Properties of the solutions of delocalised coagulation and inception problems with outflow boundaries
Well posedness is established for a family of equations modelling particle populations undergoing delocalised coagulation, advection, inflow and outflow in a externally specified velocity field. Very general particle types are allowed while the spatial domain is a bounded region of -dimensional space for which every point lies on exactly one streamline associated with the velocity field. The problem is formulated as a semi-linear ODE in the Banach space of bounded measures on particle position and type space. A local Lipschitz property is established in total variation norm for the propagators (generalised semi-groups) associated with the problem and used to construct a Picard iteration that establishes local existence and global uniqueness for any initial condition. The unique weak solution is shown further to be a differentiable or at least bounded variation strong solution under smoothness assumptions on the parameters of the coagulation interaction. In the case of one spatial dimension strong differentiability is established even for coagulation parameters with a particular bounded variation structure in space. This one dimensional extension establishes the convergence of the simulation processes studied in [Patterson, Stoch. Anal. Appl. 31, 2013] to a unique and differentiable limit.
Keywords: Coagulation, advection, existence, uniqueness, regularity, Banach ODE, propagator, boundary
AMS 2010: 34G20, 35A01, 35A02, 35F61, 82C22
Smoluchowski  introduced equations for the concentrations of particles of different sizes undergoing coagulation in a spatially homogeneous population.
where is the concentration of particles of size at time and is a symmetric function defining the ‘reaction’ rates. The Smoluchowski coagulation equations can be regarded as describing a system of binary reactions involving an infinite number of species, but with a very structured, although non-sparse set of rates and (1) abstractly written . The model therefore extends naturally to a reaction–transport problem for spatially inhomogeneous populations of coagulating particles of the general form for some transport operator .
Since coagulation is a binary reaction in which every possible pair of particles may coagulate, the equations are, even in the spatially homogeneous case, non-linear and more significantly non-local in particle size (size may here be generalised to ‘type’). The first existence results for the Smoluchowski coagulation equation and its extensions were based on convergent sub-sequences of approximating stochastic processes. The first convergence result of this kind with simple diffusive transport of particles is due to Lang and Xanh , generalisations were achieved by Norris [12, 11], Wells  and Yaghouti et al. . This is quite a natural approach, because the equations are based on a microscopic stochastic model and related stochastic processes have also proved fruitful for numerical purposes going back to Marcus  and Gillespie .
The results just mentioned are essentially compactness results and say nothing about uniqueness of the limiting trajectories, much less of uniqueness for the solutions to the Smoluchowski equation and its extensions. Convergence and uniqueness were proved together by Guiaş  who modelled diffusion as a random walk on a lattice and used a more functional analytic approach. Going further in this direction one is led to regard the Smoluchowski equation and its extensions as an ODE on a Banach space and to proceed via a locally Lipschitz source term and a Picard iteration method to show existence and uniqueness in some functional setting. The general strategy is presented in chapters 5&6 of . Applications to Smoluchowski problems are given by [20, 1, 3] and the works cited therein.
Especially when approaching the Smoluchowski equation from the point of view of stochastic particle systems it is natural to think of measure valued solutions. A particle system is identified with its empirical measure and thus instead of functional solutions one is led to look at measure valued solutions in a weak setting. To give the concrete example that will be the focus of this work: A solution (with a given initial condition) is a flow of measures on positions in and particle types (sizes and potentially additional details) in satisfying
for all in a class of functions to be specified below. Here the problem has been moved from the strong formulation of (1) to a weak setting; a transport operator (the dual of the mentioned above) has been introduced and the delocalisation of the coagulation specified via a function , which may be regarded as a mollifier. A particle source term has also been added, which is relevant for many real-world applications as discussed later.
Signed measures can be regarded as Banach space under a wide range of norms and equation (2) interpreted as a Banach space valued ODE and Picard-like fixed point strategies introduced. An important insight of the monograph  was to exploit duality of linear operators and norms between measures and appropriate spaces of test functions in pursuit of this programme. In this way one performs most calculations for operators on test function spaces, which are a little easier to work with than operators on spaces of measures. Measure valued solutions are also the topic of , which also uses a linear operator approach, but uses approximation rather than duality arguments and deals with unbounded coagulation kernels.
All the work discussed so far deals with diffusing particles (contrast (2)) and solutions either with a zero gradient boundary conditions, which excludes outflow or defined on the whole of so that outflow is thereby excluded. For numerical reasons motivated by applications in engineering, the present author has been interested in the Smoluchowski equation with advective transport and a delocalised coagulation interaction [15, 10]. In particular for engineering applications particle gain and loss terms are important—industrial equipment is designed to take in material, alter it and then send it on either as waste or product. This gives the problem as formulated in [15, 10] and other applied works a different structure to those studied in previous mathematical works. For example, individual particles experience irreversible processes, but nevertheless the system is expected to reach a steady state in the large time limit under a wide range of conditions. Measure valued processes (which can be interpreted as particle processes) with an inflow term although no interaction were also studied in .
For (2) specific problem an initial existence result via the compactness of approximating stochastic processes was given in . In that work however convergence of the approximating processes could not be proved, only sequential compactness, because the number of distinct limit points was unknown. This was not only mathematically frustrating, but also a major obstacle hindering the numerical analysis of the associated simulation methods.
The purpose of the present work is to establish uniqueness of measure valued solutions for (2). Additionally Lipschitz continuity in the initial conditions is shown and the same Picard iteration method that proves uniqueness of solutions provides a purely analytic existence proof. The result can thus be characterised as one of “well posedness”. Formally there are some new existence results—the assumption of only one spatial dimension in  is relaxed, but with the assumptions used in this work the proof in that paper could easily be extended. The existence of a differentiable strong solution is of interest, because it opens the way to a study of the way in which the solution approaches a solution of the corresponding equation with a local coagulation interaction, see for example .
2 Statement of Main Results
In order to make a precise statement it is first necessary to go into details regarding the various objects appearing in (2). The basic spaces are the particle position and type spaces and respectively. The type space, which carries information about the mass and any other internal details of a particle is assumed to be a locally compact, second countable Hausdorff space on which coagulation is represented by a commutative + operator. The particle position space is assumed to be a simply connected, relatively compact subset of , which is equipped with Lebesgue measure and a derivative . Both and are given their respective Borel s and is given the product topology and .
Throughout this work will be given the usual Euclidean norm, which will be written . Linear operators between two normed spaces are given the operator norm
2.1 Properties of the Flow and Spatial Domain
Particles are assume to be transported in a time dependent velocity field defined on the closure of such that , satisfying
viewing the matrices as linear operators,
It is assumed that the spatial domain is simply connected and has a regular boundary that can be decomposed into three parts, each with outward normal :
where for all ,
Further but .
Define as the position at time of a particle moving with the velocity field starting from at time . It is assumed that
There exists a such that, for all and one has , that is, an upper bounded on the residence time.
For every and there exist unique , such that and either or (the possibility of both is not excluded). This defines a start position for each point in the flow and .
and are differentiable in and . A bound for the derivative of is given in the appendix.
The set forms a differentiable dimensional manifold that divides into two disjoint simply connected components.
2.2 Test Function Spaces
Let be the space of bounded measurable functions on with the supremum (not essential supremum) norm, which will be written .
Let be the space of bounded measurable functions on with the supremum (not essential supremum) norm, which will be written .
Let be the space of -dimensional vector valued functions with components in . This will be given the norm , where is the Euclidean norm on .
To handle the derivative in (2) and associated boundary condition introduce
The norm is
2.3 Solution Spaces
A particle distribution is at a minimum a measure on the product of the particle position and type spaces, that is on . The solution processes must accordingly take values in the following spaces, which are built from the space of measures on particle types :
Let be the normed space of signed bounded measures on with the total variation norm
Let be the vector space of bounded signed measures on .
with the norm
where a measure is identified with its density.
The and dual norms on will play a role in this work
As the notation suggests, the norm is the total variation norm on . For calculations the point of view is emphasised, however the main results are stated in terms of TV. When dealing with processes the following abbreviation is useful
Let and then
It is now possible to set out the assumptions on the coagulation dynamics specified by and in (2). is assumed to be non-negative and measurable with some bound such that .
The delocalisation must be measurable and non-negative. For fixed write and for the the functions given by and . It will be assumed that neither nor are identically zero—this would lead to a trivial problem with no coagulation.
H2: H1 holds and with positive, and of special bounded variation (derivative in plus atoms) for all and with the number of atoms in the weak derivatives bounded. Further one has and its symmetric counterpart
H3: H1 holds and the are in with. It should be noted that H3 implies H2 (Proposition 60 is helpful here).
The function parametrises the numerical methods that lie behind this work . H2 is describes the case where the spatial domain is partitioned into cells and coagulation is only simulated between particles that are in the same cell. From a software point of view this is somewhat simpler than dealing with functions satisfying H3. In one dimension, which was the case simulated in , H2 is a weak integrability condition on the derivative of .
Particles are added to the system with intensity given by signed measures .
I1: and .
I2: I1 holds, the are non-negative measures and for every
with also with the respective norms uniformly bounded for all time and with some such that for all and .
I3: I2 holds, has a time and space derivative so that and has an -derivative which is
2.6 Statements of the Theorems
These results progress from local existence and uniqueness of a measure valued solution to a global result and then existence followed by differentiability of a density for the measures.
Assume H2 or H3 holds and that , then there exists a such there is a unique solution to (2) in with initial condition and this solution is in .
Additionally, there is no time interval on which more than one TV-bounded solution exists for a given initial condition. If solutions exist on a common compact time interval for at two or more initial conditions, then the solutions are Lipschitz continuous with respect to the initial data in the TV-norm on this compact time interval.
In the physically reasonable setting of non-negative particle numbers, the previous result holds for all time:
The from the previous theorem is if and the are non-negative measures.
Assume H2 or H3 holds, that is in the positive cone of , and that I2 is satisfied, then (2) has a unique solution, which is in and therefore has a density in starting from .
Assume that is consistent with the boundary condition given below, that I3 is satisfied, and further that either H3 holds and has a sufficiently regular boundary or , H2 holds and is bounded away from 0, then (2) has a unique solution with a density in , satisfying the boundary condition and with initial condition .
As a corollary of the preceding two results an earlier result by the author, which demonstrated the existence of converging sub-sequences of stochastic approximations to solutions (2) can be extended to a full convergence result:
3 Dual Operator Estimates
Introduce the more compact notation for and . It is now helpful to seek a generator for the evolution given in (2), that is an operator such that
This is in fact a dual generator, because it acts on the functions not the measures.
The author emphasises his dependence on Kolokoltsov  for the material in this section and the first half of the next. The first novelty in this section is the boundary condition associated with the finite domain and outflow, which required careful treatment, but is not covered by the existing work. Also the consideration of coefficients of bounded variation (H2) is essential to treating the motivating example from  and even under this relatively weak assumption differentiability of the solutions in one spatial dimension is established. An additional variation from  appears in Proposition 26 where some additional problem structure is exploited and enables the fixed point methods to be applied in the -norm, rather than the weaker -norm used in .
3.1 The Generators
Because (2) is quadratic in the same must be true of the expression , which is achieved by including the path as a parameter of . It is technically convenient to parametrise by the entire path, not just , because one eventually deals with propagators where the dependence cannot be expressed in terms of at any finite set of time points. One notes that where is the transport operator and is the coagulation operator.
Let and assume H1 holds. The coagulation generator parametrised by is defined by
This is not the only possible definition for , other versions also yield the desired expression (the coagulation term from (2)) for . Each definition would lead to characterising the solutions as fixed points of a different mapping; the definition given here seems to be the one that minimises the technical difficulties in the following analysis.
Let and and assume H1 holds, then the operator norm of as a mapping satisfies
Let and then the transport generator is given by
One can now define as a linear operator .
3.2 The Propagators
Propagators are generalisations of semi-groups to deal with time dependent generators. For a detailed discussion the reader is referred to [7, Chapter 2] or [17, Chapter 5]. The key idea (given in the dual setting appropriate to this section) is that a generator generates a family of linear operators such that and
The goal of this section is to construct such a family of propagators for the generator from the previous section.
Let , then the transport propagator preserves . The following operator norm estimates hold:
where is an indicator function and
The -norm of is immediate. Use the chain rule and Proposition 60 in the appendix for the derivative of . ∎
The required propagator is now constructed as a perturbation of the transport propagator by the bounded coagulation generator:
Let , and , and define (compare [7, Theorem 2.9]):
It is now necessary to establish estimates for the operator norm of on and . For this it is shown that the infinite sum just given is absolutely convergent in both operator norms. During this analysis it is convenient to use some additional notation:
Under the assumptions of Definition 20 let and ; define both and
This allows one to write
Proceed by induction. ∎
The -operator norm estimate now follows:
Let , and and assume H1 holds, then is a locally bounded propagator on satisfying
and is -continuous in for every and (this is known as ‘strong continuity’). Further, for any and almost all
where one recalls .
The (left) generator can be found differentiating the series in Proposition 20 term by term and observing that the resulting series is again absolutely convergent. ∎
Differentiating with respect to in Proposition 23 is not possible, because does not necessarily preserve . This is addressed in the next few propositions by making stronger smoothness assumptions on , the spatial delocalisation of the coagulation interaction introduced in §2.4 and used in the definition of (Definition 15).
Let , , and either H3 hold or H2 hold but with bounded for all (not just Lebesgue almost all) times, then is a propagator on and there is a such that
Further, for any and almost all
These results concerning the dual propagators are concluded by showing Lipschitz continuity in the measure valued path parameter.
Let , suppose , and H1 holds, then
Write and show by induction that
This result exploits a small amount of additional problem structure to adapt the method set out in the proof of Theorem 2.12 in . The key is that the parameterisation only affects the coagulation () part of the propagator, which has a bounded generator, while the transport () part of the propagator, which has an unbounded generator is independent of the parameterisation by and .
4 Operators on the Space of Measures
Under the duality pairing of and given by as used above, (dual) operators define (pre-dual) operators with the same operator norms.
Let and . For the pre-duals of and write and respectively and note the reversal of the time indices. For the pre-dual of write .
It is emphasised that acts on functions while acts on measures, but both are parameterised by a measure-valued path .
The existence of the dual operators and their norm estimates is immediate, see for example [7, Thrm 2.10]. The duality relations yield:
Let , and assume H1 holds, then
Duality and Proposition 23. ∎
Let , , , and assume H1 holds, then for almost all
Duality and Proposition 23 ∎
4.1 The Fixed Point Mapping
This section presents a Picard iteration method for (2) highlighting the roles of the and norms on the space of measures. The mapping that will be shown to have a fixed point is:
Suppose , , let and suppose H1 holds. Define by
Under the assumptions of Definition 30 one has with
The time derivative exists for almost all with
and if then
boundedness is a consequence of Proposition 28 and continuity follows from the continuity in of .
Suppose , , H2 or H3 holds and is a bounded solution to (2) with initial condition , then is a fixed point of .
Proposition 32 is the only place where one requires H2 or H3 in the existence and uniqueness analysis. This is in order to invoke Proposition 25 and more fundamentally so that preserves ; otherwise one cannot give meaning to . Without this result it still follows that the mapping has unique fixed point with all the advertised properties (in particular solving (2)), but one cannot rule out the possibility that there are additional (possibly less regular) solutions to (2). These conclusions are stated more formally in Proposition 35 for which two preparatory results are needed.
Let , be large enough to satisfy
define and assume H1 holds. Then there exists a such that preserves .
Let and be as in Proposition 33 and assume H1 holds, then there is a such that