A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation

In general, the non-conservative approximation of coagulation-fragmentation equations (CFEs) may lead to the occurrence of gelation phenomenon. In this article, it is shown that the non-conservative approximation of CFEs can also provide the existence of mass conserving solutions to CFEs for large classes of unbounded coagulation and fragmentation kernels.


Introduction
An area of substantial concern in engineering and science is the phenomenon of particulate coagulation and fragmentation in chemical and biochemical process systems such as crystallization, fluidization and activated sludge flocculation. The basic reactions taken into consideration are coalescing of two particles to form a larger particle and the breakage of particles into two daughter fragments. The coagulation-fragmentation equations (CFEs) are widely used to describe the evolution of the particle size distribution in the above mentioned processes. The coagulationfragmentation equations are integro-partial differential equations which describe the particle size distribution g(y, t) of particles of volume y > 0 at time t ≥ 0 and read as, see [7,8,14,19,20], ∂g(y, t) ∂t = 1 2 y 0 K(y − z, z)g(y − z, t)g(z, t)dz − ∞ 0 K(y, z)g(y, t)g(z, t)dz F (y, z)g(y + z, t)dz − 1 2 y 0 F (y − z, z)g(y, t)dz := ρ(g) (say), (1.1) with the initial datum g(y, 0) = g in (y) ≥ 0 a.e., (1.2) where ρ(g) := ρ 1 (g) − ρ 2 (g) + ρ 3 (g) − ρ 4 (g). For i = 1, 2, 3, 4, ρ i (g) represent the first, second, third and fourth terms respectively on the right-hand side to (1.1). The coagulation kernel, K(y, z), describes the rate at which particles of volumes y unite with particles of volume z to produce larger particles of volume (y + z).
The kernel F represents the rate at which particles of volume (y + z) breakup into those of volumes y and z. This fragmentation kernel F becomes F (y − z, z) = 0, if y < z. It will be assumed throughout the paper that kernels K and F are nonnegative measurable unbounded functions with K(y, z) = K(z, y) and F (y, z) = F (z, y) for all y > 0 and z > 0, i.e., symmetric.
The first integral ρ 1 (g) on the right-hand side of (1.1) represents the formation of particles of volume y after coalescence of particles of volumes y − z and z, whereas the second integral ρ 2 (g) shows the disappearance of particles of volume y after combining with particles of volume z. The third and fourth integrals, ρ 3 (g) and ρ 4 (g), respectively describe the appearance and disappearance of particles of volume y due to fragmentation events.
An important property of the solution to CFEs (1.1)-(1.2) is known as mass conservation i.e. the total mass of particles remains conserved in the system, i.e.
We know that the total mass of particles is neither created nor destroyed during the coagulationfragmentation events. Therefore, it is expected that the total volume (mass) remains conserved during these events. However, when coagulation kernel increases sufficiently rapidly compared to the fragmentation kernel for large volume of particles, a runaway growth takes place to produce an infinite gel (super particle) in finite time which are removed from the system. Therefore, the total mass (volume) of the system breaks down, this phenomenon is known as gelation, see [17,18] and the time at which this process starts is known as gelation time.  [20] for unbounded kernels K and F , where kernels K and F satisfy the following conditions Several authors have also discussed the existence of mass-conserving solutions, when K(y, z) ≤ A(1 + y + z) for some A > 0, under various assumptions on fragmentation kernels, see [6,15]. Particularly, in [15], Laurençot and Mischler have considered fragmentation kernels as F (y, z) ≤ k(1 + y + z) to the existence of solution to the discrete version of (1.1)-(1.2). In [7], Escobedo et. al. have also shown the existence of mass-conserving solutions to (1.1)-(1.2) under strong fragmentation for bilinear growth condition on coagulation kernels. The proof relies on the weak L 1 Compactness method which is originally introduced by Stewart [19]. In [6], Dubovski and Stewart have discussed the existence of mass conserving solutions to CFEs by using a different approach. The classes of coagulation and fragmentation kernels, which they have considered, also cover the hypotheses (H1) − (H4) considered in this article. In addition, they have shown the uniqueness of solution to CFEs under the following additional restriction on fragmentation kernels Due to the unavailability of a uniqueness result to (1.1)-(1.2) for K and F satisfying (H3) and (H4) respectively, it is unclear that the solution to (1.1)-(1.2) provided by a non-conservative approximation is mass conserving or not. In general, it is known that a non-conservative approximation to Smoluchowski coagulation equations (SCEs) may lead to the gelation phenomenon. In 2004, Filbet and Laurençot [10] have developed a finite volume scheme to demonstrate the occurrence of gelation to SCEs by using a non-conservative approximation. Moreover, they have seen from some experimental results that for the large computational domain the loss of mass is decreased. Therefore, it is expected that the non-conservative approximation may also give a mass conserving solution to SCEs as the upper limit of the domain of truncation tends to infinity. Further, they have provided a mathematical proof for this observation in [9]. Later in 2008, Bourgade and Filbet [4] have generalized the finite volume scheme of [10] to CFEs. By performing some numerical computations, they have also studied the occurrence of gelation to CFEs which appeared due to the finite interval of computations. In case of CFEs, they have concluded similar observations as in [10] that the loss of mass can be decreased by taking a sufficiently large computational domain. This gives a positive hope to achieve a mass conserving solution for CFEs as well by considering a non-conservative truncation. Therefore, the aim of this article is to show mathematically that the mass lost due to the non-conservative truncation converges to zero in the limiting case. This result is not very surprising but certainly important from the point of view of mathematical clarity and correctness. The motivation of the present work is from [2], [4], [9] and [10].
Let us provide a brief plan of the article. In section 2, we mention some hypotheses, definitions, conservative and non-conservative approximations to CFEs. In addition, the existence of massconserving solution to CFEs by considering a conservative approximation is recalled. This section also contain the existence of solutions of the non-conservative truncation of CFEs. These solutions may not satisfy the mass conserving property. At the end of section 2, the main result on the existence of mass conserving solution to CFEs with non-conservative approximation is stated. In section 3, the Dunford-Pettis theorem is applied to the family of solutions of nonconservative truncations to CFEs. Further, equicontinuity argument with respect to time helps us to use a refined version of Arzelà-Ascoli theorem. Moreover, the main existence theorem is proved in this section.

Preliminaries and main result
In order to prove the Theorem 2.3 for the existence of mass conserving solution to (1.1)-(1.2), we consider the following hypotheses.
and the last integral on the right-hand side to (2.1) can also be written in the following way: is the space of all infinitely continuously differentiable functions with compact support. Now we define the characteristic function χ E on a set E as Next, we construct a mass conserving solution relying on the conservative approximation to CFEs (1.1)-(1.2), which is defined as: for a given natural number n ∈ N, we set which gives the following conservative approximation to (1.1)-(1.2) as: with the truncated initial conditioñ Considering (H1) − (H4) and g in n ∈ L 1 1 (0, ∞), we may show as in [19], there exists a unique Again using (H1) − (H4), there is a subsequence (g n k ) of (g n ) such that Hence, g is indeed a solution to (1.1)-(1.2). Consequently, g is mass conserving, i.e Here, the space of all weakly continuous functions from [0, T ] to L 1 There is a possibility to have different approximations to CFEs (1.1)-(1.2) which are not the conservative one i.e. (2.4), see [4]. However, in order to study the gelation phenomenon, a non-conservative approximation of coagulation and a conservative truncation of fragmentation is required which can be constructed as follows: for a given n ∈ N, we define with the truncated initial condition g n (y, 0) = g in n , for y ∈]0, n[, where ρ n := ρ n 1 − ρ n 2 + ρ n 3 − ρ n 4 , and ρ n 1 , ρ n 2 , ρ n 3 and ρ n 4 represent the first, second, third and fourth integrals respectively on the right-hand side to (2.6). Now, it can easily be verified from (2.6)-(2.7) that the total mass may not remain conserved. i.e. From (2.8), we interpret that the total volume (mass) may not be conserved. Similarly, we show the existence and uniqueness of a non-negative solution g n ∈ C([0, ∞); L 1 (0, n)) to (2.6)-(2.7) by using a classical fixed point theorem, but this solution g n does not satisfy mass conserving property.
For the sake of information, we would like to mention that there is also another non-conservative approximation which is different from above. In this approximation, non-conservative form of coagulation and non-conservative form of fragmentation are considered, see [4]. The nonconservative coagulation and non-conservative fragmentation equation is given by In the non-conservative fragmentation part, one can see that the mass of the system with respect to time increases which is not realistic in nature but in case of non-conservative coagulation the total mass decreases in time. Now, the sign of the addition between these two increasing and deceasing masses is very difficult to determine. So, here we have considered the non-conservative coagulation and conservative fragmentation. Now, we are in a position to state the main theorem of this paper.

Theorem 2.3. (Main Theorem)
Assume that hypotheses (H1) − (H4) hold with the initial datum g in ∈ L 1 1 (0, ∞). For n ≥ 1, we denote g n the solution to (2.6)-(2.7). Then there is a subsequence (g n k ) of (g n ) and a solution g to (1.1)-(1.2) such that g n k → g in C([0, T ]; w − L 1 1 (0, ∞)) for each T > 0 satisfying the formulation (2.1). Moreover, it satisfies the mass conserving property, i.e. In order to prove the Theorem 2.3, let g in ∈ L 1 1 (0, ∞), from a refined version of de la Vallée-Poussin theorem (see [5,9]), we are cognizant that there exist two non-negative convex functions σ 1 and σ 2 in C 2 [0, ∞) (space of all twice continuously differentiable functions) such that their derivatives, σ Let us state some properties of non-decreasing convex function with concave derivatives, which are required to prove our main result.
Proof. Let us simplify the following integral by using (2.9) as n 0 (1 + y)g n (y, t)dy = G ω (y, z)K(y, z)g n (y, s)g n (z, s)dzdyds Now, the first integral on the right-hand side of (3.2) can be split into following six sub-integrals From the definition of G ω (y, z) and ω(y) = χ ]0,1[ (y), one can easily show that all the above sub-integrals on the right-hand side to (3.3) are less than or equal to zero, which ensures the non-positivity of the first integral on the right-hand side to (3.2). Next, the second integral of (3.2) can be written as Using the transformation y − z = z ′ , y = y ′ and the symmetry of F to the first integral on the right-hand side to (3.6), this can easily be observed that both integrals on the right-hand side are equal. Therefore, (3.6) becomes In order to prove the next lemma, we require one more important property of convex functions, i.e. the non-decreasing convex function σ 2 satisfies for r, s ∈ (0, ∞) Note that (3.9) follows from (2.12) and the convexity of σ 2 . In the following lemma, the equi-integrability of {g n } n≥1 ⊂ L 1 1 (0, R) is shown by using the convex function σ 2 .