Stationary non-equilibrium solutions for coagulation systems

We study coagulation equations under non-equilibrium conditions which are induced by addition of a source term for small cluster sizes. We consider both discrete and continuous coagulation equations, and allow for a large class of coagulation rate kernels, with the main restriction being boundedness from above and below by certain weight functions. The weight functions depend on two powerlaw parameters, and the assumptions cover, in particular, the commonly used free molecular and diffusion limited aggregation coagulation kernels. Our main result is to show that the two weight function parameters already determine whether there exists a stationary solution under the presence of a source term or not. In particular, we find that the diffusive kernel allows for stationary solutions while there can be no such solutions for the free molecular kernel. The argument to prove the non-existence of solutions relies on a novel powerlaw lower bound, valid in the appropriate parameter regime, for the decay of stationary solutions with a constant flux. We obtain optimal lower and upper estimates of the solutions for large cluster sizes, and prove that the solutions of the discrete model behave asymptotically as solutions of the continuous model.


Introduction
Atmospheric cluster formation processes [15], where certain species of the gas molecules (called monomers) can stick together and eventually produce macroscopic particles, are an important component in cloud formation and radiation scattering and, hence, also in quantitative understanding of weather and climate. However, a reliable, accurate control of these phenomena is still lacking. In fact, as discussed in [4], it is the largest single source of uncertainty in the related practical predictions.
The above cluster formation processes are modelled with the so-called General Dynamic Equation (GDE) [15]. Under atmospheric conditions, the particle clusters are often aggregates of various molecular species and formed by collisions of several different monomer types, cf. [34,35] for more details and examples. Accordingly, in the GDE one needs to label clusters not only by the total number of monomers in them but also by counting each monomer type. This results in multicomponent labels for the concentration vector, with nonlinear interactions between the components. Another feature of the GDE which has been largely absent from previous mathematical work on coagulation equations, is the presence of an external monomer source term. Such sources are nevertheless important for atmospheric phenomena: for more details about the chemical and physical origin and relevance of the sources we refer to [10,21,24].
In this work, we focus on the effect the addition of a source term has on solutions of standard one-component coagulation equations. This is by no means to imply that multicomponent coagulation equations would not have interesting new mathematical features but these will be the focus of a separate work. Here, we consider only one species of monomers, and we are interested in the distribution of the concentration of clusters formed out of these monomers. Let n α ≥ 0 denote the concentration of clusters with α ∈ N + monomers.
Considering the regime in which the precise spatial structure and loss of particles by deposition are not important, the GDE yields the following nonlinear evolution equation for the concentrations n α : ∂ t n α = 1 2 0<β<α K α−β,β n α−β n β − n α β>0 K α,β n β + β>0 Γ α+β,α n α+β − 1 2 0<β<α Γ α,β n α + s α . (1.1) The coefficients K α,β describe the coagulation rate joining two clusters of sizes α and β into a cluster of size α + β, as dictated by mass conservation. Analogously, the coefficients Γ α,β describe the fragmentation rate of clusters of size α into two clusters which have sizes β and α − β. We denote with s α the (external) source of clusters of size α. In applications, typically only monomers or small clusters are being produced, so we make the assumption that the function α → s α has a bounded, non-empty support. In the following, we make one further simplification and consider only cases where also fragmentation can be ignored, Γ α,β = 0; the reasoning behind this choice is discussed later in Sec. 1.1. An overview of the currently available mathematical results for coagulation-fragmentation models can be found in [7,22]. Therefore, we are led to study the evolution equation ∂ t n α = 1 2 β<α K α−β,β n α−β n β − n α β>0 K α,β n β + s α . (1.2) In this paper, we are concerned with the existence or nonexistence of steady state solutions to (3.25) for fairly general coagulation rate kernels K, including in particular the standard kernels discussed in Sec. 1.1. The source is here assumed to be localized on the "left boundary" of the system which have small cluster sizes. Such source terms often lead to nontrivial stationary solutions towards which the time-dependent solutions evolve as time increases. These stationary solutions are nonequilibrium steady states since they involve a steady flux of matter from the source into the system. Our main result is to address the question of existence of such stationary solutions to (3.25). We prove that for a large class of kernels-including in particular the diffusion limited aggregation kernel given in (1.9)-stationary solutions to (3.25) yielding a constant flux of monomers towards clusters with large sizes exist. On the contrary, for a different class of kernels-including the free molecular coagulation kernel with the form (1.7)-stationary solutions to (3.25) yielding a constant flux of monomers towards larger cluster sizes do not exist.
In the case of collision kernels for which stationary nonequilibrium solutions to (3.25) exist, we can even compute the rate of formation of macroscopic particles, which we identify here with infinitely large particles, from an analysis of the properties of these stationary solutions, cf. Section 2.1. We find that in this case the main mechanism of transport of monomers to large clusters corresponds to coagulation between clusters with comparable sizes, cf. Lemma 6.1, Section 6.
The non-existence of such stationary solutions under a monomer source is at the first sight somewhat counterintuitive since our result will cover systems for which the dynamics is known to be well-posed without the source term. Hence, one needs to explain what will happen at large times to the monomers injected into the system. Our results imply that a constant current with the second class of kernels is incompatible with the assumption that the loss term β>0 K α,β n β in (3.25) is finite. Therefore, we are led to conclude that for such kernels the aggregation of monomers with large clusters is so fast that it cannot be compensated by the constant addition of monomers described by the injection term s α . Moreover, we conjecture that the time-dependent solution decays to zero in any fixed finite range of cluster sizes, including also the sites at which monomers are injected into the system. The solution would then weakly converge to zero as t → ∞, even though zero is not a stationary solution.
The free molecular kernel derived from kinetic theory is commonly used for microscopic computations involving aerosols [34] but, as we prove here, there then cannot be any stationary distributions, at least not without an addition of a fragmentation or a deposition term. On the other hand, atmospheric experiments suggest that, as long as the sources and other atmospheric conditions remain constant, the aerosol size distribution will be nonzero and stationary. Based on our results, in order to explain the experimental observations at all scales, it seems likely that some additional physical phenomena should be included in the evolution equation. These could include sedimentation of large particles or the onset of different aggregation effects for large particles in contrast to the free molecular aggregation of the smallest particles. In numerical simulations focusing on large particles, a diffusive coagulation kernel and a loss term (dry deposition) have been used, and indeed these simulations yield a quasistationary picture which matches the observations reasonably well, apart from small cluster sizes [25,Fig. 4].
In this paper we consider, in addition to the stationary solutions of (3.25), also the stationary solutions of the continuous counterpart of (3.25), (1.3) In fact, we will allow f and η in this equation to be positive measures. This will make it possible to study the continuous and discrete equations simultaneously, using Dirac δfunctions to connect f (ξ) and n α via the formula f (ξ)dξ = ∞ α=1 n α δ(ξ − α)dξ. In most of the mathematical studies of the coagulation equation to date, it has been assumed that the injection terms s α and η (x) are absent. In the case of homogeneous kernels, i.e., kernels satisfying K(rx, ry) = r γ K(x, y) (1.4) for any r > 0, the long time asymptotics of the solutions of (1.3) with η (x) = 0 might be expected to be self-similar for a large class of initial data. This has been rigorously proved in [27] for the particular choices of kernels K(x, y) = 1 and K(x, y) = x + y. In the case of discrete problems, the distribution of clusters n α has also been proved to behave in self-similar form for large times and for a large class of initial data if the kernel is constant, K α,β = 1, or additive, K α,β = α + β [27]. For these kernels it is possible to find explicit representation formulas for the solutions of (3.25), (1.3) using Laplace transforms. For general homogenenous kernels construction of explicit self-similar solutions is no longer possible. However, the existence of self-similar solutions of (1.3) with η = 0 has been proved for certain classes of homogeneous kernels K(x, y) using fixed point methods. These solutions might have a finite monomer density (i.e., ∞ 0 xf (x, t) dx < ∞) as in [12,16], or infinite monomer density (i.e., ∞ 0 xf (x, t) dx = ∞) as in [2,3,29,30]. Similar strategies can be applied to other kinetic equations [18,23,28].
Problems like (3.25), (1.3) with nonzero injection terms s α , η (x) have been much less studied both in the physical and mathematical literature. In [8] it has been observed using a combination of asymptotic analysis arguments and numerical simulations that solutions of (3.25), (1.3) with a finite monomer density behave in self-similar form for long times and for a class of homogeneous coagulation kernels, even considering source terms which depend on time following a power law t ω . Coagulation equations with sources have also been considered in [26] using Renormalization Group methods and leading to predictions of analogous self-similar behaviour. In [9], the existence of stationary solutions has been obtained in the case of bounded kernels. Well-posedness of the time-dependent problem for a class of homogeneous coagulation kernels with homogeneity γ ∈ [0, 2] has been proven in [11]. For the constant kernel, also the stability of the corresponding solutions is proven there using Laplace transform methods. Convergence to equilibrium for a class of coagulation equations containing also growth terms as well as sources has been studied in [19,20].
In this paper we study the solutions of (3.25), (1.3) for coagulation kernels satisfying for some c 1 , c 2 > 0 and for all x, y. The weight function w depends on two real parameters: the homogeneity parameter γ and the "off-diagonal rate" parameter λ. The parameter γ yields the behaviour of kernel K under the scaling of the particle size while the parameter λ measures how relevant the coagulation events between particles of different sizes are. However, let us stress that we do not assume the kernel K itself to be homogeneous, even though the weight functions are that.
In the existence result summarized in Theorem 2.3, we will assume |γ + 2λ| < 1, and show that under some additional mild assumptions there exists at least one nontrivial stationary solution to the problem (1.3). In contrast, if |γ + 2λ| ≥ 1, Theorem 2.4 will imply that no such stationary solutions can exist. Note that the parameters γ and λ may be negative or greater than one here.

On the choice of coagulation and fragmentation rate functions
Although we do not keep track of any spatial structure, the coagulation rates K α,β do depend on the specific mechanism which is responsible for the aggregation of the clusters. These coefficients need to be computed for example using kinetic theory and the result will depend on what is assumed about the particle sizes and the processes yielding the motion of the clusters.
For instance, in the case of electrically neutral particles with a size much smaller than the mean free path between two collisions between clusters, the coagulation kernel is (cf. [15]) where V (α) and m(α) are respectively the volume and the mass of the cluster characterized by the composition α. We denote as k B the Boltzmann constant, as T the absolute temperature, and if m 1 is the mass of one monomer, we have above m(α) = m 1 α. In the derivation, one also assumes a spherical shape of the clusters. If the particles are distributed inside the sphere with a uniform density ρ, assumed to be independent of the cluster size, we also have V (α) = ρα. Changing the time-scale we can set all the physical constants to one. Finally, it is possible to define a continuum function K(x, y) by setting α = x, β = y in the above formula. We call this function the free molecular coagulation kernel, given explicitly by It is now straightforward to check that with the parameter choice γ = 1 6 , λ = 1 2 there are c 1 , c 2 > 0 such that (1.5) holds for all x, y > 0. Since here γ + 2λ = 7 6 > 1, the free molecular kernel belongs to the second category which has no stationary state.
Another often encountered example is diffusion limited aggregation which was studied already in the original work by Smoluchowski [32]. Suppose that there is a background of non-aggregating neutral particles producing cluster paths resembling Brownian motion between their collisions. Then one arrives at the coagulation kernel where µ > 0 is the viscosity of the gas in which the clusters move.
As before, we then set V (α) = ρα and define a continuum function K(x, y) by setting α = x, β = y on the right hand side of (1.8). The constants may then be collected together and after rescaling time one may use the following kernel function which we call here diffusive coagulation kernel. In this case, for the parameter choice γ = 0, λ = 1 3 there are c 1 , c 2 > 0 such that for all x, y > 0 (1.5) holds. Since here 0 < γ + 2λ = 2 3 < 1, the diffusive kernel belongs to the first category which will have some stationary solutions.
Several other coagulation kernels can be found in the physical and chemical literature. For instance, the derivation of the free molecular kernel (1.6) and the Brownian kernel (1.8) is discussed in [15]. The derivation of coagulation describing the aggregation between charged and neutral particles can be found in [33]. Applications of these three kernels to specific problems in chemistry can be found for instance in [34].
Concerning the fragmentation coefficients Γ α,β , it is commonly assumed in the physics and chemistry literature that these coefficients are related to the coagulation coefficients by means of the following detailed balance condition (cf. for instance [34]) where ∆G ref,α is the Gibbs energy of formation of the cluster α and P ref is the reference pressure at which these energies of formation are calculated. Since we assume the coagulation kernel to be symmetric, K α,β = K β,α , the fragmentation coefficients then satisfy a symmetry requirement Γ α+β,α = Γ α+β,β for all α, β ∈ N d + . In the processes of particle aggregation, usually the formation of larger particles is energetically favourable, which means that Under this assumption, it follows from (1.10) that Γ α+β,β ≪ K α,β , and then we might expect to approximate the solutions of (3.21) by means of the solutions of (3.25). The description of the precise conditions on the Gibbs free energy ∆G ref,α which would allow to make this approximation rigorous is an interesting mathematical problem that we do not address in the present paper. Therefore, we restrict our analysis here to the coagulation equations (3.25) and (1.3).

Notations and plan of the paper
Let I be any interval such that I ⊂ R + = [0, ∞). We reserve the notation R * for the case I = (0, ∞). We will denote by C c (I) the space of compactly supported continuous functions on I and by C b (I) the space of functions that are continuous and bounded on I. Unless mentioned otherwise, we endow both spaces with the standard supremum norm. Then C b (I) is a Banach space and C c (I) is its subspace. We denote the completion of C c (I) in C b (I) by C 0 (I) which naturally results in a Banach space. For example, then C 0 (R + ) is the space of continuous functions vanishing at infinity and C 0 (I) = C c (I) = C b (I) if I is a finite, closed interval.
Moreover, we denote by M + (I) the space of nonnegative Radon measures on I. We recall that for such a Radon measure compact sets have a finite measure, even though the measure of the whole space can be infinite. Since I is locally compact, M + (I) can be identified with the space of positive linear functionals on C c (I) via Riesz-Markov-Kakutani theorem. For measures µ ∈ M + (I), we denote its total variation norm by µ and recall that since the measure is positive, we have µ = µ(I). Unless I is a closed finite interval, not all of these measures need to be bounded. The collection of bounded, positive measures is denoted by M +,b (I) := {µ ∈ M + (I) | µ(I) < ∞}. We recall that the total variation norm is indeed a norm in the space of complex measures on I and this space is a Banach space which can be identified with the dual space C 0 (I) * = C c (T ) * using the duality mapping ϕ, µ = I ϕ(x)µ(dx). In addition, M +,b (I) is a norm-closed subset of C 0 (I) * but it can also be endowed with the * -weak topology inherited from C 0 (I) * . Since M +,b (I) = ∩ ϕ∈C 0 (I),ϕ≥0 ϕ, · ← (R + ), M +,b (I) is also * -weak closed in C 0 (I) * . Both topologies will appear in the following.
We will use indistinctly η(x)dx and η(dx) to denote elements of these measure spaces. The notation η(dx) will be preferred when performing integrations or when we want to emphasize that the measure might not be absolutely continuous with respect to the Lebesgue measure. In addition, "dx" will often be dropped from the first notation, typically when the measure eventually turns out to be absolutely continuous.
For the sake of notational simplicity, in some of the proofs we will resort to a generic constant C which may change from line to line.
The paper is structured as follows. In Section 2 we discuss the types of solutions considered here and we state the main results. In Section 3 we prove the existence of steady states for the coagulation equation with source in the continuum case (1.3) assuming |γ +2λ| < 1. We prove the complementary nonexistence of stationary solutions to (1.3) for |γ + 2λ| ≥ 1 in Section 4. The analogous existence and nonexistence results for the discrete model (3.25) are collected into Section 5. In Section 6 we derive several further estimates for the solutions of both continuous and discrete models, including also estimates for moments of the solutions. These estimates imply in particular that the only relevant collisions are those between particles of comparable sizes. Finally, in Section 7 we prove that the stationary solutions of the discrete model  The stationary solutions to the discrete equation (3.25) satisfy: where α ∈ N and s α is supported on a finite set of integers. Analogously, in the continuous case, the stationary solutions to (1.3) satisfy where the source term η (x) is compactly supported in [1, ∞). Although we write the equation using a notation where f and η are given as functions, the equation can be extended in a natural manner to allow for measures. The details of the construction are discussed in Sec. 3 and the explicit weak formulation may be found in (2.15). We remark that equation (2.1) can be written as where we define J 0 (n) = 0 and, for α ≥ 1, we set On the other hand, for sufficiently regular functions f equation (2.2) can similarly be written as This implies that the fluxes J α (n) and J(x; f ) are constant for α and x sufficiently large due to the fact that s is supported in a finite set and η is compactly supported, and we prove in Lemma 2.7 that this property continues to hold even when f is a measure. If s α or η x decay sufficiently fast for large values of α or x then J α (n) or J(x; f ) converges to a positive constant as α or x tend to infinity. Given that other concepts of stationary solutions are found in the physics literature, we will call the solutions of (2.1) and (2.2) stationary injection solutions. In this paper we will be mainly concerned with these solutions. The physical meaning of these solutions, when they exist, is that it is possible to transport monomers towards large clusters at the same rate at which the monomers are added into the system.
For comparison, let us also discuss briefly other concepts of stationary solutions and the relation with the stationary injection solutions. One case often considered in the physics literature are constant flux solutions. These are solutions of (2.2) with η ≡ 0 satisfying where J 0 ∈ R + and J(x; f ) is defined in (2.5). For homogeneous kernels K these solutions can be directly obtained from (2.2) by means of some transformations of the domain of integration that were introduced by Zakharov (cf. [36,37]). This method has been applied to coagulation equations in [6]. Alternatively, we can obtain power law solutions of (2.6) using the homogeneity γ of the kernel (cf. (1.4)). Indeed, suppose that f (x) = C c (x) −α for some C 0 positive and α ∈ R. Using the homogeneity of the kernel K we obtain under the assumption that Using (2.6), we then obtain α = (3 + γ)/2 and C c = J 0 G(α) . Therefore, (2.7) holds if and only if |γ + 2λ| < 1. Notice that (2.7) yields a necessary and sufficient condition to have a power law solution of (2.6). However, one should not assume that all solutions of (2.6) are given by a power law; indeed, we have preliminary evidence that there exist smooth homogeneous kernels satisfying (1.5) for which there are non-power law solutions to (2.6).
Finally, let us mention one more type of solutions associated with the discrete coagulation equation (2.1) that have some physical interest. This is the boundary value problem in which the concentration of monomers is given and the coagulation equation (2.1) is satisfied for clusters containing two or more monomers (α ≥ 2). The problem then becomes where c 1 > 0 is given. Notice that if we can solve the injection problem (2.1) for some source s = s 1 δ α,1 with s 1 > 0, then we can solve problem boundary value (2.8) for any c 1 > 0. Indeed, let us denote by N α (s 1 ), α ∈ N, the solution to (2.1) with source s = s 1 δ α,1 . Then equation (2.1) reduces to This implies that 0 < N 1 (s 1 ) < ∞. Then the solution to (2.8) is given by Moreover, if we can solve (2.1) for some s 1 > 0, then we can solve (2.1) for arbitrary values of s 1 . Indeed, the solution of (2.1) with s α =s 1 δ α,1 is given by In this paper we will consider the problems (2.1) and (2.2) in Sections 2 to 6 and the problem (2.6) in Section 7. We will not discuss solutions to (2.8).

Definition of solution and main results
We restrict our analysis to the kernels satisfying (1.5), or at least one of the inequalities there. To account for all the relevant cases, let us summarize the assumptions on the kernel slightly differently here. We always assume that and for all x, y, K(x, y) ≥ 0 , K(x, y) = K(y, x) . (2.10) We also only consider kernels for which one may find γ, λ ∈ R such that at least one of the following holds: there is c 1 > 0 such that for all (x, y) ∈ R 2 and/or there is c 2 > 0 such that for all (x, y) ∈ R 2 * K (x, y) ≤ c 2 x γ+λ y −λ + y γ+λ x −λ . (2.12) The class of kernels satisfying all of the above assumptions includes many of the most commonly encountered coagulation kernels. It includes in particular the Smoluchowski (or Brownian) kernel (cf. (1.9)) and the free molecular kernel (cf. (1.7)).
The source rate is assumed to be given by η ∈ M + (R * ) and to satisfy Note that then always η (R * ) < ∞, i.e., the measure η is bounded.
We study the existence of stationary injection solutions to equation (1.3) in the following precise sense: Definition 2.1 Assume that K : R 2 * → R + is a continuous function satisfying (2.10) and the upper bound (2.12). Assume further that η ∈ M + (R * ) satisfies (2.13). We will say that is a stationary injection solution of (1.3) if the following identity holds for any test function ϕ ∈ C c (R * ): Remark 2.2 Definition 2.1, or a discrete version of it, will be used throughout most of the paper (cf. Sections 2 to 6). In Section 7, we will use a more general notion of a stationary injection solution to (1.3), considering source terms η which satisfy supp η ⊂ [a, b] for some given constants a and b such that 0 < a < b. Then we require that f ∈ M + (R * ) and f ((0, a)) = 0, in addition to (2.14). Note that for such measures we have R * f (dx) = [a,∞) f (dx). The generalized case is straightforwardly reduced to the above setup by rescaling space via the change of variables x ′ = x/a.
The condition f ((0, 1)) = 0 is a natural requirement for stationary solutions of the coagulation equation, given that η ((0, 1)) = 0. As we show next, the second integrability condition (2.14) is the minimal one needed to have well defined integrals in the coagulation operator.
First, note that all the integrals appearing in (2.15) are well defined for any ϕ ∈ C c (R * ) with supp ϕ ⊂ (0, L], because we can then restrict the domain of integration to the set where C L depends on ϕ, γ, and λ. Then, the assumption (2.14) in the Definition 2.1 implies that all the integrals appearing in (2.15) are convergent.
We now state the main results of this paper. The flux of mass from small to large particles at the stationary state is computed in the next lemma for the above measure-valued solutions. In comparison to (2.5), then one needs to refine the definition by using a closed interval for the first integration and an open interval for the second integration, as stated in (2.16) below.
Lemma 2.7 Suppose that the assumptions of Theorem 2.3 hold. Let f be a stationary injection solution in the sense of Definition 2.1. Then f satisfies for any R > 0 (2.16) Remark 2.8 Note that if R ≥ L η , the right-hand side of (2.16) is always equal to J = [1,Lη] xη(dx) > 0. Therefore, the flux is constant in regions involving only large cluster sizes.
Proof: If R < 1, both sides of (2.16) are zero, and the equality holds. Consider then some R ≥ 1 and for all ε with 0 < ε < R choose some Then for each ε we may define ϕ(x) = xχ ε (x) and thus obtain a valid test function ϕ ∈ C c (R * ). Since then (2.15) holds, we find that for all ε (2.17) The first term can be rewritten as follows 1 2 We readily see that the terms involving χ ε on the right hand side tend to zero as ε tends to zero due to the fact that for Radon measures µ the integrals [a−ε,a) dµ and (a,a+ε] dµ converge to 0 as ε tends to zero. Then we obtain from (2.17) Rearranging the terms we obtain which implies (2.16) using a symmetrization argument.
The following Lemma will be used several times throughout the paper to convert bounds for certain "running averages" into uniform bounds of integrals; note that the bound on the right hand side of (2.19) does not depend on R. The function ϕ below is included mainly for later convenience.
The iterated integral satisfies the assumptions of Fubini's theorem, and thus it can be written as an integral over the set Therefore, after using Fubini's theorem to obtain an integral where z-integration comes first, we obtain The integral over z yields ln(b −1 ) > 0, and thus [a,bR]

Existence results: Continuous model
Our first goal is to prove the existence of a stationary injection solution (cf. Theorem 2. 3) under the assumption |γ + 2λ| < 1. This will be accomplished in three steps: We first prove in Proposition 3.6 existence and uniqueness of time-dependent solutions for a particular class of compactly supported continuous kernels. Considering these solutions at large times allows us to prove in Proposition 3.10 existence of stationary injection solutions for this class of kernels using a fixed point argument. We then extend the existence result to general unbounded kernels supported in R 2 * and satisfying (2.10)-(2.12) with |γ + 2λ| < 1. Compactly supported continuous kernels are automatically bounded from above but, for the first two results, we will also assume that the kernel has a uniform lower bound on the support of the source. To pass to the limit including the more general kernel functions, it will be necessary to control the dependence of the solutions on both of the bounds and on the size of support of the kernel. To fix the notations, let us first choose an upper bound L η for the support of the source, i.e., a constant satisfying (2.13). In the first two Propositions, we will consider kernel functions which are continuous, non-negative, have a compact support, and for which we may find R * ≥ L η and a 1 , a 2 such that 0 < a 1 < a 2 and K(x, y) ∈ [a 1 , a 2 ], for (x, y) ∈ [1, 2R * ] 2 . This allows us to prove first that the time-evolution is well-defined, Proposition 3.6, and then in Proposition 3.10 the existence of stationary injection solutions for this class of kernels using a fixed point argument. The proofs include sufficient control on the dependence of the solutions on the cut-off parameters to remove the restrictions and obtain the result in Theorem 2.3.
In fact, not only do we regularize the kernel, but we also introduce a cut-off for the coagulation gain term which guarantees that the equation is well-posed and has solutions whose support never extends beyond the interval [1, 2R * ]. To this end, let us choose ζ R * ∈ C (R * ) such that 0 ≤ ζ R * ≤ 1, ζ R * (x) = 1 for 0 ≤ x ≤ R * , and ζ R * (x) = 0 for x ≥ 2R * . We then regularize the time evolution equation (1.3) as (3.1) As we show later, this will result in a well-posedness theory such that any solution of (3.1) has the following property: f (·, t) is supported on the interval [1, 2R * ] for each t ≥ 0. Let us also point out that since we are interested in solutions f such that f ((0, 1) , t) = 0, the above integral (0,x] (· · ·) can be replaced by [ Assumption 3.1 Consider a fixed source term η ∈ M + (R * ) and assume that L η ≥ 1 satisfies (2.13). Suppose R * , a 1 , a 2 , and T are constants for which R * > L η , 0 < a 1 < a 2 , and T > 0. Suppose K : R 2 * → R + is a continuous, non-negative, symmetric function such that K(x, y) ≤ a 2 for all x, y, and we also have We will now study measure-valued solutions of the regularized problem (3.1) in an integrated form. To this end, we use a fairly strong notion of continuous differentiability although uniqueness of the regularized problem might hold in a larger class. However, since we cannot prove uniqueness after the regularization has been removed, it is not a central issue here.
We also drop the normed space Y from the notation if it is obvious from the context, in Clearly, if f ∈ C 1 ([0, T ], S; Y ), the functionḟ above is unique and it can be found by requiring that for all t ∈ (0, T ) and then taking the left and right limits to obtain the valuesḟ (0) andḟ (T ). What is sometimes relaxed in similar notations is the existence of the left and right limits.
Remark 3.4 Note that for any such solution f , automatically by continuity and compactness of [0, T ] one has 3) Thus the derivate on the left hand side of (3.2) is defined in the usual sense and, in fact, it is equal to R * ϕ (x, t)ḟ (dx, t). In addition, there is sufficient regularity that after integrating (3.2) over the interval [0, t] we obtain We can define also weak stationary solutions of (3.1). It is straightforward to check that if f 0 (dx) is chosen as f in Definition 3.5, then setting f (dx, t) = f 0 (dx) yields a constant solution satisfying Definition 3.3.
Definition 3.5 Suppose that Assumption 3.1 holds. We will say that f ∈ M + (R * ), satisfying f ((0, 1) ∪ (2R * , ∞)) = 0 is a stationary injection solution of (3.1) if the following identity holds for any test function ϕ ∈ C c (R * ): Proposition 3.6 Suppose that Assumption 3.1 holds. Then, for any initial condition f 0 sat- 6) and the following estimate holds Remark 3.7 We remark that the lower estimate K(x, y) ≥ a 1 > 0 will not be used in the Proof of Proposition 3.6. However, this assumption will be used later in the proof of the existence of stationary flux solutions.
clearly have f ∈ X R * if and only if ϕ(x)f (dx) = 0 for all ϕ ∈ C 0 (R * ) whose support lies in (0, 1) ∪ (2R * , ∞). Therefore, X R * is a closed subset both in the * −weak and norm topology of C 0 (R * ) * . For the rest of this proof, we endow X R * with the norm topology which makes it into a complete metric space. We look for solutions f in the subset X : By the uniform limit theorem, also X is then a complete metric space.
We now reformulate (3.1) as the following integral equation acting on X R * : we define for and using this we obtain a measure Notice that in the above, the definition (3.8) indeed is pointwise well defined and yields a function (x, s) → a [f ] (x, s) which is continuous and non-negative for any f ∈ X. Moreover, we claim that if f ∈ X, then (3.9) defines a measure in M + (R * ) for each t ∈ [0, T ], and we have in addition T [f ] ∈ X. The only non-obvious term is the term on the right-hand side containing We first explain how this term defines a continuous linear functional on C c (R * ). Define g(x, s) = Here the right-hand side of (3.10) is well defined since f (·, s) ∈ X R * for each s ∈ [0, t] . Moreover, this operator defines a continuous linear functional from C c (R * ) to R, and thus is associated with a unique positive Radon measure. Finally, if ϕ(x) = 0 for 1 ≤ x ≤ 2R * , then g(x + y, s)ϕ (x + y) = 0 or x + y < 1, which implies that the right hand side of (3.10) is zero. Therefore, the measure belongs to X R * for all t. Continuity in t follows straightforwardly. The operator T [·] defined in (3.9) is thus a mapping from C([0, T ], X R * ) to C([0, T ], X R * ) for each T > 0. We now claim that it is a contractive mapping from the complete metric space to itself if T is sufficiently small. This follows by means of standard computations using the assumption K (x, y) ≤ a 2 , as well as the inequality |e −x 1 − e −x 2 | ≤ |x 1 − x 2 | valid for Therefore, there exists a unique solution of f = T [f ] in X T assuming that T is sufficiently small. Notice that f ≥ 0 by construction.
In order to show that the obtained solution can be extended to arbitrarily long times we first notice that if f = T [f ], then f ∈ C 1 ([0, T ] , X R * ) and the definition in (3.9) implies that f satisfies (3.1). Integrating this equation with respect to the x variable, we obtain the following estimates: whence (3.7) follows. We can then extend the solution to arbitrarily long times T > 0 using standard arguments. After this, the uniqueness of the solution in C 1 ([0, T ] , M +,b (R * )) follows by a standard Grönwall estimate.
Remark 3.8 Notice that using the inequality K(x, y) ≥ a 1 > 0 we can strengthen (3.11) into the estimate Inspecting the sign of the right hand side this implies an estimate stronger than (3.7), namely, .
We now prove that solutions obtained in Proposition 3.6 are weak solutions in the sense of Definition 3.3. Proposition 3.9 Suppose that the assumptions in Proposition 3.6 hold. Then, the solution f obtained is a Weak Solution of (3.1) in the sense of Definition 3.3.
Proof: Multiplying (3.1) by a continuous test function ϕ ∈ C 1 ([0, T ] , C (R * )) with T > 0 we obtain, using the action of the convolution on a test function in (3.10): As mentioned earlier, the left-hand side can be rewritten as t) .
We will use in the following the semigroup notation S (t) for the map where f is the solution of (3.1) obtained in Proposition 3.6. Note that by uniqueness S (t) has the following semigroup property: (3.14) The operators S (t) define a mapping We can now prove the following result: (3.16) Moreover, U M is compact in the * −weak topology due to Banach-Alaoglu's Theorem (cf. [1]), since it is an intersection of a * −weak closed set X R * and the closed ball f ≤ M . Consider the operator S(t) : X R * → X R * defined in (3.13). We now endow X R * with the * −weak topology and prove that S(t) is continuous. Due to Proposition 3.9 we have that (3.4) and subtracting the corresponding equations for f andf , we obtain For the derivation of (3.17), we have used symmetry properties under the transformation x ↔ y: clearly, K (x, y) ϕ (x + y, s) χ {x+y≤R * } (x, y) − ϕ (x, s) − ϕ (y, s) is then symmetric and s) is antisymmetric, and hence their product integrates to zero.
Consider then an arbitrary ψ ∈ C c (R * ). Our goal is to find a test function Given such a function ϕ, equation (3.17) implies Therefore, if such a function ϕ exists for any ψ ∈ C c (R * ), we would find that the estimate at time t, R * ψ (x) (f (dx, t) −f (dx, t)) , will become arbitrarily small if the estimate at time , is made sufficiently small. In particular, this property can be used to prove that for every f t = S(t)f 0 in a * −weak open set U one can find a * −weak open neighbourhood V of f 0 such that for anyf 0 ∈ V one has S(t)f 0 ∈ U . Hence, the * −weak continuity of S (t) would then follow. In order to conclude the proof of the continuity of S(t) in the * −weak topology it only remains to prove the existence of ϕ ∈ C 1 ([0, T ] , C c (R * )) satisfying (3.18). This can be readily seen using a fixed point argument. Notice that our assumptions on K (x, y) and ζ R * (x + y) imply that the support of ϕ ( We next prove that also t → S (t) f 0 is continuous in the * −weak topology. Let t 1 , t 2 ∈ [0, T ] with t 1 < t 2 . Let ϕ ∈ C c (R * ) . Using (3.4) we obtain: Thus using the bound f T < ∞ we obtain where the constant C does not depend on t 1 , t 2 or ϕ. Therefore, the mapping t → S (t) f 0 is continuous in the * −weak topology. We can now conclude the proof of Proposition 3.10. As proven above, for any fixed t, the operator S(t) : U M → U M is continuous and U M is convex and compact when endowed with the * −weak topology. Using Schauder fixed point theorem, for all δ > 0, there exists a fixed pointf δ of S(δ) in U M . In addition, U M is metrizable and hence sequentially compact, and thus we can find a sequence δ n → 0 andf ∈ U M such thatf δn →f in the * −weak topology.
To complete the proof of the Proposition, we now prove that then S(t)f =f for all t. By definition S(0)f =f . Fix t > 0, and let r n ∈ [0, 1) denote the fractional part of t δn . Then to each n, there is a natural number m n ≥ 0 such that t = m n δ n + r n δ n . By the semigroup property, for any n then S(t)f δn = S(r n δ n )S(δ n ) mnf δn = S(r n δ n )f δn . Since r n δ n ≤ δ n → 0 as n → ∞, we can set t 2 = r n δ n and t 1 = 0 in (3.20) and conclude that S(r n δ n )f δn →f as n → ∞. On the other hand, by continuity of S(t), we have S(t)f δn → S(t)f . Therefore, S(t)f =f and thusf is a stationary injection solution to (3.1).
Notice that the dependence of the function Φ on x is due to the fact that we are not assuming the kernel K(x, y) to be an homogeneous function.
We use two levels of truncations: where: where A is a large constant independent of ε, that we can take A = 1 when Φ is unbounded and sufficiently large in a way that will be seen in the proof if Φ is bounded. Concerning β we take β = 0 if p ≤ 0 for any γ, β > 0 arbitrary small if p > 0 and γ ≤ 0 and 0 < β < p γ if p > 0 and γ > 0. We then have: The second level of truncation is: where ω R * ∈ C ∞ 0 (R 2 + ) and Notice that, if γ ≤ 0 the truncation in min (x + y) γ , 1 ε in (3.23) does not have any effect, because we are only interested in the region where x ≥ 1 and y ≥ 1, due to the fact that the solutions we construct satisfy f ([0, 1)) = 0.
Using that Since x ≥ 2z/3 in the domain of integration, we obtain where C ε is a numerical constant depending on ε but independent on R * . From Lemma 2.9 it follows that whereC ε is a constant independent on R * . In order to consider the limit of objects defined in R + we extend the measures f ε,R * as zero for x > R * . Then, the estimate (3.30) implies, that taking a subsequence if needed, there exists f ε ∈ M + (R + ) such that f ε ([0, 1)) = 0 and: Let f ε be as in (3.31). Notice that (3.30) implies the estimate: (3.32) Taking the limit in (3.27) we obtain, using that for any (x, y) ∈ (R + ) 2 we have lim R * →∞ ζ R * (x + y) = 1 as well as (3.32) that, for any bounded continuous test function Let δ > 0, z ∈ [0, ∞) and χ δ ∈ C ∞ (R + ) satisfy χ δ (x) = 1, x ≤ z and χ δ (x) = 0, x ≥ z +δ. Choosing a test function ϕ(x) = xχ δ (x) and following the same steps as in the proof of Lemma 2.7 we obtain where c, which is defined in (3.28), is independent of ε. We now observe that (3.23) implies: Combining this estimate with (3.34) as well as the fact that Therefore, we obtain the following estimates for the the measures f ε (dx): whereC is independent on ε. This estimate yields * −weak compactness of the family of measures {f ε } ε>0 in M + (R + ) . Therefore, there exists f ∈ M + (R + ) such that: f εn ⇀ f as n → ∞ in the * −weak topology (3.37) for some subsequence {ε n } n∈N with lim n→∞ ε n = 0. It only remains to take the limit ε n → 0 in (3.33). Suppose that ϕ ∈ C c (R + ) . Then, in the term containing ϕ(x + y) we have that the integrand is different from zero only in a bounded region. Using then that lim ε→0 K ε (x, y) = K (x, y) uniformly in compact sets as well as (3.37) we obtain that the limit of that term is: The terms containing ϕ (x) or ϕ (y) can be treated analogously due to the symmetry under the transformation x ↔ y. We then consider the limit of the term containing ϕ (x) where ϕ ∈ C c (R + ) . Our goal is to show that the contribution to the integral due to regions {y ≥ M } where M is very large, can be made arbitrarily small as M → ∞, uniformly in ε. Suppose that M is chosen sufficiently large, so that the support of ϕ is contained in [0, M ]. We then have the following identity: Given that x ≥ 1 and x is in a bounded region contained in [1, M ] we obtain, using (3.35), the estimate: Using (3.36) we obtain: Therefore, the second term in the right hand side of (3.38) tends to zero as ε → 0. In order to estimate the first term we need to consider separately different ranges of the values of the exponents p and γ.
We claim that for 1 ≤ x ≤ M, y ≥ M the following estimates hold: 1. If γ ≤ 0 and p ≤ 0 we have 2. If γ > 0 and p ≤ 0 we have 40) where χ U is the characteristic function of the set U.
3. If γ ≤ 0 and p > 0 we have 4. If γ > 0 and p > 0 we have The estimate (3.39) follows since (x + y) γ Φ ε x x+y , x is bounded for the considered range of values of x, y, γ and p. The estimate (3.40) is obtained distinguishing the ranges of values in which each of the terms in the minimum are the dominant ones as well as the estimate (3.25). Note that in this case β = 0. To get estimate (3.41) we use the fact that, due to (3.24), The estimate (3.42) follows using (3.24) as well as the fact that since β < γ p we have that min{(x + y) γ , 1 ε } = (x + y) γ if Φ ε ( x x+y , x) = 0. We can now estimate the first term in the right hand side of (3.38) in all the cases. We observe that in the cases (3.39), (3.41), (3.42) we only integrate in regions where min{z γ , 1 ε } = z γ (or the cutoff does not act at all if γ ≥ 0). Then, in these three cases we have that in the region of integration and using the previous Lemma 2.9, we obtain: with b > 0 since |γ + 2λ| < 1. Therefore, this term can be made arbitrarily small taking M → ∞.

Nonexistence result: Continuous model.
Assume that there exists a small δ such that 0 < δ < 1 and the following inequality holds:
Proof: Since F is non-increasing and right-continuous, we have: We first consider the case of a > 0. We use a comparison argument. To this end, we construct an auxiliary function R a with B > 0 to be determined. We choose B in order to have: Therefore, we need to impose: This inequality follows using that, for 0 < δ < 1 sufficiently small (independent of R), R > 1/δ and y ∈ [1, δR] we have: [1,δR] y 1+b f (dy) .
To prove (4.3) we argue by contradiction. Suppose that there exists R 1 ≥ R 0 such that F (R 1 ) < B (R 1 ) a . Then, using that F (R) is decreasing, we obtain that We define Combining (4.2) and (4.6) we obtain that: (4.9) Using (4.8) we obtain: Notice that since F * (R) and B 2 1 (R 1 ) a are continuous functions we have that G is right continuous and (4.5) implies: We define R 2 as: Since G is right-continuous, then G(R 2 ) ≤ 0. From (4.10) and (4.11), G(R 2 ) ≥ G(R − 2 ) ≥ 0. Therefore, necessarily G(R 2 ) = 0. From (4.10) we also have that R 2 > R 1 1−δ and For y ∈ [1, δR 2 ], we have that (R 2 − y) ∈ [R 1 , R 2 ), therefore G(R 2 − y) > 0. This implies: which contradicts (4.9). Then R 2 = ∞ whence G (R) ≥ 0 for all R ≥ R 1 . Therefore: However, this inequality implies that F (R) < 0 for R large enough, but this contradicts the definition of F , henceforth which concludes the proof. We now consider the case a = 0. We construct an auxiliary function with B > 0 to be determined. We choose B in order to have: Therefore, we need to impose: This inequality follows using that, for 0 < δ < 1 sufficiently small, R > 1/δ and y ∈ [1, δR] we have: For R > 1/δ we obtain that [1,δR] is a well-defined positive constant due to b + 1 < a, (4.1) and f = 0. Therefore, choosing we obtain that (4.14) follows.
To prove (4.4) we argue by contradiction. Suppose that there exists R 1 ≥ R 0 such that F (R 1 ) < B log(R 1 ). Then, using that F (R) is decreasing, we obtain that Combining (4.2) and (4.13) we obtain that: Using (4.15) we obtain: Notice that since F * (R) and 3B log(R 1 ) are continuous functions we have that G is right continuous and (4.5) implies: We define R 2 as: Using the same reasoning as in the case a < 0 we obtain that R 2 = ∞ whence G (R) ≥ 0 for all R ≥ R 1 . Therefore: However, this inequality implies that F (R) < 0 for R large enough, but this contradicts the definition of F , henceforth which concludes the proof.
Assume that there exists a small δ such that 0 < δ < 1 and the following inequality holds: where R 0 > 1/δ and C > 0. Then for some constant B > 0.
Proof: Since F is non-decreasing and right-continuous, we have: We now use a comparison argument. To this end, we construct an auxiliary function where C ε > 0 is a small constant to be determined. We choose C ε in order to have Therefore, we need to impose: This inequality follows using that, for 0 < δ < 1 sufficiently small (independent of R), for ε > 0 sufficiently small and for y ∈ [1, δR] we have: where A > 0. For R > 1/δ we obtain that [1,δR] y 1+b f (dy) ≤ D, is a well-defined positive constant due to b + 1 < a, (4.19) and f = 0. Therefore, choosing A = C Da and C ε = εA we obtain that (4.24) follows. Next we will prove (4.21). We define Combining (4.20) and (4.23) we obtain that: Since F is increasing and F (R 0 ) > 0, then F (R) > 0 for all R ≥ R 0 . Moreover since F * is continuous then given a small δ > 0, there exists ε > 0 such that Since F * (R) is continuous we have that G is right continuous and (4.22) implies: We define R 2 as: (4.28) For y ∈ [1, δR 2 ], we have that (R 2 − y) ∈ [R 1 , R 2 ), therefore G(R 2 − y) > 0. This implies: which contradicts (4.25). Then R 2 = ∞ whence G (R) ≥ 0 for all R ≥ R 0 . Therefore:

Proof of Theorem 2.4 (non-existence).
We argue by contradiction. Suppose that f ∈ M + (R * ) satisfies f ((0, 1)) = 0 as well as (2.14) and it is a stationary injection solution of (1.3) in the sense of Definition 2.1. Then, from Lemma 2.7 and using also that f ((0, 1)) = 0 we obtain (4.29) Then we introduce a function J : R + → R + defined by means of Suppose that η is different from zero. Then (4.29) implies that J (L η ) = R + xη (dx) > 0. If (γ + 2λ) ≥ 1, we define a := γ + λ and b := −λ, else, if (γ + 2λ) ≤ −1, we define a := −λ and b := γ + λ. The assumption |γ + 2λ| ≥ 1 becomes a − b ≥ 1 in both cases. By assumption (cf. (2.14)) we have: We now prove that the main contribution to the integral in (4.30) as R → ∞ is due to the portion of the region of integration where x is close to R and y is order one. To this end, given δ > 0 small, we define the domains: We then write: We estimate first J 2 (R) for large values of R. Using (2.12) we obtain: Using that (a − b) > 0 we obtain that in the region D (2) δ we have x a y b ≤C δ y a x b . Therefore: 1+δ , ∞ , whence: Given that (a − b) ≥ 1 we obtain, taking into account (4.32): Moreover, using again (4.32), it follows that This implies that the contribution due to J 2 vanishes in the limit R → ∞, namely lim R→∞ J 2 (R) = 0.
Therefore, (4.31) implies that We now notice that in Σ R ∩ D (1) δ we have x ≤ R. In this region we have also y a x b ≤ δ |a−b| x a y b . Combining (2.12) and the fact that x ≤ R we obtain: Notice that, for R > 1/δ, for R ≥ R 0 with R 0 large enough. The rest of the proof is divided into two cases: a ≥ 0 and a < 0. Suppose first that a ≥ 0. Due to (4.32) we may define the function: Note that the function R → F (R) is right continuous, i.e. F (R) = F (R + ) = lim ρ→R + F (ρ) . Moreover, F is non-increasing and F (R) ≥ 0, for all R ≥ 0. Using (4.34) we can rewrite (4.33) as: From Lemma 4.1, it then follows: and for some constant B > 0.
In the case where a > 0, we use (4.32) and (4.35) to obtain: x a f (dx) + B .
By taking the limit R → ∞ we obtain that B ≤ 0 which leads to a contradiction.
In the case where a = 0, (4.36) yields By taking the limit R → ∞ we obtain using (4.32) that the left-hand side converges to zero, while the right hand-side diverges, which leads to a contradiction. Suppose now that a < 0. We define the function: The function R → F (R) is right continuous and non-decreasing. Since f = 0, then F (R) > 0, for all R ≥ R 0 , for R 0 large enough. Using (4.33) we can rewrite (4.37) as:

From Lemma 4.2, it follows that
where we used (4.39). Since a < 0, then R a → 0 as R → ∞, which implies that B ≤ ε, leading to a contradiction.
5 Existence and non-existence results: Discrete model
We consider coagulation kernels K α,β : N 2 + → N + defined on the integers satisfying the same conditions as before: and Similarly to the continuous case, we will try to construct steady states for the coagulation equation (5.1) yielding the transfer of particles to infinity. More precisely, we consider stationary injection solutions to the discrete coagulation equation (5.1).
Proof: The proof of this statement relies on classical arguments of the theory of ordinary differential equations. We just outline the main steps. To simplify the notation we define Then, we can rewrite (5.8) as ∂ t n α = g α (n 1 , . . . , n R * ), with initial condition n α (0) . We observe that the functions g α are polynomials, therefore they are locally Lipschitz continuous functions. Thus, due to the Picard-Lindelöf theorem there exists a unique solution differentiably continuous {n α (t)} α∈I on a maximal time interval [0, T * ). Moreover, since K α,β ≥ 0, s γ ≥ 0 by assumption and n α (0) ≥ 0 it easily follows that n α ≥ 0 in [0, T * ) for any α = 1, . . . , R * . The fact that the solutions of (5.8) are globally defined in time follows from the fact that We define next time-dependent solution and stationary injection solution to (5.8) in the weak sense.
Definition 5.5 Assume that 1 < R * < ∞, T > 0 and that K : N 2 + → R + is a continuous function satisfying (5.3) and (5.5). Assume further that s = {s α } α∈N + satisfies (5.2). Let {n α (0)} α∈N + be the initial condition. We will say that • {n α (t)} α∈N + with n α : [0, T ] → R + continuously differentiable for any α ∈ I is a timedependent solution of (5.8) if the following identity holds for any test function {ϕ α } α∈I such that ϕ α : [0, T ] → R is continuously differentiable for any α: (5.10) • {n α } α∈I ∈ R R * is a stationary injection solution of (5.8) if the following identity holds for any test function {ϕ α } α∈I ∈ R R * : Proof: We multiply n by a test function ϕ α (t) and add in α. Then: where c 0 = β≤R * s β ϕ β . Notice that a 1 > 0 because we assume that (5.4) holds. We then obtain the invariant region with M ≥ 2c 0 a 1 . Moreover, U M is compact. Consider the operator S(t) : R R * → R R * defined by n α (t) = S(t)n α (0). This operator is continuous by standard continuity results on the initial data for the solutions of ODEs (cf. [5]). Since the functions n α (t) solve a first order ODE, they are also continuous in time. Then, the mapping t → S(t)n α (0) is continuous.
We can now conclude the proof of Proposition 5.7. The operator S(t) : U M → U M is compact and the set U M is convex. Then, Brouwer's Theorem (cf. [13]) implies that for all δ > 0, there exists a fixed-pointf δ of S(δ) in U M . Using the semi-group property, we have that if S(δ)f δ =f δ then S(nδ)f δ =f δ for all n ∈ N. Choose δ n = t/n. By letting n → ∞, we obtainf δn →f , for somef ∈ M + ([1, R * ]). Then by continuity, S(t)f δn → S(t)f . By the semi-group property, S(t)f = S(nδ n )f =f , which implies thatf is a stationary injection solution to (5.8).
We now prove the Theorem 5.2. Proof of Theorem 5.2 (existence). We just sketch the argument since it is an adaptation of Theorem 2.3. We can rewrite the kernel K α,β = K(α, β) in the form (3.21) where now x, y ∈ N + . The function Φ(s, x) is defined in a subset of the rational numbers contained in the interval (0, 1) and satisfies (3.22) in this domain of definition. We then define the kernel K ε (x, y) as in (3.23) and K ε,R * (x, y) as in (3.26). Hence, using Proposition 5.7 there exists a stationary injection solution n ε,R * satisfying for any test function ϕ : N + → R compactly supported. Choosing ϕ α = αψ α we obtain Symmetrizing we arrive at Let us assume that ψ(α) = 0 for α ≥ R * .
The indicator function defines a range for α and β given by α < z and β > z − α, which leads to We now integrate the equation in z and obtain: where we used (5.15). We can then argue as in the proof of (3.30) to obtain α< 2R * 3 n ε,R * (α) ≤C ε .
This can be made arguing exactly as in the proof of Theorem (2.3) distinguishing the cases (3.39)-(3.42) and replacing the integrals by sums. Moreover, taking the limit of (5.18) as ε → 0 we arrive at: which implies (5.6) using that −1 < γ + 2λ < 1 .
Remark 5. 8 We notice that in the paper [31] it has been proved that there exists a unique stationary solution of a problem that can be reformulated as a solution of (5.1) for the explicit kernel K α,β = αβ.
Proof: First we notice that since K satisfies (5.5) and K is a continuous interpolation of K, then K satisfies (2.12). Also, η satisfies (2.13) with L η = L s . So K and η satisfy the assumptions of Definition 2.1.
Let C ⊂ M + (R + ) be the set of positive bounded Radon measures supported on the natural numbers, i.e., Proof of Theorem 5.3 (non-existence). From Theorem 2.3 there is no stationary injection solution to (1.3) in the sense of Definition 2.1. In particular, from Lemma 5.9 there is no solution in the subset of discrete measures C , which concludes the proof.

Estimates and regularity
In order to define upper and lower estimates for the measure f we need detailed estimates for the fluxes J defined in (2.5). To this end we introduce some auxiliary notation. We define On the other hand, for any δ > 0, we introduce a partition of (R 2) and we then define for j = 1, 2, 3 3) The following Lemma will be used to prove that the contribution to the integral defining the fluxes due to the points contained in Σ 1 (δ) and Σ 3 (δ) are small for δ sufficiently small.
In the case where γ + 2λ < 0 exchanging the exponents γ + λ and −λ yields and the right-hand side converges to zero as θ goes to ∞.
In the case of region Σ 3 we have that Integrating (6.3) in z from R to 2R we obtain Notice that in the region of integration we have R/4 ≤ x ≤ z ≤ 2R, if δ is small enough.
In this Section we use the assumptions of Theorem 2.3 as stated next which guarantee the existence of a stationary injection solution f in the sense of Definition 2.1.
Integrating (6.11) with respect to z in [R, 2R], using the upper estimate (6.8) as well as Lemma 6.1 we obtain that for δ > 0 sufficiently small depending only on γ, λ and on the constants c 1 , c 2 in (2.11)-(2.12), the following chain of inequalities holds A simple geometrical argument shows that there exists a constant b > 0 depending only on δ (and therefore on γ, λ, c 1 and c 2 ) such that z∈ [R,2R] (Ω z ∩ J 2 (z, δ)) ⊂ ( Moreover, for every (x, y) ∈ ( √ bR, R/ √ b] 2 we have xK(x, y) ≤ CR γ+1 , with C depending only on γ, λ, c 1 and c 2 . Then JR −(γ+3)/2 for R ≥ L η . Thus (6.9) follows after substi- In the next Corollary we obtain the moment estimates for a stationary injection solution, when it exists. Corollary 6.4 Suppose that Assumption 6.2 holds. Then we have the following moment estimates: Proof: a) The boundedness of moments of order µ for µ < γ+1 2 has already been obtained in the proof of Theorem 2.3 in Section 3 equation (3.45). Notice also that a) is an easy consequence of (6.8).
b) Using the lower bound (6.9) and multiplying by z (γ+3)/2 we obtain for some constant C > 0. In particular, for any natural number n satisfying b −n ≤ L η / √ b and for z = b −n we have that C 2 √ J ≤ C (b n+1 ,b n ] x (γ+1)/2 f (dx). Summing in n we finally obtain the result b).
Remark 6.5 Notice that for γ > 1 Corollary 6.4 a) implies that the first moment R + xf (dx) is finite. Therefore the stationary injection solutions can be interpreted in this case as solutions having a finite number of monomers for which the source of monomers η(x) is balanced with the flux of monomers towards infinity. This is closely related to the phenomenon of gelation, which takes place for γ > 1, in which it is possible to have solutions with a finite number of monomers having a flux of monomers towards infinity. Notice that for γ < 1 we have that Proof: Using (2.15) we obtain where α(x) = R + K(x, y)f (dy). (6.16) The continuity and the lower estimate for the kernel K (cf. (2.11) and (2.9)) imply that α(x) ≥ α L 0 > 0, for all x ∈ [ 1 8 , L 0 + 1]. Using an approximation argument as in (4.29), we may use in (6.15) a test function ϕ(x) = χ [x 0 −r,x 0 +r] (x), where χ is the indicator function. Using the boundedness of η we obtain (6.17) We now use a geometrical argument to show that for every x 0 ∈ [ 1 4 , L 0 + 1] and r < ρ 2 there exists a set {ξ ℓ } ℓ∈J ⊂ R + such that #J ≤ L 0 +1 r and {(x, y) | |x + y − x 0 | ≤ r} ⊂ ℓ∈J Q ℓ with Q ℓ = [ξ ℓ − 2r, ξ ℓ + 2r] × [x 0 − ξ ℓ − 2r, x 0 − ξ ℓ + 2r] and ξ ℓ ≤ x 0 − 1 for all ℓ ∈ J. Then using the boundedness of K we obtain where C depends on K and L 0 . Using (6.13) it follows that Q ℓ f (dx) f (dy) ≤ 4A 2 r 2 . Then, since #J ≤ L 0 +1 r , we get [x 0 −r,x 0 +r] f (dx) ≤ 1 α L 0 2 η L ∞ r + 4A 2 C(L 0 + 1)r .

Convergence of discrete to continuous model
We start by defining constant flux solution (cf. Section 2.1).
Definition 7.1 Assume that K : R 2 + → R + is a continuous function satisfying (2.10) and (2.12). We will say that f ∈ M + (0, ∞) , satisfying: is a constant flux solution of (1.3) with η ≡ 0 if the following identity holds for some constant J ≥ 0 and for any z > 0: Our goal is to prove that for a large class of kernels K α,β satisfying (5.3)-(5.5), the stationary injection solutions to the discrete problem (5.1) can be approximated for large cluster sizes by constant flux solutions of the continuous problem (1.3) in the sense of Definition 7.1. To this end, for each R > 0 we construct stationary injection solutions f R to (1.3) with some suitable kernel K R and η R satisfying supp η R ⊂ [1/R, L η /R] (cf. Remark 2.2).
Proof: We first substitute the expressions for f R , K R and η R in the weak formulation (2.15) and perform a change of variables ξ = Rx and θ = Ry. We then obtain an expression where all the terms are multiplied by the same factor R. Using then that for m ∈ N, ζ ε (m) = 1, m = 0 and ζ ε (m) = 0, m = 0 we obtain that the weak formulation of the continuous problem (2.15) reduces to the weak formulation of the discrete problem (5.7).
Theorem 7.4 Let {n α } α a solution of the stationary problem (5.1) in the sense of Definition 5.1.Let f R , K R and η R be as in (7.3), (7.4) and (7.5), respectively. Assume that there exists K ∈ C((R + ) 2 ) such that K R → K as R → ∞ uniformly on compact sets of (0, ∞) 2 . Consider the family of stationary injection solutions defined above (f R ) R>0 . Then for any sequence {R n } n∈N such that lim n→∞ R n = ∞ there exists a subsequence {R n k } k∈N and f ∈ M (0, ∞) (that might depend on the subsequence) such that ∀ϕ ∈ C 0 (0, ∞), and f is a constant flux solution to (1.3) in the sense of Definition 7.1 with J = ∞ α=1 αs α .
Remark 7.5 Note that apriori we may expect that the only constant flux solutions in the sense of Definition 7.1 are power laws. We will see in [14] that there are homogeneous kernels K that satisfy the upper and lower bounds (2.11)-(2.12) for which this is not true. Therefore the limit measure f can be different for different subsequences {f n k } k in (7.6).

Remark 7.6
The assumption K R → K as R → ∞ means that the discrete kernel K α,β behaves like the continuous kernel K for large values of α, β.