Weak solutions to the continuous coagulation model with collisional breakage

A global existence theorem on weak solutions is shown for the continuous coagulation equation with collisional breakage under certain classes of unbounded collision kernels and distribution functions. This model describes the dynamics of particle growth when binary collisions occur to form either a single particle via coalescence or two/more particles via breakup with possible transfer of mass. Each of these processes may take place with a suitably assigned probability depending on the volume of particles participating in the collision. The distribution function may have a possibility to attain an algebraic singularity for small volumes.


Introduction
Coagulation and breakage processes arise in the different fields of science and engineering, for instance, chemistry (when a matter (water vapor) changes from its gas phase to a liquid phase by condensation process, the molecules in the gas start to come together to form bigger and bigger droplets (dew drops) of the liquid phase), astrophysics (formation of the planets), atmospheric science (raindrop breakup), biology (aggregation of red blood cells), etc. The basic reactions between particles taken into account are the coalescence of a pair of particles to form bigger particles and the breakage of particles into smaller pieces. In general, coagulation event is always a nonlinear process. However, the breakage process may be divided into two different categories on the basis of breakage behaviour of particles, (i) linear breakage and (ii) collisional or nonlinear breakage. Due to the external forces or spontaneously (that depends on the nature of particles), linear breakage occurs whereas the collisional breakage happens due to the collision between a pair of particles. It is worth to mention that the smaller particles are only produced due to the linear breakage process while the collisional breakage allows some transfer of mass between a pair of particles and might produce particles of mass larger than one of each colliding particles. Here, the volume (or size) of each particle is denoted by a positive real number. Now, let us turn to the mathematical model considered in this work. We first take a closed system of particles undergoing binary collisions such that any number of particles are produced by the collision, subject to the constraint that the sum of the volumes of the product particles is equal to the sum of the volumes of the two original particles. The following three possible outcomes may arise in such a process; • if only one particle is produced by the collision, then a coagulation event occurs, • if the collision process gives two particles, then the collision was either elastic or volume (or mass) was exchanged between the original particles, • if three or more particles emerge from the collision, then a breakage event takes place.
Hence, (1.1) can also be written in the following equivalent form ∂g ∂t = C 1 (g) − B 3 (g) + B 1 (g), (1.2) with the following initial data g(z, 0) = g 0 (z) ≥ 0 a.e. (1.3) Here, Φ(z, z 1 ) denotes the collision kernel, which describes the rate at which particles of volumes z and z 1 are colliding and E(z, z 1 ) is the probability that the two colliding particles aggregate to form a single one. If they do not (an event which occurs with probability 1 − E(z, z 1 )) they undergo breakage with possible transfer of mass. In addition, both the collision kernel Φ and collision probability E are symmetric in nature, i.e. Φ(z, z 1 ) = Φ(z 1 , z) and E(z, is a distribution function describing the expected number of particles of volume z produced from the breakage event arising from the collision of particles of volumes z 1 and z 2 . The first integral C 1 (g) and the second integral C 2 (g) of (1.1) represent the formation and disappearance, respectively, of particles of volume z due to coagulation events. On the other hand, the third integral B 1 (g) represents the birth of particles of volumes z due to the collisional breakage between a pair of particles of volumes z 1 − z 2 and z 2 , and the last integral B 2 (g) describes the death of particles of volume z due to collisional breakage between a pair of particles of volumes z and z 1 . The factor 1/2 appears in the integrals C 1 (g) and B 1 (g) to avoid double counting of formation of particles due to coagulation and collisional breakage processes.
The distribution function P has the following properties: (i) P is non-negative and symmetric with respect to z 1 and z 2 , i.e. P (z|z 1 ; z 2 ) = P (z|z 2 ; z 1 ) ≥ 0, (ii) The total number of particles resulting from the collisional breakage event is given by z 1 +z 2 0 P (z|z 1 ; z 2 )dz = N, for all z 1 > 0 and z 2 > 0, P (z|z 1 ; z 2 ) = 0 for z > z 1 + z 2 , (1.4) (iii) A necessary condition for mass conservation during collisional breakage events is z 1 +z 2 0 zP (z|z 1 ; z 2 )dz = z 1 + z 2 , for all z 1 > 0 and z 2 > 0. (1.5) From the condition (1.5), the total volume z 1 + z 2 of particles remains conserved during the collisional breakage of particles of volumes z 1 and z 2 .
Next, let us mention some particular cases of the continuous coagulation and collisional breakage equation. When E(z, z 1 ) = 1, then equation (1.2) becomes the continuous Smoluchowski coagulation equation [2,3,10,11]. Another case taken into consideration is the collision between a pair of particles of volumes z and z 1 that results in either the coalescence of both into of volumes (z + z 1 ) or into an elastic collision leaving the incoming clusters unchanged. In both cases P (z|z; z 1 ) = P (z 1 |z; z 1 ) = 1 and P (z * |z; z 1 ) = 0 if z * / ∈ {z, z 1 } which again, reduces (1.2) into the continuous Smoluchowski coagulation equation with (E(z, z 1 )Φ(z, z 1 )) as the coagulation rate. Now, by substituting E = 0 and P (z|z 1 ; (1.2), it can easily be seen that (1.2) becomes the pure nonlinear breakage model which has been extensively studied in many articles, [5,6,8,12,13]. In these articles, the authors have been considered when a pair of particles collide, one particle fragment into smaller pieces without transfer of masses from other one. The continuous nonlinear breakage equation reads as where Ψ(z, z 1 ) = Ψ(z 1 , z) ≥ 0 is the collisional kernel and B(z|z 1 ; z 2 ) denotes the breakup kernel or breakage function, which represents particle of volume z obtained by collision between particles of z 1 and z 2 and satisfies the following property Finally, we define moments of number density g. Let M r (t) denotes the r th moment of g which is defined as The zeroth and first moments represent the total number of particles and the total mass of particles, respectively. In collisional breakage events, the total number of particles, i.e. M 0 (t), increases whereas M 0 (t) decreases during coagulation events. In addition, it is expected that the total mass of the system remains constant during these events. However, sometimes the mass conserving property breaks down due to the rapid growth of coagulation kernels, (EΦ), compare to the breakage kernels, ([1 − E]Φ). Hence, gelation may appear in the system.
In this work, we mainly address the issue on the existence of weak solutions to the continuous coagulation and collisional breakage equation (1.2)-(1.3). The existence and uniqueness of solutions to the classical coagulation-fragmentation equations have been discussed in several articles by applying various techniques, see [2,3,10,15,16]. However, best to our knowledge, the mathematical theory on the continuous coagulation and collisional breakage equation has not been rigorously studied. Although there are a few articles available which are devoted to (1.2)-(1.3), see [4,13,14,17,19]. This model has been described in [14,19]. In particular, in [13], the existence of mass conserving weak solutions to the discrete version of (1.2)-(1.3) has been shown by using a weak L 1 compactness method. Moreover, they have also studied the uniqueness of solutions, long time behaviour in some particular cases and occurrence of gelation transition. In [4], the structural stability of the continuous coagulation and collisional breakage model is studied by applying both analytical method and numerical experiment. Later in [17], the partial analytical solutions to the discrete (1.2)-(1.3) is studied for the constant collision kernel. Moreover, this solution is also compared with Monte-Carlo simulation. In addition, there are a few articles in which analytical solutions to the continuous nonlinear breakage equations have been investigated for some specific collision kernels only, see [5,6,8,12]. However, in general, it is quite delicate to handle the continuous nonlinear breakage equation mathematically because here the small sized particles have the tendency to fragment further into very small sized clusters which leads to the formation of an infinite number of clusters in a finite time. In order to overcome this situation, we consider a fully nonlinear continuous coagulation and collisional for large classes of unbounded collision kernels and distribution function.
The paper is organized in the following manner: In Section 2, we state some definitions, assumptions and lemmas, which are essentially required in subsequent sections. The statement of main existence theorem is also given at the end of this section. Section 3 contains the rigorous proof of the existence theorem which relies on a weak L 1 compactness method.

Definitions and Results
Let us define the following Banach space S + as We can also define the norms in the following way: Next, we formulate weak solutions to (1.2)-(1.3) through the following definition: Now, throughout the paper, we assume the following conditions on collision kernel Φ, distribution function P , and the probability function E: E satisfies the following condition locally: where N is given in (1.4), (Γ 4 ) for each W > 0 and for z 1 ∈ (0, W ), 0 < α ≤ β < 1 (introduce in (Γ 2 )) and any measurable subset U of (0, 1) with |U | ≤ δ, we have where |U | denotes the Lebesgue measure of U and χ U is the characteristic function of U given by Let us take the following example of distribution function P which satisfies (Γ 4 )-(Γ 5 ).
For ν = 0, this leads to the case of binary breakage and for −1 < ν ≤ 0, we get the finite number of particles, which is denoted by N and written as N = ν+2 ν+1 . But, for −2 < ν < −1, we obtain an infeasible number of particles and for the case of ν = −1, an infinite number of daughter particles are produced. It is clear from (1.4). Now, (Γ 4 ) is checked in the following way: for z 1 ∈ (0, W ) and W > 0 is fixed, For α < 1, and applying Hölder's inequality, we get This implies that In order to verify the (Γ 5 ), for z 1 + z 2 > W and W > 0 is fixed, we have Now we are in the position to state the following existence result: hold and assume that the initial value g 0 ∈ S + . Then,

Existence of weak solutions
In order to construct weak solutions to (1.2)-(1.3), we follow a weak L 1 compactness method introduced in the classical work of Stewart [15].
To prove theorem 2.2, we first write (1.2)-(1.3) into the limit of a sequence of truncated equations obtained by replacing the collision kernel Φ by their cut-off kernels Φ n [15], where for n ≥ 1 and n ∈ N. Here χ A denotes the characteristic function on a set A. Considering (Γ 1 )-(Γ 5 ) and g 0 ∈ S + , for each n ≥ 1, we may employ the argument of the classical fixed point theory, as in [15,Theorem 3.1] or [18], to show that with the truncated initial data g n 0 (z) : has a unique non-negative solutionĝ n ∈ C ′ ([0, ∞); L 1 ((0, n), dz)) s.t. the truncated version of mass conservation holds, i.e.
Now, we extend the truncated solutionĝ n by zero in

Weak compactness
Lemma 3.1. Assume that (Γ 1 )-(Γ 5 ) hold and fix T > 0. Let g 0 ∈ S + and g n be a solution to (3.2)- (3.3). Then, the followings hold true: (iii) for a given ǫ > 0, there exists δ ǫ > 0 (depending on ǫ) such that, for every measurable set Proof. (i) Let n ≥ 1 and t ∈ [0, T ], where T > 0 is fixed. For n = 1, the proof is trivial. Next, for n > 1 and then taking integration of (3.2) from 0 to 1 with respect to z and by using Leibniz's rule, we obtain The first term on the right-hand side of (3.6) can be simplified by using Fubini's theorem and the transformation z − z 1 = z ′ and z 1 = z ′ 1 as Using Fubini's theorem, the third term on the right-hand side of (3.6) can be written as Let us manipulate I n 1 , by changing the order of integrations, using (1.4) and applying the transformation z 1 − z 2 = z ′ 1 and z 2 = z ′ 2 , as Next, simplifying I n 2 , by using Fubini's theorem and (1.4), as Again using Fubini's theorem and applying transformation z − z 1 = z ′ and z 1 = z ′ 1 into (3.9), we get Substituting the estimates on I n 1 and I n 2 into (3.8), we evaluate Inserting (3.7) and (3.10) into (3.6), we obtain d dt Applying (Γ 3 ) to the first and the second integrals and then using the negativity of the first, second and third terms on the right-hand side of (3.11), we have d dt Applying (Γ 2 ), (3.3) and g 0 ∈ S + to (3.12), we obtain d dt [z α z 1 + z β z 1 ]g n (z, t)g n (z 1 , t)dz 1 dz Again, taking integration of (3.13) from 0 to t with respect to time and then applying Gronwall's inequality, we have where (ii) The second part of Lemma 3.1 can be easily proved in similar way as given in Giri et al. [10].
Next, by applying Fubini's theorem twice, (Γ 4 ), (Γ 5 ) and Hölder's inequality for p > 1 such that pτ 2 < 1, we estimate J n 2 as Again repeated application of Fubini's theorem, (Γ 2 ), Lemma 3.1 (i), z − z 1 = z ′ and z 1 = z ′ 1 , we have Gathering the above estimates on J n 1 , J n 2 and inserting them into (3.16), and applying Leibniz's rule, we obtain Integrating the above inequality with respect to t and taking supremum over all U such that U ⊂ (0, W ) with |U | ≤ δ ǫ , we estimate An application of Gronwall's inequality finally gives This shows that sup n {r n (δ ǫ , t)} → 0 as δ ǫ → 0. Hence, from Dunford-Pettis theorem, we have (g n ) n∈N is a relatively compact subset of S + for each t ∈ [0, T ].

Equicontinuity with respect to time in weak sense
By showing the following lemma, we check the time equicontinuity in weak sense of the family {g n (t), t ∈ [0, T ]} in L 1 (R + , dz).
Next, consider the following integral, by using triangle inequality, as Using (3.2), the first integral on the right-hand side to (3.19) can be estimated as × Φ n (z 1 − z 2 , z 2 )g n (z 1 − z 2 , s)g n (z 2 , s)dz 2 dz 1 dzds =: where J n i represents the first, second and third terms, respectively, on the right-hand side to (3.20), for i = 3, 4, 5. Next, J n 3 can be evaluated by applying Fubini's theorem, and using z − z 1 = z ′ and z 1 = z ′ 1 , (Γ 2 ) and Lemma 3.1 (i), we have Similarly, J n 4 can be estimated by using (Γ 2 ) and Lemma 3.1 (i), as Now, J n 5 can be estimated by applying Fubini's theorem twice, and using (1.4) and (Γ 5 ), we obtain Again, applying Fubini's theorem and using z 1 − z 2 = z ′ 1 and z 2 = z ′ 2 , (Γ 2 ) and Lemma 3.1 (i), we estimate (3.21) as t+h t a 0 a 0 Φ n (z 1 , z 2 )g n (z 1 , s)g n (z 2 , s)dz 2 dz 1 ds Inserting estimates on J n 3 , J n 4 and J n 5 into (3.20), we thus obtain Next, using Lemma 3.1 (i), the last term on the right-hand side to (3.19) can be estimated as

Using (3.22) and (3.23) into (3.19), we have
where h is arbitrary. This completes the proof of Lemma 3.2.
Then according to a refined version of the is the space of all weakly continuous functions from [0, T ] to L 1 (R + , dz). This implies that converges uniformly for t ∈ [0, T ] to some g ∈ C w ([0, T ]; L 1 (R + , dz)).
In order to show that B n 1 (g n ) ⇀ B 1 (g) in L 1 ((0, a), dz) as n → ∞, it is sufficient to prove the following integral tends to zero as n → ∞.
where we choose b with n > b > a large enough for a given ǫ > 0, such that Now, let us simplify the following integral by applying Fubini's theorem, as where H n i , for i = 1, 2, 3, 4, are preceding integrals on the right-hand side to (3.28). Each H n i , for i = 1, 2, 3, 4 is evaluated individually. Let us first estimate H n 1 , by using the repeated applications of Fubini's theorem and the transformation z 1 − z 2 = z ′ 1 and z 2 = z ′ 2 , as Next, the following integral can be estimated, by using Lemma 3.1 (i), (1.4) and (Γ 2 ), for each z 2 ∈ (0, a), as Further, H n 2 can be simplified by using Fubini's theorem, and the transformation z 1 − z 2 = z ′ 1 and z 2 = z ′ 2 , as Next, we evaluate the following integral by using (Γ 5 ), (Γ 2 ) and Lemma 3.1 (i), as a) for each z 2 ∈ (0, a). (3.33) Similarly, by using (Γ 5 ), (Γ 2 ) and Lemma 3.1 (i), we estimate the following term, as Let us now consider H n 3 , after implementing Fubini's theorem twice, Again, using (1.4) and (Γ 2 ), we evaluate the following integral, as × g(z 2 , t)[g n (z 1 , t) − g(z 1 , t)]dzdz 2 dz 1 .
We conclude that g is a solution to (1.2)-(1.3) on [0, ∞). This completes the proof of the existence Theorem 2.2.