Fast fusion in a two-dimensional coagulation model

In this work, we study a particular system of coagulation equations characterized by two values, namely volume $v$ and surface area $a$. Compared to the standard one-dimensional models, this model incorporates additional information about the geometry of the particles. We describe the coagulation process as a combination between collision and fusion of particles. We prove that we are able to recover the standard one-dimensional coagulation model when fusion happens quickly and that we are able to recover an equation in which particles interact and form a ramified-like system in time when fusion happens slowly.

2 The case of fast fusion 10 2.1 Existence of a limit of solutions of coagulation equations with fast fusion . . . 10 2.2 Reduction to the one-dimensional coagulation model in the case of fast fusion .20

Introduction
Most of the works on coagulation equations assume that the particles are characterized by a single variable, usually the particle volume (or equivalent quantities like polymer length), see for instance [11,[13][14][15].Nevertheless, other parameters that might provide insight about the geometry or other features of the particles are usually omitted.In a previous work (see [2]), we study the mathematical properties of a class of coagulation equations in which the aggregating particles are characterized by two degrees of freedom, namely the volume v and the surface area a.This type of models was introduced in [9,10].More precisely, the model considered is the following: where In this model, f is the density of particles in the space of area and volume for any given time t ≥ 0. The coagulation operator K[ f ] is the classical coagulation operator that was introduced by Smoluchowski (see [15]) and gives the coagulation rate of particles which evolve according to the following mechanism: It is assumed that the particles attach to each other at their contact point and therefore in this way both the total area and volume of the particles involved in the process are preserved.
On the other hand, the fusion term ∂ a [r(a, v)(c 0 v 2 3 − a) f (a, v, t)] describes an evolution of the particles towards a spherical shape.The dynamics generated by this term preserves the total number and volume of the particles.The term c 0 v 2 3 − a indicates that the area of the particles tends to be reduced as long as it is larger than that of a sphere c 0 v 2 3 (see Figure 1 for a description of the complete coagulation process assumed in (1.1)).
Fig. 1 Coagulation process: collision of particles followed by fusion Additionally, r(a, v) will indicate the fusion rate and describes how quickly the particles evolve towards the spherical shape and thus has units of the inverse of the fusion time.If the fusion kernel r is very large compared with the coagulation rate, we expect that the particles become spherical in very short times.Therefore, it should be possible to approximate the solutions of (1.1) by means of solutions of a coagulation model depending on only the variable v, i.e. an one-dimensional coagulation equation.On the contrary, in the particular case when r ≡ 0, fusion does not occur and particles attach at contact points forming a ramified-like system in time.Thus, when r is very small compared with the coagulation rate, we can approximate the solutions of (1.1) by means of solutions of a two-dimensional coagulation model without fusion depending on two variables, a and v.
More precisely, we analyse the following model as Λ → 0 and as Λ → ∞: We remark that the particles must satisfy the isoperimetric inequality, therefore the density f Λ should be supported in the region where {a ≥ c 0 v 2 3 }.Moreover, the evolution generated by (1.1) (or by (1.2)) has the property that it preserves the set of measures supported in this region.
We assume r(a, v) behaves like a power law of a and v.For the coagulation kernel K, we assume that it has a weak dependence on the surface area of the interacting particles, but it can have a power law behavior in the volume of the coalescing particles.
Since collision does not change if we permute the colliding particles, i.e. (a, v) ↔ (a , v ), the coagulation kernel must satisfy the following symmetry property: for all (a, v, a , v ) ∈ (0, ∞) 4 .
In order to control the mass of the solutions when |a − c 0 v 2) if Λ is small, we require the following technical assumptions on the fusion kernel r: , and for some constant B > 0. A particular case used in applications that satisfies the above mentioned properties is when r(a, v) = Ra µ v σ , with µ ≥ −1 and R ∈ [R 0 , R 1 ].The condition (1.5) is not optimal and it would be possible to impose weaker conditions on the fusion kernel.However, this would imply more involved arguments in the proofs later on.We impose the stronger condition (1.5) as our main goal is that the statements of our theorems hold for fusion kernels that behave as power laws, case which is included in condition (1.5).
In comparison to [2], since in this paper we are not interested in the long-time behavior of solutions, we do not assume homogeneity of neither the fusion nor coagulation kernel and we simply assume that they behave like power laws, see (1.4) and (1.6).In [2], we restricted the analysis to coagulation and fusion kernels which rescale in a similar manner when the size of the particles is changed without modifying their geometry.This was needed for the study of the long-time behavior of particles as the fusion term was chosen in a desire for particles to form a spherical shape in time.In particular, it meant that, if the particle volume is scaled by a factor λ, then the diameter is scaled with a factor λ 1 3 and the area scales like λ 2 3 and that we needed to impose the additional assumption that 2  3 µ + σ = γ − 1, where γ is the homogeneity of the coagulation kernel and µ, σ are as in (1.4).In this work, we treat the more general case of arbitrary µ, σ ∈ R. In order to deal with this case, we will need to obtain additional moment estimates.
Our main goal for this paper is to prove that all solutions of equation (1.2) which satisfy some very general moment estimates concentrate their mass around the isoperimetric line {a = c 0 v 2 3 } as Λ → 0 and tend in an appropriate sense to a measure which can be computed by solving a suitable one-dimensional coagulation equation.Moreover, we prove that solutions of (1.2) satisfying these moment estimates do exist.The limit measure f of the sequence { f Λ } acts like a Dirac-like measure in the area variable, namely f (v, t) = δ(a − c 0 v 2 3 )F(v, t), with F satisfying the standard one-dimensional coagulation equation.We can then use the known results for the one-dimensional coagulation equations (for example, it was proven mathematically in [1, Proposition 10.2.1] for solutions in C([0, ∞); L 1 (R >0 ))) to prove that the average volume of the particles increases in time.
The reason we can reduce the evolution equation for the two-dimensional system to a onedimensional one is that, as Λ → 0, the fusion process takes place much faster than the collision process and then the particles are transported close to the isoperimetric line {a = c 0 v 2 3 } almost instantaneously.Equivalently, fusion happens immediately after collision.
On the other hand, if we let Λ → ∞ in equation (1.2), we recover a two-dimensional coagulation model, in which particles attach to each other at a contact point, forming a ramified-like system in time.A physical interpretation of this is that, as Λ → ∞, the effect of the fusion term becomes negligible (see Figure 2).

Fig. 2 System of particles under different fusion times
The evolution of a system of coagulation equations which can be described by area and volume and where the particles undergo a fusion process after they come in contact has been described in [9,Chapter 12].The specific problem under consideration was the study of aerosol flame reactors.A heuristic analysis of the shapes for the resulting particles for different values of the ratio between the average fusion time and the average collision time can be found in there.
This ratio is given by the parameter Λ in our model in (1.2).In the types of models considered in [9,Chapter 12], it is seen that, for small particles or high temperatures, the parameter Λ is small.On the contrary, for sufficiently large particles or after the gas has been cooled, we must assume that Λ is very large.The results in this paper provide a precise mathematical formulation of the behavior that has been suggested in [9,Chapter 12].The results in this work are complementary to those in [2], in which we consider a particular form of the fusion term for which Λ ≈ 1 for arbitrary times.
Coagulation equations for particle distributions characterized by a single variable have been extensively studied.In particular, the long-time behavior for coagulation equations for which solutions can be explicitly computed has been studied in [12].The existence of self-similar solutions for general classes of kernels has been obtained in [4,8].
Multi-dimensional coagulation equations have not been studied as much in the mathematical literature as their one-dimensional counterpart.Several discrete multi-component coagulation problems which are relevant in aerosol physics have been mentioned in [17].A discrete version of the model in (1.1) has been studied in [18].The model considered in there includes coagulation of particles and an effect similar to the fusion of particles in (1.1), which has been termed compaction.The diameter of the particles is restricted by the total number of monomers as well as by the isoperimetric inequality.The coagulation and the fusion rates are assumed to be constant.Due to this, the model considered in [18] is explicitly solvable using generating functions.The long-time behavior of the solutions which depends on the ratio between the fusion and coagulation kernels has been then analysed using the explicit formulas of the solutions.
In [5][6][7], the mathematical properties of some classes of coagulation equations describing clusters that are composed of several types of monomers with different chemical composition are analysed.More recently, uniqueness of the solutions for the models of multi-component coagulation equations considered in [5][6][7] has been studied in [16].
More precisely, it has been proven in [5,6] that time-dependent solutions for the multidimensional coagulation equation concentrate along a line in the space of cluster concentrations for long times for coagulation kernels for which the scaling properties of each of the components are the same for all the species that compose the system.However, as the surface area and volume appear in a less symmetric manner in our model, it does not seem feasible to adapt the proof in [5,6] to obtain our result, even in the absence of the fusion term.
Another difference between our model and the one in [5,6] is that the proof in the latter relies on the conservation of mass for each of the types of monomers.Due to the fusion term, we do not have two conserved quantities for (1.2), but only the volume is conserved.In addition, the fusion term in (1.1), (1.2) yields a non-trivial evolution of the distribution of particles.The solutions in [5][6][7] concentrate along a line with the orientation fixed by the initial distribution of cluster compositions or the source term.Thus, the solutions in [5][6][7] can concentrate along different lines depending on the initial distribution of particles.On the contrary, in one of the situations considered in this paper, the solutions concentrate always near the isoperimetric line, independently of the initial data.When Λ → 0 in (1.2), we have that the coagulation operator transports particles away from the isoperimetric line but these are transported extremely fast towards the isoperimetric region due to the fusion term.

Notations and plan of the paper
For I ⊂ [0, ∞) 2 , we denote by C c (I) and C 0 (I) the space of continuous functions on I with compact support and the space of continuous functions on I which vanish at infinity, respectively, both endowed with the supremum norm.M + (I) will denote the space of non-negative Radon measures, while M +,b (I) will be the space of non-negative, bounded Radon measures, which we endow with the weak- * topology.
We make in addition the following simplifications: • We use the notation η := (a, v).We will use interchangeably both notations for convenience.
• We will use the notation f (a, v)dvda or f (η)dη for Radon measures, independently of the fact the measure may not be absolutely continuous with respect to the Lebesgue measure.
• For a suitably chosen ϕ : R 2 >0 → R and for (a, v, a , v ) ∈ (0, ∞) 4 , we will denote: • We use C to denote a generic constant which may differ from line to line and depends only on the parameters characterizing the kernels K and r.
• We use the symbols and when the inequalities hold up to a constant, i.e. f g if and only if f ≤ Cg, for some C > 0.
The structure of the paper is as follows.In the rest of this section, we establish the setting and state the main definitions and results.
In Section 2, we prove that there exists a limit for the sequence of solutions of equation (1.2) as Λ → 0. To this end, we first prove that the mass of solutions concentrates around the isoperimetric line.This is done by looking at the adjoint equation of (1.2).The fact that the measures take small values if we are at a positive distance from the line {a = c 0 v 2 3 } together with the fact that we can control large values of the area a suffices to prove the equicontinuity in time of solutions and conclude that a limit of the sequence exists.We then prove that the found limit is a solution for the standard one-dimensional coagulation equation.This is since now the a variable acts like c 0 v 2 3 and we can omit the fusion term by testing (1.2) with functions only depending on the v variable.
In Section 3, we deal with the case when Λ → ∞ in equation (1.2).We prove that a limit exists as Λ → ∞ and that the limit satisfies a two-dimensional coagulation equation where the interaction of particles consists of particles which attach at a contact point.The proof of this result is straightforward after obtaining suitable moment estimates for the solutions, which are independent of the value of Λ.

Setting and main results
We work with non-negative continuous kernels on (0, ∞) 4 that, in addition to the properties already stated, i.e. (1.3), have the following bounds: for some K 1 , K 0 > 0, for all a, v, a , v and for the following coefficients: α > 0 and β ∈ (0, 1) such that β − α ∈ (0, 1). (1.7) Notice that condition (1.6) implies that the kernel has a weak dependence on the area variable, but K is not necessarily independent of the area variable.
Since we work with physically relevant particles, i.e. the particles for which the isoperimetric inequality is satisfied, it is helpful to define the following space The superscript I stands for isoperimetric.We endow the newly-defined space with the weak- * topology on M + (R 2 >0 ).Similarly, we denote )).We say that f Λ is a solution for the weak version of the time-dependent Λ-fusion problem if, for every T > 0, Remark 1.2.Functions f Λ as in Definition 1.1 exist and the methods to prove their existence are similar to the ones used to prove existence of self-similar solutions in [2] (in order to derive some moment estimates in [2], ideas from the one-dimensional case in [3,4] were adapted).A sketch for proving their existence will be shown in Proposition 2.1.

The case of fast fusion
Remark 1.3.From now on, in order to simplify the notation, we replace Λ by in (1.2) when we consider the case Λ → 0 and we replace Λ by 1 in (1.2) when we consider the case Λ → ∞.
Remark 1.5.Theorem 1.4 holds true also in the case µ ≤ 0 if we assume instead that (0,∞) 2 (v −1 + v 2 +a) f in (a, v)dvda < ∞ (plus some additional moment bound of the form M 0,d , with d depending on σ, which does not offer much qualitative information), which in turn will imply that there exists a constant C(T ) > 0, which is independent of ∈ (0, 1), such that sup The methods to prove the two cases, µ > 0 and µ ≤ 0, are similar up to minor technicalities and thus we restrict our attention to the case µ > 0 for simplicity of notation.
Remark 1.6.Theorem 1.4 says that we can construct a sequence of functions { f } ∈(0,1) such that a limit exists.However, Theorem 1.4 (and all the results stated below in this subsection) is valid for any sequence of functions { f } ∈(0,1) for which (1.11) holds.Note that uniqueness of coagulation equations is, with the exception of some particular choices of coagulation kernels, still an open problem.Condition (1.11) is however a rather strong condition and an upper bound for the moments for the initial condition is a sufficient condition to construct a sequence for which (1.11) is true.We expect (1.11) to hold under weaker assumptions than the ones stated.This will be considered in a future work.

Existence of a limit of solutions of coagulation equations with fast fusion
Let T > 0 and t ∈ [0, T ].We look at the equation for ∈ (0, 1), ϕ ∈ C 1 0 (R 2 >0 ) and with ).We begin by remembering the truncated functions used to prove the existence of solutions for truncated versions of coagulation equations allowing fusion of particles, which is done using a fixed point argument.More details can be found in [2].
We define K R : (0, ∞) 4 → [0, ∞) to be a continuous function such that: where K satisfies the upper bound in (1.6) and take ξ R : R >0 → [0, ∞) to be continuous and defined in the following manner: Then, for ϕ ∈ C 1 0 (R 2 >0 ), we denote by For the fusion term, we use the following truncation: for δ ∈ (0, 1) and some fixed L > 0. L was chosen in [2] to be L := 12 R 0 (1−γ) , where R 0 is as in (1.4), in order to obtain existence of self-similar profiles for equation (1.1).
For functions and such that sup for all times T ∈ [0, ∞), we define the space )), f satisfies (2.7) and (2.8)}. (2.9) Let T > 0 and fix ∈ (0, 1).Then there exists an f ∈ C([0, T ]; Remark 2.2.In order to prove the next proposition, we will need that sup t∈[0,T ] (0,∞) 2 a µ+3 f (dη, t) ≤ C(T ). (2.11) While the estimates in (2.10) suffice for the existence of solutions as in Definition 1.1, in order to obtain an upper bound for moments involving higher powers of the area, we need the additional assumption that sup The proof of (2.11) relies on the fact that the terms of the form a µ+2 v β , which appear due to the form of the coagulation kernel, can be bounded by For more details, we refer to [2, Subsection 3.3].
We first prove the existence and uniqueness of functions ).For this, we repeat the arguments used in [2, Proposition 3.1].We then define for every ϕ ∈ C 0 (R 2 >0 ).Notice that the functions f ,R defined in this manner will satisfy equation (2.14).For more details, see [2,Proposition 3.1].
We are now left to prove uniform estimates for f ,R in order to finish the proof.Due to the choice of the space U ˜ ,R , we can test (2.14) with ϕ(a, v) = a and ϕ(a, v) = v d , with d ∈ R. When d ≤ 1, we obtain: (2.17) Using the fact that f ,R ∈ U ˜ ,R , we can extend (2.17) to hold for functions ϕ ∈ C c (R 2 >0 ) and then for all ϕ ∈ C 0 (R 2 >0 ).For details, see [2,Proposition 3.18].Combining the found equicontinuity in (2.17) with the uniform moment estimates, which are independent of ˜ , R and δ, we conclude using Arzelà-Ascoli theorem that there exists a subsequence of { f ,R }, which we do not relabel, and an f ∈ C([0, T ]; )), such that f ,R (t) converge to f (t) in the weak- * topology as ˜ → 0, R → ∞ and δ → 0, for every t ∈ [0, T ].
Thus, we can use standard arguments found in the study of coagulation equations in order to pass to the limit as ˜ → 0 and R → ∞ (and δ → 0 if µ > 0) in (2.14).
Proof.Define xA,V (t) := x A,V ( t).Then xA,V (0) = A and (2.18) This implies that lim →0 x A,V (t) = lim →0 xA,V ( t ) = lim t→∞ xA,V (t).In (2.18) notice that Notice that, at least for small times, the coagulation term in (1.10) gives a small contribution when we are away from the line {a = c 0 v 2 3 }.We thus look to control the contribution coming from the fusion term.For this, we first look at a simplified form for the adjoint problem of (1.10).
Proof.For Statement 1 we use Proposition 2.6, the fact that we can find an explicit solution for equation (2.20) and then we let → 0. Statement 2 follows directly from the fact that at time T we have ϕ (T, •) = χ(•) ∈ C 1 b (S) and by integrating along the characteristics in equation (2.20).
For Statement 3, we notice that, for a ≥ c 0 v 2 3 , we have that where for the last inequality in (2.21), we used (1.5).
Remark 2.8.In the case µ ≤ 0, the condition (1.5) is more general due to the fact that we will not modify the fusion term r in Proposition 2.7.

.23)
Proof.Assume for simplicity that ϕ ∈ C 1 b (R 2 >0 ).We construct a sequence of functions The idea is to use Lebegue's dominated convergence theorem in (1.10) for the functions ϕ n = ζ n ϕ.We thus show below only the needed estimates for the proof.The term with the coagulation kernel in (1.10) can be bounded directly by In order to control the fusion term in (1.10), notice that we can construct ζ n such that aζ n (η) ≤ C, for some constant independent of n ∈ N.Moreover, we know that the fusion kernel satisfies (1.4) and that a ≥ c 0 v 2 3 .Thus Using the above inequality, we can bound from above the fusion term up to a multiplicity constant.We then use Young's inequality to deduce that (0,∞) 2 Thus, the moment estimates in (2.23) suffice to conclude our proof for ϕ ∈ C 1 b (R 2 >0 ).To prove that equation (2.1) holds for every ϕ ∈ C 1 ([0, T ]; C 1 (R 2 >0 )) with sup s∈[0,T ];η∈R 2
Proof of Proposition 2.3.Let σ > 0 and δ 1 , δ 2 ∈ (0, 1).Assume The extension of the result from functions compactly supported in the v variable to functions that do not necessarily have compact support is straightforward using moment estimates, and thus we omit the details.Suppose in addition that Φ is such that Φ(η) = 0 when a < c 0 v 2 3 + δ 1 .Let, in addition, t 1 , t 2 , with t 2 ≥ σ, and t 1 such that t 2 − t 1 = τ > 0. Notice that τ = τ(δ 2 ) depends on τ 2 , but we will not write this dependence explicitly in order to simplify our notation.Let M > 0, sufficiently large (and also depending on δ 2 ), to be fixed later, and let ϕ be the solution found in Proposition 2.7, if µ > 0, associated to the measure f such that ϕ (η, t 2 ) = Φ(η).We want to prove (2.13).
Notice first that, from Proposition 2.7, Statement 2, there exists a constant independent of , such that sup s∈[t 1 ,t 2 ],η∈S |ϕ(η, s)| ≤ C.Then, for t 1 , t 2 ∈ (0, T ] such that t 2 − t 1 = τ, we have where we made use of the fact that [M 0,1 + M 0,−1 ]( f ) is uniformly bounded from above, independently of ∈ (0, 1).In order to estimate the term with the fusion kernel, we notice that For the first term in (2.25), we make use of the exponential decay of ϕ proven in Statement 3 of Proposition 2.7 in order to obtain 1 We can then use Statement 2 of Proposition 2.7, the upper bound (1.4) and (2.12) in order to control the region containing large values of a, namely Combining the estimates (2.25), (2.26) and (2.27), we deduce that (2.28) Proposition 2.9 gives us that we can test (1.10) with continuous functions that are not necessarily compactly supported in the a variable, as long as we work with functions in C 1 b (R 2 >0 ).We then make use of the fact that ϕ satisfies (2.20) and then use the estimates in (2.24) and (2.28) in order to deduce that, for every ≤ δ 1 ,δ 2 , with δ 1 ,δ 2 (notice here that M and τ depend on δ 2 ) as in Proposition 2.7, Statement 1, we have where we used that suppϕ ⊆ {a ≥ M} from Statement 1 of Proposition 2.7 and the fact that sup t∈[0,T ] M 1,0 ( f (t)) ≤ C(T ), with C(T ) being independent of .We choose M such that C(T )CM −1 + Cτ ≤ δ 2 .
Thus, we obtain that for every t 2 ≥ σ since we can choose τ to be sufficiently small.
We are now able to prove the equicontinuity in time of the sequence { f } ∈(0,1) .We proved in Proposition 2.3 that if we are at a positive distance from the line a = c 0 v 2 3 , the measure f takes values close to zero.Near the line a = c 0 v 2 3 , we use the fact that a function of the form ϕ(a, v) can be approximated in terms of a function depending only on v, making negligible the contribution coming from the fusion term.
Proof of Proposition 2.4.Fix σ > 0. Let δ 1 , δ 2 > 0 as in Proposition 2.3.We define a continuous function χ δ 1 : R 2 >0 → [0, 1] to be equal to one in the region where {c 0 v ) is then straightforward using moment estimates and the fact that we can approximate a function in C c (R 2 >0 ) with a C 1 c (R 2 >0 ) function on a compact set.From Proposition 2.3, we deduce that there exists δ 1 ,δ 2 ∈ (0, 1) such that for all ≤ δ 1 ,δ 2 , the following holds (2.31) From the support of χ δ 1 , the fact that the measure f is supported in the region {a ≥ c 0 v 2 3 } and the continuity of ϕ, we can find a function ϕ : R >0 → R depending only on v, ϕ ∈ C 1 b (R >0 ), such that |ϕ(a, v) − ϕ(v)| ≤ δ 3 , for δ 3 sufficiently small, and for any (a, v) In order to control the contribution of the region R 2 >0 \K, we use moment estimates.For example, for the region {a > M}, we obtain that We can use the same argument for the region {v We are now in a position to prove Theorem 1.4.
Proof of Lemma 1.7.We pass to the limit in Proposition 2.3.

Reduction to the one-dimensional coagulation model in the case of fast fusion
We now consider the behavior of the solutions of (1.2) as Λ → 0. We first show that we can extend F in Lemma 1.7 to be continuous at time t = 0 as mentioned in Theorem 1.10.Proposition 2.10.Let T > 0. Let F ∈ C([0, T ]; M + (R >0 )) be as in Definition 1.9.Then the sequence {F } ∈(0,1) is equicontinuous in time, and thus we can deduce that there exists a limit F ∈ C([0, T ]; M + (R >0 )) as → 0 for a subsequence of {F } ∈(0,1) , which we do not relabel.
Step 1.We first prove that, for every t ≥ σ, we have that We can prove that I 2 + I 3 gives a small contribution due to moment estimates.For I 1 , we use the fact that |a − c 0 v M 0,0 ( f (t)) 2 , (2.37) where was chosen small but arbitrary and M 0,0 ( f ) is uniformly bounded independently of ∈ (0, 1).Notice that δ 1 was chosen to be sufficiently small to satisfy both (2.35) and (2.37).
Step 1. b) Let f , F be as in Lemma 1.7.We prove now that

2 3 |
≤ δ 1 , the continuity of K and that we work on a compact set, to deduceI 1 ≤ C sup t∈[0,T ]