Agglomeration of particles in a fluid environment (e.g., a chemical reactor or the atmosphere) is an integral part of many industrial processes (e.g., Goldberger [
In his original work Smoluchowski considered the agglomeration equation in a discrete form. Later it was considered in a continuous form by Mller [
Let R be the real numbers, 
I is a finite, infinite, or semi-infinite open interval}, and for
,
If A is a subspace of a vector space B we write
. These function spaces are vector spaces and 
To develop the discrete model, assume that all particles are a multiple of a particle of smallest size (volume), say
. Thus a particle made up of i smallest-sized particles has size
. In polymer chemistry, the particle is called an i-mer. The initial time is 
where I0 is the largest time interval of interest. We indicate this by the extended interval notation
. We also let 
and
. Unless otherwise specified, we assume
. Now for each 
let 
be a real-valued function (either in
) that approximates the number of i-mers in the reactor at time t. Since there are an infinite number of sizes, initially, we take the state (or phase) space to be
. Assume the initial number density 
is known.
As time passes, particles collide, agglutinations occur, and larger particles result. The net rate of increase in ni(t) with time, dni/dt, is the rate of formation minus the rate of depletion (conservation of mass). For 
we consider as a possible Σ space (i.e., the designated space where we look for solutions) either 
for the analytic context or 
for the continuous context where
Functions in 
are continuous, but functions in 
are not as we have not established a topology on R∞. They are componentwise continuous.
For 
we may define 
. The derivatives dni/dt exist and are in C(I,R). However, we can not assert that 
as we have no topology on R∞.
Let 
be the set of “infinite matrices”. The kernel (which measures adhesion or “stickiness”), 
, is a doubly infinite array of real-valued functions of time either in
(analytic context) or in
(continuous context). As with
, we establish no topology on
.
The resultant Discrete Agglomeration Model or Discrete Agglomeration Problem (DAP) is an IVP consisting of an infinite system of Ordinary Differential Equations (ODE’s) each with an Initial Condition (IC) that may be written in scalar (componentwise) form as:
IVP
where for i = 1 the empty sum on the right hand side of (1) is assumed to be zero. The first sum in the scalar (componentwise) discrete agglomeration Equation (1) is the (average) rate of formation of i-mers by agglutinations of 
with j-mers. The 1/2 avoids double counting. The second sum is the (average) rate of depletion of i-mers by the agglutinations of i-mers with all particle sizes. We model a stochastic process as deterministic. The physical system is often stationary so that each 
is time independent and the model is said to be autonomous. In a physical context, we require
. However, we will address DAP as a mathematical problem where we allow the initial number of particles
, the components of the kernel
, and the components of the solution, 
, to be negative. The physical context will be a special case.
Smoluchowski found in the physical context that when 
is a constant, that
where
uniquely satisfies DAP on its interval of validity
. If we assume
, then
.
The requirement on 
in (4) and the infinite sum in (1.1) motivate consideration of the Banach spaces
where
(Martin [16, p. 3]) with norm 
(and hence a metric and a topology). Equality of two vectors in 
requires the metric (the norm of their difference) to be zero. This is equivalent to both vectors being in 
and being componentwise equal. If 
, then 
defines a norm on 
(Naylor and Sell [17, p. 58]). To insure that 
exists (even for negative initial conditions)we will require 
so that
.
We are particularly interested in the time-varying kernel 
which depends on time, but not on particle size. In the continuous context where
the problem parameters are
. In the analytic context where
the problem parameters are
. For any kernel, solution requires that both sides of (1) are continuous in the continuous context and analytic in the analytic context.
The ith depletion coefficient associated with 
and the distribution 
is defined formally by the infinite series
The only direct dependence of 
on t is through
. If (5) converges for all
, then 
maps 
to
. We may view 
as a function of an infinite number of real variables or as a function of time and a size distribution. Regardless, if
, and we have convergence, the composition 
maps I to R.
Implicit in (1) is that for solution in the continuous context, we must have for all
, that 
. That is, DAP requires us to first find 
such that for all 
and
, 
exists (i.e., converges) and defines a function in
. If, in addition, 
(the Σ space) and satisfies (1) on I and (2), then it solves DAP on I. This formulation of DAP does not require mathematics beyond calculus and is often used by engineers and scientists.
For DAP with a time varying kernel, 
, in the analytic context, Moseley [
where
, satisfies DAP uniquely on its interval of validity
or the physical context where
, again we have 
and require 
. The formula (6) satisfies (1) on I and (2) in the continuous context as well where we now allow 
However, since (6) was not derived using equivalent equation operations, uniqueness has not been proved rigorously for 
. Unless otherwise stated, for the rest of the paper, we focus on the continuous context.
Moseley [
To rearrange terms in infinite series we will need
If all sums exist, we add all of the elements in 
in two different ways. Since we use them often, we will use 
to mean “for all” and 
to mean “there exists” (with apologies to the logicians). If y = n(t), we use any of n, n(t), y(t) and n(
) to denote the function. Also, we denote the restriction of a function to a smaller domain by the same symbol. The context will make it clear.
). If the 
set is a vector space, we say Σ space. A problem is (set-theoretically) well-posed if it has exactly one solution in its 
set. (In this paper, we will not consider continuity with respect to problem parameters.) A well-developed model of dynamics using an IVP is well-posed (exactly one event happens). As modelers, we expect our models to be well-posed. As mathematicians, we require rigorous proof. Often, we solve equations by using equivalent equation operations to isolate the unknown(s). This yields uniqueness, and, as all steps are reversible, existence. (Squaring both sides of an equation is not an equivalent equation operation and may lead to extraneous roots.) For linear ODE’s, we may guess the form of a solution and prove existence and uniqueness by using the linear theory. For nonlinear problems, we may prove existence by substituting back into the equation. Uniqueness then becomes an issue.
. If a solution is unique in B, and it is in A, then it is unique in A. If A is the Σ set for the problem and contains only one solution, then the solution is unique in B. Being in A is a requirement for existence. In the continuous context, for
, we look for solutions to
space
. Thus, as is usually done, we require solutions to (8) to not only exist, but to also have continuous derivatives. We also require 
where 
and the range of y(t) is in U for y(t) in the 
space. Placing these additional constraints avoids dealing with pathology, but narrows the space where a known solution is to be shown to be unique. There may be (pathological) solutions to (8) where the derivative exists, but is not continuous.
where a solution exists is the interval of validity for the solution (i.e., the domain). We say that we have shown global existence on I if, given 
, we prove that there exists a solution on I (i.e., a solution in
). Suppose a solution on 
goes through the point where
. It is said to be locally unique at 
if there exists 
such that it is the only solution on I1. It is locally unique on I if it is locally unique at every point in I. Obviously, if a solution exists globally on 
, and is locally unique on I, then it is globally unique on I. That is, it is the only solution in the 
space
).
space 
and say that
satisfies (1) on I if
, the composition 
exists (converges) and is in 
and ni(t) satisfies (1) on I. Since composition of continuous functions is continuous, we expect 
if in some sense 
from 
given by (5) is continuous. But we do not have a topology on 
and hence not one on
. Instead of requiring
, 
as a separate condition for solution, we may incorporate it into the Σ space. We refer to DAP with the Σ space
, instead of
, we may choose the state space as 
which has a norm (and hence a metric and a topology). A solution on I is then a time-varying infinite-dimensional “state vector”
. Later we will choose an appropriate 
space and write DAP in vector form. We refer to this formulation of DAP as the Vector Discrete Agglomeration Problem (VDAP). As with SDAP, VDAP may be in the continuous or analytic context. If SDAP is well-posed, and its solution is in the (smaller) 
space for VDAP, then SDAP and VDAP are equivalent except for the space where local uniqueness is proved. That is, by choosing a smaller 
space, VDAP requires proving local uniqueness in a smaller space than does SDAP. If we do not worry about pathology, and redefine the 
space for SDAP to be the same as for VDAP, the two problems are equivalent. The question is: How do we choose an appropriate (smaller) 
space? But first we consider an equivalent scalar problem and 
spaces.
that 
converges 
where
. Now define the functions
. For these functions, as with 
the only explicit dependence on t is through
. For 
we may now write (1) as the system of ODE’s
to 
(which we denote by the same symbol) converges 
and is continuous on 
with respect to the norm topology, we write
. That is, 
.
and investigate 
, and
. Note 
is just a finite sum involving Ki,j(t) and components of
, 
is just the product of 
with a component of
, and 
is just the difference of 
and
.
and 
. Then
, 
, and 
are all in
.
sufficiently small so that
. For example, if 
, then the projection function 
is continuous since if
, then 
satisfies a Lipschitz condition (Bartle [18, p. 161]) and hence is continuous on ℓ1 i.e., is in
. Since it is a constant function of t, it is in
. We investigate continuity and differentiability in ℓp in more detail in the next section.
. If the composition 
converges 
and is continuous on I, we write
. Previewing the next section, we define the function spaces 
and 
as the componentwise continuous functions that have codomain ℓ1, and claim that 
.
and 
. If
, then the compositions 
and 
are all in
.
are continuous. ■
, then SDAP given by (12) and
is equivalent to the infinite system of scalar (componentwise) Voltera integral equations
is a solution to (13) if it is in the 
space
, 
satisfies (6). (We require
and not just that the integral in (6)
is a solution of SDAP in 
if and only if it is a solution of ISDAP in
.
is a solution of SDAP in
. We have by the definition of a solution of SDAP, that
, that
, that (13) is satisfied on I, and that (2) is satisfied. Since both sides of (13) are continuous, we may integrate from t0 to 
to obtain
is a solution of (13) in
. Substituting in t0 we obtain (2).
, we have 
that
so that the integrand,
, is continuous. Since ni(t) is written as an integral, it is differentiable so that
. Differentiating we see that (11) is satisfied. ■
as the 
space for both problems, the problems remain equivalent as any solution to (13) in 
is in fact in
. That is, there are no solutions to (13) in
. These results can also be established in the analytic context.
has a norm (and hence a metric) we have a topology on the subspace 
of
. Many of the limit laws can be extended to
. For example, if
and
, then
. We also have if
and
, then
is continuous at 
with respect to the norm topology if
in
; that is, given ε > 0, 
implies
. If it is continuous
, it is continuous on I. Similarly, a function 
is continuous at 
with respect to the norm topology if 
in I;
such that
implies
. If it is continuous
, it is continuous on
. Similarly for the functions
, 
, and
.
and
as well as
, 
and
. If
, then we may assume
. For
, the range is restricted to the set B whereas, for
, it is allowed to be in the larger set C. Since 
. However, 
has a norm (and hence a metric and a topology), but 
does not. (We could establish a topology for
, but this is not necessary if the system states are all in
.) We will use 
for functions that are componentwise continuous with codomain 
and write
,
, and
. Also, if
, we write 
if
; that is, we use the same symbol for the restriction of a function to a smaller domain.
to be in 
.
Proof. We show that
. That is, if
, then 
is componentwise continuous. As 
is a vector space, by our previous comments 
follows. Let 
and
. Then
in
. That is, given 
implies
. Since
, given
such that 
implies
. Hence
, 
in R so that
. Hence
. ■
, then
.
, then
. If
, then
.
(or any normed linear space)then the triangle inequality 
implies 
so the norm function
satisfies a Lipschitz condition on 
and hence is continuous on 
i.e., is in
. We say it is Lipschitz continuous on
. Now let
. Since 
is the composition of the norm function with
, 
implies
. For
, let
. Since
, 
, then so that M0(A) is Lipschitz continuous on 
so that
.
) is continuous, 
and for 
let
as each ni(t) is continuous and
,
. However, 
, 
but for
,
. Hence 
either does not exist or is greater then or equal to 1. Hence 
in not contiuous at 
as 
does not exist. Hence 
does not exist in
. Hence
. Hence the relations
are proper.
and for any t let
and 
otherwise.
, we have 
and
. Obviously 
so 
even though 
as
,
.
, we need its range to be in
,
, and
. However, all of these do force 
to be in
.
, 
,
and
.
, 
and
. Also, 
, 
,
are all in
. Let
. Then
.
. Since
, we can choose N sufficiently large so that
. Since
, 
such that 
implies 
so that
. Since
, 
choose δi so that 
implies
. Hence
.
. Hence 
implies
.
. ■
, it may be easier to check that for each
, 
,
since 
and 
map from I to R. SimilarlyCorollary 2.7.
and
.
and
, then
.
and
. Choose δ1 such that
implies 
implies
. Let
.
implies
and
, then
. If 
and
, then
. If 
and
, then
.
is differentiable
, if
exists in
. If 
exists 
and is in
, then
. We define integration componentwise. Following Theorem 2.6, we have Theorem 2.10. If
, then
. Also,
.
follows from considering the limit for components. A proof of
can be obtained following the proof for scalar valued functions in calculus books (e.g., Stewart [19, p. 88]). The description of 
follows from Theorem 2.6. ■
, n(t) has an infinite number of derivatives and equals it’s Taylor series, 
, then
.
,
,
,
,
.
, then
, Part 1. (16)
, then
. Part 2. (17)
, Moseley [
by considering the Taylor series coefficients obtained from the initial conditions and the differential equation. However, he chose a smaller Σ space containing only distributions where if 
is in the Σ space, then
. He then obtained the explicit formula (6) for the (analytic) solution when A(t) is analytic. He did not rigorously isolate the unknown so he established global existence by showing that the solution given by the formula (6) was in the Σ space, checking the initial conditions (2), and then substituting the formula into (1). Since global existence holds, local uniqueness implies global uniqueness.
. Global existence may be obtained as before. However, local uniqueness is not as easy as it was in the analytic context. McLaughlin, Lamb, and McBride [
.
. If
,
converges
, then
maps 
to
. We say that (the restriction of) 
(to
) is in
if
, (the restriction of)
(to
) is in 
and write
. Furthermore, when
, we write
if
.
, and
, then
.
)
and
, then
. (See Corollary 2.2.) ■
so that
. Then for
in (any subspace of) 
we have
. Then the convergence and continuity condition on 
need not be explicitly stated for the Σ space or as a condition for solution (except as required for interpreting (1)). We begin with three classes of kernels:
,
and
.
and 
, if we can prove that for
, we have
, then for all kernels in these three classes, if
, we have 
. However, for clarity, we proceed class by class.
and
, then for all 
we have
so that
is the zeroth moment of the sequence. In the physical context, 
so that,
is the total number of particles and 
is the total mass of the particles (which should not change) where 
is the first moment of the solution and ρ is the mass density. Treat [
. Againwe view a solution as a time-varying infinite-dimensional “state vector”
.
so that
,
. If 
, then 
exists (converges absolutely) and
. If
, then 
. If
, then 
and
.
so that
,
. By Theorem 2.5, if
, then 
exists (converges absolutely) and
. If
, then 
and 
so that
exists (converges absolutely). By Corollary 2.9,
. Now let
. By Theorem 2.6 
so that 
exists (converges absolutely)
. Since 
and
, 
and
exist (converge absolutely)
. As compositions and products of continuous functions
) are continuous, 
, and
are in
. ■
. Then
exists (converges absolutely)
. If
, then
.
. (Note that 
is a constant function of t for this kernel.) Then 
such that
. Let
and
. Then
exists (converges absolutely). Now let 
and
. Then
is Lipschitz continuous and hence in
. Since 
is a constant function of time,
. Hence
. If 
then 
and
. ■
. Then 
exists (converges absolutely) and
. If
, then
.
. Then
and
(by Theorem 2.15). By Corollary 2.9
. If
, then
. ■
and
. For
, let
. We briefly review the Weierstrass M-test and succeeding theorems on absolute and uniform convergence (Kaplan [21, pp. 436-444]). This should be familiar to engineers and scientists. We then consider a fourth class of kernels. Let
(let
and
). We say that K(t) is locally uniformly bounded in time and size.
, 
and
. Suppose
such that
, 
and
, then
and 
exist
and are uniformly convergent on J so that theyare in
. Since t1 was arbitrary, 
so 
and 
and 
are in 
so that
. If 
and 
such that
, 
and
, then 
and 
exist (converge absolutely)
and are uniformly convergent on J so that they are in C(J,R). Since t1 was arbitrary, 
so 
and 
and 
are in 
so that
. Also,
.
, to insure
, we will require
to satisfy a stronger local uniform boundedness condition in time.
.
is locally uniformly bounded at t1 if
, such that
, 
and
.
is locally uniformly bounded on I if it is locally uniformly bounded
.
,
and
.
(which he denoted by
) as the Σ space in the analytic context. We have
.
with
and ni(t) increasing. Now for
, let 
and
. Then
. Hence
.
where 
is given by (1.6) is in 
where
.
. Then 
and 
are in C(I,R) and
. If 
then
, 
and
.
and
. Then 
such that
, 
and
. Hence
,
so that
. By Theorem 2.17, 
and 
are in C(J,R). Since t1 was arbitrary, 
and 
. If
, then, again by Theorem 2.17, 
and
. Hence by Theorem 2.6, 
and so
. ■
is contained in
, we have
. Similarly,
.
.
.
.
and
, then
. We use the local uniform boundedness of 
and
. Let
and 
are vector spaces and
. For
, and
, 
exists (converges absolutely) and
. If 
, then
.
. Then
, and 
such that for all 
and 
Let 
and 
where
. Then 
exists (converges absolutely for
). Since t1 was arbitrary 
exists for
. Also, since
and
, by Theorem 2.17 we have that 
and fixed
,
. Hence
. However, we do not have
. We may (or may not) be able to prove this with a further extension of the Weirstrauss M-test. Instead we let
. Then for 
and 
not only do we have 
such that for all 
and 
but also 
such that
and
. Then
. Hence
. ■
and we choose a subspace of 
as our Σ space, we obviate the need to explicitly require 
as a condition for 
to be a solution (except to interpret (1.1)) or as a specific condition for the Σ space.
converges 
the functions
, 
, and 
all map 
and that
maps
. Now let
,
, and
. These three functions map
. As with
, the only explicit dependence on t is through K(t). If 
and
, then
,
, and 
are all in 
(see Theorem 2.1). Now let 
Then by Theorem 2.16,
. If we can show that 
implies
, 
, and 
are in
, then these functions can be thought of as functions from 
to 
instead of from
. We indeed show that if we restrict 
to
, then the restrictions of
, 
, and 
to 
all map to 
so these functions are all in
. Let
is a vector space. We show that if
, then
, 
, and 
are all in
. We have
. Then, for
the images
, 
, and 
are all in
. Also,
,
, and
. If
, the
, and
,
, and 
are all in
.
and
. Then 
such that
. Hence for 
where
, from (2.12) we have
so that
, 
is in
. Furthermore, since
and
, by Theorem 2.17, for fixed
,
.
.
. Also, since
. Since
,
. Hence
. Since 
is a vector space (note
)we have that
.
. Then by Theorem 2.16,
. By using Theorem 2.1 and above, 
, 
and 
are in 
. However, assuming
we consider the Vector Problem (VP):
. That is, we now require the derivative to be defined with respect to the norm topology,
in
, and equality as equality in
. For VP, we take our
.
we now show that VP is equivalent to
(the Σ space) to be a solution of (21), we require
, that (12) holds; that is, integration is componentwise. Equality is in
. We refer to this problem as the Integral Vector Problem (IVP)
is a solution of VP in 
if and only if it is a solution of IVP in
.
. Now assume that 
is a solution of VP so that (19) and (20) are satisfied, and the right hand side of (19) is in
. We may integrate from t0to t using (17) to obtain the vector equation
is a solution of (21). Then substitute 
to obtain (20). Since
, and
, differentiating
, then
and
. If
, then
and
. On the other hand, if
, then
and
. If, in addition,
, then 
and
.
, then by Theorem 2.16,
. By Theorem 2.21,
. If
, then
and
. Now let
. Then by Theorem 2.21,
and
. If, in addition,
, then 
and
.
and
. Then for
, we would have 
and
. When
, we do have
, but have only shown that 
so that for
, we have 
and
. We refer to this problem with Σ space 
as VDAP1. When
, we settle for
and
so that if
, then 
and
. We refer to this problem with Σ space 
as VDAP2. As
, if we take
as our Σ space for SDAPISDAP, and VDAP1or VDAP2, then they are all equivalent if they have the same problem parameters
.
) in the analytic context, the problem parameters are
. For this problem, Moseley [
. However, he chose the smaller Σ space 
containing only distributions where (for a time-varying kernel) if 
is in
, then the depletion coefficients 
are in
, He then obtained the explicit formula (6) for the (analytic) solution. He did not rigorously isolate the unknown so he established global existence by showing that the solution given by the formula (6) was in the Σ space, checking the initial conditions, and then substituting the formula into (1). Since global existence holds, local uniqueness in the analytic context implies global uniqueness.
as our Σ space, then SDAP, ISDAP, VDAP1, and IVDAP2 are all equivalent in the continuous context if they have the same problem parameters
. If
and
, we have
, so that we need not specify this condition separately. For the time varying kernel, the solution given by (6) is in
where 
in the continuous context. However, we have not shown (local) uniqueness in the continuous context. To do this we have (at least) four choices:
in the scalar problems SDAP and ISDAP.
in the vector problems VDAP1 and VDAP2.