widely postulated in scientific models. Examples
1
v ariseinthestudyofaerosols, ofphaseseparationinliquidmixtures, inpolymerization
5 and astronomy. Typically, it is argued that the rate at which pairs of particles
4
coagulate depends, for physical reasons, in a given way, on some positive parameter
1
1 associated to each particle, such as size or mass. In the models we shall consider, it
0
is further argued that the effects of spatial fluctuations in the size or mass density
8
9 are negligible—for example, by supposing that the particles perform independent
/ random motions on a time scale faster than the process of coagulation.
h
t The firstmathematical modelofthissortofprocess was proposedbySmoluchowski
a
m [vS16] in 1916, see also Chandrasekhar [Cha43]. Smoluchowski argued that particles
: of radius r would perform independent Brownian motions of variance proportional
v
to 1/r, so pairs of particles of radii r and r would meet at a rate proportional to
i 1 2
X
r (r +r )(1/r +1/r ).
a 1 2 1 2
Expressed in terms of masses, this leads to the coagulation kernel
K(x,y) = (x1/3 +y1/3)(x 1/3 +y 1/3)
− −
for particles of masses x and y. Then, making some implicit assumptions about er-
godic averages, Smoluchowski wrote down the following infinite system of differential
This research was supported financially by the European Union under contract FMRX CT96
0075A and by the Mathematical Sciences Research Institute. Research at MSRI is supported in
part by NSF grant DMS-9701755.
1
2 JAMES R. NORRIS
equations for the evolution of densities µ(x) of particles of mass x = 1,2,3,...
x 1
d 1 − ∞
µ (x) = K(y,x y)µ (y)µ (x y) µ (x) K(x,y)µ (y).
t t t t t
dt 2 − − −
y=1 y=1
X X
Here, the first sum on the right corresponds to coagulation of smaller particles to
produce one of mass x, whereas the second sum corresponds to removal of particles
of mass x as they in turn coagulate to produce larger particles.
More generally, in other models, such systems of equations are considered for many
different coagulation kernels K, see for example [Ald]. Also, analogous integro-
differential equations are considered which allow for a continuum of masses x. It
is known by now that, for a suitable initial mass distribution µ , Smoluchowski’s
0
original equations have a unique solution. Much progress has been made in deter-
mining when existence and uniqueness holds for more general coagulationkernels, see
[McL62], [McL64], [Whi80], [BC90], [Hei92] for discrete mass distributions. Never-
theless many fundamentalquestions remainopen, even forcertaincoagulationkernels
studied extensively in applied sciences.
In this paper we give some new positive results on the existence and uniqueness
problem for Smoluchowski-type equations. In particular:
we prove existence of solutions for continuous coagulation kernels K such that
•
K(x,y)/xy 0 as (x,y)
→ → ∞
extending a result of Jeon [Jeo] for the discrete case;
we prove local existence and uniqueness of solutions when K(x,y) ϕ(x)ϕ(y)
• ≤
for some continuous sublinear function ϕ : E (0, ), provided that the initial
→ ∞
mass distribution µ satisfies
0
ϕ(x)2µ (dx) < ;
0
∞
Z(0, )
∞
this allows us to treat the case where K(x,y) blows up as x 0 or y 0, also
• → →
to prove uniqueness in some cases when the mass distribution has no second, or
even first, moment;
we can do without any local regularity conditions on K;
•
we do not have to assume that the initial mass distribution is discrete, nor that
•
it has a density with respect to Lebesgue measure.
We also construct in 3 an example of a coagulation kernel K and an initial mass
§
distribution µ , such thatSmoluchowski’s equation hasatleast two distinct solutions,
0
both of which are conservative, in the sense that
xµ (dx) = xµ (dx) <
t 0
∞
Z(0, ) Z(0, )
∞ ∞
for all t.
SMOLUCHOWSKI’S COAGULATION EQUATION 3
Then in 4 we consider a stochastic system of coagulating particles, where particles
§
ofmasses xandy coagulateatarateproportionaltoK(x,y). Weshowthat, when we
can establish uniqueness in Smoluchowski’s equation, the particle system, suitably
normalized, converges weakly to the solution of the deterministic equation. Thus
we obtain a statistical derivation of Smoluchowski’s equation. This goes some way
towards resolving Problem 10 in Aldous’ survey article [Ald].
2. Existence and uniqueness in Smoluchowski’s coagulation equation
Let E = (0, ) and let K : E E [0, ) be a symmetric measurable function,
the coagulation∞kernel. Denote b×y M→= M ∞the space of signed Radon measures on
E
E, that is to say, those signed measures having finite total variation on each compact
subset of E. Denote by M+ the set of (non-negative) measures in M. If µ M+
∈
satisfies, for all compact sets B E
⊆
K(x,y)µ(dx)µ(dy) < ,
∞
ZB E
×
then we define L(µ) M by
∈
1
f,L(µ) = f(x+y) f(x) f(y) K(x,y)µ(dx)µ(dy)
h i 2 { − − }
ZE E
×
for all bounded measurable functions f of compact support.
We consider the following weak form of Smoluchowski’s coagulation equation
t
(2.1) µ = µ + L(µ )ds.
t 0 s
Z0
We admit as a local solution any map
t µ : [0,T) M+
t
7→ 7→
where T (0, ], such that:
∈ ∞
(i) we have
x1 µ (dx) < ;
x 1 0
≤ ∞
ZE
(ii) for all compact sets B E, the following map is measurable
⊆
t µ (B) : [0,T) [0, );
t
7→ → ∞
(iii) we have, for all t < T and all compact sets B E
⊆
t
K(x,y)µ (dx)µ (dy)ds < ;
s s
∞
Z0 ZB E
×
4 JAMES R. NORRIS
(iv) for all bounded measurable functions f of compact support and also for f(x) =
x1 , for all t < T
x 1
≤
t
(2.2) f,µ = f,µ + f,L(µ ) ds.
t 0 s
h i h i h i
Z0
In the case T = , we have a solution.
∞
The condition that, for f(x) = x1 , we have f,µ < and that (2.2) holds is
x 1 0
≤ h i ∞
a boundary condition, expressing that no mass enters at 0. We obtain an equivalent
condition on replacing f by any non-vanishing sublinear function E [0, ) of
→ ∞
bounded support, which is linear near 0. A function f : E [0, ) is sublinear if
→ ∞
f(λx) λf(x) for all x E,λ 1.
≤ ∈ ≥
Note that such a function f is always subadditive:
f(x+y) f(x)+f(y) for all x,y E.
≤ ∈
Hence f,L(µ) 0 for allµ M+. Notealso that, if ϕ : E [0, ) is anysublinear
h i ≤ ∈ → ∞
function and if
nxϕ(n 1), 0 < x n 1,
− −
≤
ϕ (x) = ϕ(x), n 1 < x n,
n −
(0, x > n, ≤
then ϕ (x) ϕ(x) for all x, and ϕ is sublinear of bounded support, linear near 0.
n n
↑
So, for t < T
t
ϕ ,µ ϕ ,µ = ϕ ,L(µ ) 0.
n t n 0 n s
h i−h i h i ≤
Z0
Hence, using monotone convergence on the left and Fatou’s lemma on the right
t
(2.3) ϕ,µ ϕ,µ ϕ,L(µ ) ds.
0 t s
h i ≥ h i− h i
Z0
In particular, ϕ,µ is non-increasing in t. In particular, the total mass density
t
h i
xµ (dx)
t
ZE
is non-increasing in t; if it is finite and constant, we say that (µ ) is conservative.
t t<T
Throughout this section we make the basic assumption that
(2.4) K(x,y) ϕ(x)ϕ(y) for all x,y E
≤ ∈
where ϕ : E (0, ) is a continuous sublinear function. We also assume that the
→ ∞
initial measure µ satisfies
0
(2.5) ϕ,µ < .
0
h i ∞
We call any local solution (µ ) such that
t t<T
t
ϕ2,µ ds < for all t < T
s
h i ∞
Z0
SMOLUCHOWSKI’S COAGULATION EQUATION 5
a strong local solution.
Here is a summary of what is known so far about existence, uniqueness and con-
servation of mass in Smoluchowski’s equation. The picture is more complete for
discrete mass distributions—that is when µ is supported on N. Then, provided µ
0 0
has a finite second moment, Ball and Carr [BC90] proved existence and mass con-
servation when K(x,y) x+y, Heilman [Hei92] added uniqueness under the same
≤
hypotheses. Jeon [Jeo] has recently proved global existence when K(x,y)/xy 0 as
→
(x,y) . McLeod [McL62] long ago proved local existence when K(x,y) xy.
→ ∞ ≤
For general mass distributions µ , less is known. Dubovskii and Stewart [DS96] have
0
shown existence and uniqueness provided µ has an exponential moment and a con-
0
tinuous density with respect to Lebesgue measure, and provided K is continuous
with K(x,y) 1+x+y. Recently, Clark and Katsouros [CK] proved existence and
≤
uniqueness for a particular choice of kernel which blows up when x or y is small.
Theorem 2.1. Assume conditions (2.4) and (2.5). If (µ ) and (ν ) are local
t t<T t t<T
solutions, starting from µ , and if (µ ) is strong, then µ = ν for all t < T.
0 t t<T t t
If ϕ(x) εx for all x, for some ε > 0, then any strong solution is conservative.
≥
Moreover, if ϕ2,µ < , then
0
h i ∞
(i) there exists a unique maximalstrong solution (µ ) , with ζ(µ ) ϕ2,µ 1,
t t<ζ(µ0) 0 ≥ h 0i−
(ii) if ϕ2 is sublinear or if K(x,y) ϕ(x)+ϕ(y) for all x,y E, then ζ(µ ) = .
0
≤ ∈ ∞
The proof will occupy the remainder of this section. The method is to find an
approximation to Smoluchowski’s equation by a system depending on K and ϕ only
through their values on a given compact set. The idea of the approximation may be
readily understood in terms of the stochastic coalescent, for which a directly anal-
ogous approximation is discussed in 4. Moreover, the finite particle interpretation
§
explains certain crucial non-negativity statements, which are given less transparent
analytical proofs below.
Weremarkthatthistheoremprovidesexamples whereuniqueness holds, evenwhen
the solution fails to be conservative, in the trivial sense that the total mass density
is infinite. We have not yet found an example of a strong solution which has finite
initial mass density and which fails to conserve mass. The example of 3 shows, on
§
the other hand, that uniqueness can fail while solutions remain conservative.
Let B E be compact. We will eventually pass to the limit B E. Denote by
M the s⊆pace of finite signed measures supported on B. Note that ϕ↑ is bounded on
B
B. We define LB : M R M R by the requirement
B B
× → ×
(f,a),LB(µ,λ) =
h i
1
f(x+y)1 +aϕ(x+y)1 f(x) f(y) K(x,y)µ(dx)µ(dy)
x+y B x+y B
2 { ∈ 6∈ − − }
ZE E
×
+λ aϕ(x) f(x) ϕ(x)µ(dx)
{ − }
ZE
6 JAMES R. NORRIS
forallboundedmeasurablefunctionsf onE andalla R. Hereweusedthenotation
∈
(f,a),(µ,λ) = f,µ +aλ.
h i h i
Consider the equation
t
(2.6) (µ ,λ ) = (µ ,λ )+ LB(µ ,λ )ds.
t t 0 0 s s
Z0
We admit as a local solution any continuous map
t (µ ,λ ) : [0,T] M R
t t B
7→ → ×
where T (0, ), which satisfies the equation for all t [0,T]. When [0,T] is
∈ ∞ ∈
replaced by [0, ) we get the notion of solution.
∞
Proposition 2.2. Suppose µ M with µ 0 and that λ [0, ). The equation
0 B 0 0
∈ ≥ ∈ ∞
(2.6) has a unique solution (µ ,λ ) starting from (µ ,λ ). Moreover µ 0 and
t t t 0 0 0 t
λ 0 for all t. ≥ ≥
t
≥
Proof. Our basic assumption (2.4) remains valid when ϕ is replaced by ϕ+1, so we
may assume without loss that ϕ 1. By a scaling argument we may assume, also
≥
without loss, that
ϕ,µ +λ 1
0 0
h i ≤
which implies that
µ + λ 1.
0 0
k k | | ≤
We shall show, by a standard iterative scheme, that there is a constant T > 0,
depending only on ϕ and B, and a unique local solution (µ ,λ ) starting from
t t t T
≤
(µ ,λ ). Then we shall show, moreover, that µ 0 for all t [0,T].
0 0 t
≥ ∈
First of all, let us see that this is enough to prove the proposition. If we put f = 0
and a = 1 in (2.6), we obtain
d 1
λ = ϕ(x+y)1 K(x,y)µ (dx)µ (dy)+λ ϕ(x)2µ (dx).
t x+y B t t t t
dt 2 6∈
ZE E ZE
×
So, since µ 0, we deduce λ 0 for all t. Next, we put f = ϕ and a = 1 to see
t t
≥ ≥
that
d 1
( ϕ,µ +λ ) = ϕ(x+y) ϕ(x) ϕ(y) K(x,y)µ (dx)µ (dy) 0.
t t t t
dt h i 2 { − − } ≤
ZE E
×
Hence
µ + λ ϕ,µ +λ ϕ,µ +λ 1.
T T T T 0 0
k k | | ≤ h i ≤ h i ≤
We can now start again from (µ ,λ ) at time T to extend the solution to [0,2T],
T T
and so on, to prove the proposition.
We use the following norm on M R:
B
×
(µ,λ) = µ + λ .
k k k k | |
SMOLUCHOWSKI’S COAGULATION EQUATION 7
We note the following estimates: there is a constant C < , depending only on ϕ
and B such that, for all µ,µ M and all λ,λ R, ∞
′ B ′
∈ ∈
(2.7) LB(µ,λ) C (µ,λ) 2,
k k ≤ k k
(2.8) LB(µ,λ) LB(µ,λ) C (µ,λ) (µ,λ) ( (µ,λ) + (µ,λ) ).
′ ′ ′ ′ ′ ′
k − k ≤ k − k k k k k
We turntothe iterative scheme. Set (µ0,λ0) = (µ ,λ ) foralltanddefine inductively
t t 0 0
a sequence of continuous maps
t (µn,λn) : [0, ) M R
7→ t t ∞ → B ×
by
t
(µn+1,λn+1) = (µ ,λ )+ LB(µn,λn)ds.
t t 0 0 s s
Z0
Set
f (t) = (µn,λn)
n k t t k
then f (t) = f (0) = (µ ,λ ) 1 and by the estimate (2.7)
0 n 0 0
k k ≤
t
f (t) 1+C f (s)2ds.
n+1 n
≤
Z0
Hence
f (t) (1 Ct) 1, t C 1
n − −
≤ − ≤
for all n, so, setting T = (2C) 1, we have
−
(2.9) (µn,λn) 2, t T.
k t t k ≤ ≤
Next, set g (t) = f (t) and for n 1
0 0
≥
g (t) = (µn,λn) (µn 1,λn 1) .
n k t t − t− t− k
By the estimates (2.8) and (2.9), there is a constant C < , depending only on ϕ
∞
and B, such that
t
g (t) C g (s)ds, t T.
n+1 n
≤ ≤
Z0
Hence, by the usual arguments, (µn,λn) converges in M R, uniformly in t T, to
t t B× ≤
the desired local solution, which is also unique. Moreover, for some constant C < ,
∞
depending only on ϕ and B, we have
(µ ,λ ) C, t T.
t t
k k ≤ ≤
It remains to show that µ 0 for all t. For this we need the following result.
t
≥
8 JAMES R. NORRIS
Proposition 2.3. Let
(t,x) f (x) : [0,T] B R
t
7→ × →
be a bounded measurable function, having a bounded partial derivative ∂f/∂t. Then
for all t T
≤
d
f ,µ = ∂f/∂t,µ + (f ,0),LB(µ ,λ ) .
t t t t t t
dt h i h i h i
Proof. Fix t T and set s = (n/t) 1 ns/t and s = (n/t) 1 ns/t . Then
n − n −
≤ ⌊ ⌋ ⌊ ⌋ ⌈ ⌉ ⌈ ⌉
t t
f ,µ = f ,µ + ∂f/∂s,µ ds+ (f ,0),LB(µ ,λ ) ds
h t ti h 0 0i h ⌊s⌋ni h ⌈s⌉n s s i
Z0 Z0
and the proposition follows on letting n .
→ ∞
For t T, set
≤
t
θ (x) = exp K(x,y)µ (dy)+λ ϕ(x) ds
t s s
Z0 (cid:18)ZE (cid:19)
and define G : M M by
t B B
→
1
f,G (µ) = (fθ )(x+y)1 K(x,y)θ (x) 1θ (y) 1µ(dx)µ(dy).
t t x+y B t − t −
h i 2 ∈
ZE E
×
We note that G (µ) 0 whenever µ 0 and, for some C < , depending only on ϕ
t
≥ ≥ ∞
and B, we have
G (µ) C µ 2, G (µ) G (µ) C µ µ ( µ + µ ).
t t t ′ ′ ′
k k ≤ k k k − k ≤ k − k k k k k
Set µ˜ = θ µ . By Proposition 2.3, for all bounded measurable functions f, we have
t t t
d
f,µ˜ = f∂θ/∂t,µ + (fθ ,0),LB(µ ,λ ) = f,G (µ˜ ) .
t t t t t t t
dth i h i h i h i
Define inductively a new sequence of measures µ˜n by setting µ˜0 = µ and, for n 0
t t 0 ≥
t
µ˜n+1 = µ + G (µ˜n)ds.
t 0 s s
Z0
By an argument similar to that used for the original iterative scheme, we can show,
first, and possibly for a smaller value of T > 0, but with the same dependence, that
µ˜n is bounded, uniformly in n, for t T, and then that µ˜n µ˜ 0 as n .
k tk ≤ k t − tk → → ∞
Since µ˜n 0 for all n, we deduce µ˜ 0 and hence µ 0 for all t T. This
t ≥ t ≥ t ≥ ≤
completes the proof of Proposition 2.2
We remark that the arguments used to prove Proposition 2.2 apply with no essential
change to thecase where the coagulationkernel istime-dependent provided that(2.4)
holds uniformly in time. We remark also that, in the iterative scheme
t
µ0 = µ , µn+1 µ + (µn +µn µn)ds
t 0 t ≪ 0 s s ∗ s
Z0
SMOLUCHOWSKI’S COAGULATION EQUATION 9
for all n 0, so by induction we have
≥
∞
µn γ = µ k
t ≪ 0 ∗0
k=1
X
where µ n denotes the n-fold convolution of µ . On letting n , we see that, if
∗0 0 → ∞
(µ ,λ ) is the unique solution to (2.6), then µ γ . These remarks will be used
t t t 0 t 0
≥ ≪
in the proof of Proposition 4.2.
We now fix µ M+ with µ 0 and ϕ,µ < . For each compact set B E,
0 0 0
∈ ≥ h i ∞ ⊆
let
µB = 1 µ , λB = ϕ(x)µ (dx)
0 B 0 0 0
ZE B
\
and denote by (µB,λB) the unique solution to (2.6), starting from (µB,λB), pro-
t t t 0 0 0
≥
vided by Proposition 2.2. We shall show in Proposition 2.4 that for B B we
′
⊆
have
µB µB′, ϕ,µB +λB ϕ,µB′ +λB′.
t ≤ t h t i t ≥ h t i t
We shall also show in Proposition 2.5 that, for any local solution (ν ) of the
t t<T
coagulation equation (2.1), for all t < T
µB ν , ϕ,µB +λB ϕ,ν .
t ≤ t h t i t ≥ h ti
We now show how these facts lead to the proof of Theorem 2.1.
Set µ = lim µB and λ = lim λB. Note that
t B E t t B E t
↑ ↑
ϕ,µ = lim ϕ,µB ϕ,µ < .
h ti B Eh t i ≤ h 0i ∞
↑
So, by dominated convergence, using (2.4), for all bounded measurable functions f,
f(x+y)1 K(x,y)µB(dx)µB(dy) 0
x+y6∈B t t →
ZE E
×
and we can pass to the limit in (2.6) to obtain
d 1
f,µ = f(x+y) f(x) f(y) K(x,y)µ (dx)µ (dy) λ fϕ,µ .
t t t t t
dth i 2 { − − } − h i
ZE E
×
For any local solution (ν ) , for all t < T,
t t<T
µ ν , ϕ,µ +λ ϕ,ν .
t t t t t
≤ h i ≥ h i
Hence, if λ = 0 for all t < T, then (µ ) is a local solution and moreover is the
t t t<T
only local solution on [0,T). If (ν ) is a strong local solution, then
t t<T
t t
ϕ2,µ ds ϕ2,ν ds <
s s
h i ≤ h i ∞
Z0 Z0
for all t < T; this allows us to pass to the limit in (2.6) to obtain
d
(2.10) λ = λ ϕ2,µ
t t t
dt h i
10 JAMES R. NORRIS
and to deduce from this equation that λ = 0 for all t < T; it follows that (ν ) is
t t t<T
the only local solution on [0,T). Note that, for any local solution (ν )
t t<T
x1 ν (dx) = x1 ν (dx)
x n t x n 0
≤ ≤
ZE ZE
1 t
+ (x+y)1 x1 y1 K(x,y)ν (dx)ν (dy)ds.
x+y n x n y n s s
2 { ≤ − ≤ − ≤ }
Z0 ZE E
×
Hence, if (ν ) is strong and ϕ(x) εx for all x, for some ε > 0, then by dominated
t t<T
≥
convergence the second term on the right tends to 0 as n , showing that (ν )
t t<T
→ ∞
is conservative.
Suppose now that ϕ2,µ < and set T = ϕ2,µ 1. For any compact set
0 0 −
h i ∞ h i
B E we have
⊆
d 1
ϕ2,µB ϕ(x+y)2 ϕ(x)2 ϕ(y)2 K(x,y)µB(dx)µB(dy) ϕ2,µB 2
dth t i ≤ 2 { − − } t t ≤ h t i
ZE E
×
so, for t < T,
ϕ2,µ lim ϕ2,µB (T t) 1.
h ti ≤ B Eh t i ≤ − −
↑
Hence (2.10) holds and forces λ = 0 for t < T as above, so (µ ) is a strong local
t t t<T
solution.
If ϕ2 is sublinear, then
ϕ2,µ ϕ2,µ < .
t 0
h i ≤ h i ∞
If, on the other hand, K(x,y) ϕ(x)+ϕ(y), then
≤
d
ϕ2,µB ϕ(x)ϕ(y)(ϕ(x)+ϕ(y))µB(dx)µB(dy)
dth t i ≤ t t
ZE E
×
2 ϕ,µB ϕ2,µB 2 ϕ,µ ϕ2,µB ,
≤ h t ih t i ≤ h 0ih t i
so
ϕ2,µ exp 2 ϕ,µ t .
t 0
h i ≤ { h i }
In either case we can deduce that (µ ) is a strong solution.
t t 0
≥
Proposition 2.4. Suppose B B . Then, for all t 0,
′
⊆ ≥
µB µB′, ϕ,µB +λB ϕ,µB′ +λB′.
t ≤ t h t i t ≥ h t i t
Proof. Set
t
θ (x) = exp K(x,y)µB(dy)+λBϕ(x) ds,
t s s
Z0 (cid:18)ZE (cid:19)
π = θ (µB′ µB),
t t t − t
χ = ϕ,µB +λB ϕ,µB′ λB′.
t h t i t −h t i− t