Abstract
The continuous coagulation equation with collisional breakage explains the dynamics of particle growth when particles experience binary collisions to form either a single particle via coalescence or two/more particles via breakup with possible transfer of matter. Each of these processes may take place with a suitably assigned probability depending on the volume of particles participating in the collision. In this article, global weak solutions to the continuous coagulation equation with collisional breakage are formulated to the collision kernels and distribution functions admitting a singularity near the origin. In particular, the proof relies on a classical weak \(L^1\) compactness method applied to suitably chosen approximate equations. The question of uniqueness is also contemplated under more restricted class of collision kernels.
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1 Introduction
Coagulation-fragmentation equations (CFEs) are used as models that describe the dynamics of many physical phenomena in which two or more particles can aggregate via a collision between particles to form bigger ones or break into smaller pieces. The coagulation event occurs in different chemical, biological and physical processes such as colloidal aggregation, aggregation of red blood cells and polymerization, for instance. But, in the breakage process, at least two different cases arise that depend on the breakage behaviour of particles. The breakage of particles may happen either due to the collision between a pair of particles named as nonlinear breakage/collision-induced breakage or due to other than the interaction between particles (external forces or spontaneously) known as the linear breakage. The linear breakage may take place due to the particle-wall interaction, chemical reactions or shear fluid whereas a few examples for the occurrence of the nonlinear breakage are the rain droplet breakage, the formation of stars and planets etc. In this article, we have considered the continuous coagulation and nonlinear collisional breakage processes.
The continuous coagulation equation with collisional breakage [5, 7, 22, 25, 27] for the concentration \(g=g(\mu , t)\) of particle of volume \(\mu \in {\mathbb {R}}_{+}:=(0, \infty )\) at time \(t\ge 0\) reads as
with the following initial data
where
and \(E(\mu , \nu )+ {F}(\mu , \nu )=1\) with \(0\le E(\mu , \nu )=E(\nu , \mu ), {F(\mu , \nu )= F(\nu , \mu )} \le 1\) for all \((\mu , \nu ) \in {\mathbb {R}}_+^2\).
The first term, \({\mathcal {B}}_c\) of (1.1) describes the formation of particles of volume \(\mu \) by coalescence, and the last term, \({\mathcal {B}}_b\) of (1.1) represents the birth of particles of volume \(\mu \) due to the collisional breakage. The factor 1/2 appears in \({\mathcal {B}}_c\) and \({\mathcal {B}}_b\) to neglect the double counting for the formation of particles of volume \(\mu \) due to both coalescence and collisional breakage events respectively. The second term \({\mathcal {D}}_{cb}\) of (1.1) shows the death of particles of volume \(\mu \) due to both coalescence and collisional breakage. On the other hand, the term \({\mathcal {D}}_{cb}\) can be expressed as
Here, the collision kernel \(\Psi (\mu , \nu )\) accounts for the rate at which a particle of volume \(\mu \) and a particle of volume \(\nu \) collide which is symmetric with respect to \(\mu \) and \(\nu \) and a non-negative measurable function on \({\mathbb {R}}_{+} \times {\mathbb {R}}_{+}\). Since each collision must result in either coalescence or breakup. Thus, let \(E(\mu , \nu )\) denotes the probability that the two colliding particles of volumes \(\mu \) and \(\nu \) aggregate into a single one of volume \(\mu +\nu \) whereas \({F(\mu , \nu )}\) describes the probability that the two colliding particles of volumes \(\mu \) and \(\nu \) break into two or more daughter particles with possible transfer of mass or elastic collisions between two fragments during the collision. The distribution function \(P(\mu | \nu ; \tau )\) describes the contribution for particles of volume \(\mu \) produced from the collisional breakage event arising from the interaction between a pair of particles of volumes \(\nu \) and \(\tau \) which is also a nonnegative symmetric function in nature with respect to second and third variables, i.e. \(P(\mu | \nu ; \tau )=P(\mu | \tau ; \nu ) \ge 0\). Clearly, this distribution function satisfies
A necessary condition for the mass conservation property during the collisional breakage event is given by
In addition, let us mention another property of the distribution function providing the total number of daughter particles \(N >0\) resulting from the collisional breakage process is
where N is the size independent of \(\nu \) and \(\tau \).
Now, let us consider some special cases arising from the continuous coagulation and collisional breakage equation.
-
If \(E \equiv 1\), then Eq. (1.1) converts into the classical continuous Smoluchowski coagulation equation, see [6, 9, 16, 21].
-
By substituting \(P(\mu |\nu ;\tau )=\chi _{[\mu , \infty )}(\nu )B(\mu |\nu ; \tau ) + \chi _{[\mu , \infty )}(\tau )B(\mu |\tau ; \nu )\) and \(E \equiv 0\) into (1.1), it can easily be seen that (1.1) is transformed into the pure nonlinear breakage equation, see [11,12,13, 17].
Next, it is important to mention some physical properties i.e. moments of the concentration of particles. Let \({\mathcal {M}}_q\) denotes the \(q^{th}\) moment of the concentration g of particles which is defined as
The total number of particles and the total mass of particles are denoted by \({\mathcal {M}}_0\) and \({\mathcal {M}}_1\), respectively. In coagulation process, the total number of particles, \({\mathcal {M}}_0\) decreases whereas in collisional breakage process, \({\mathcal {M}}_0\) increases with time. However, the total mass (volume) of the system may or may not be conserved during the coagulation and collisional breakage processes that depends on the nature of the coagulation kernel (\(E \Psi \)) and breakup kernel (\({F} \Psi \)). It is worth to mention that the negative moments are also very useful in handling the case of some physical singular collision kernels such as Smoluchowski collision kernel for Brownian motion and Granulation kernel for fluidized bed etc. which have been discussed in [8,9,10, 14, 21].
1.1 Literature overview
Before getting into more details of the present work, let us first recall available literature related to the coagulation equation with collisional breakage. There is a vast literature available on the well-posedness of the continuous coagulation and linear breakage equations (CLBEs). In [4, 15, 23, 24], the authors have discussed the existence and uniqueness of solutions to the continuous CLBEs with nonsingular coagulation kernels under different growth conditions on fragmentation kernels, whereas in [3, 9, 10, 16, 21], the existence and uniqueness of solutions to the continuous CLBEs have been established for singular coagulation kernels. More precisely, the existence of self-similar solutions have been discussed in [16, 21] whereas, in [9, 10], respectively, the existence and uniqueness of weak solutions to Smoluchowski coagulation equations and CLBEs have been shown. However, there are a few number of articles in which the collisional breakage or nonlinear fragmentation model have been considered, see [11,12,13, 17]. In these articles, authors have demonstrated scaling solutions as well as asymptotic behaviour of solutions to the pure nonlinear breakage models. Moreover, they have also found analytical solutions for some specific collision and breakup kernels. In 1972, Safronov has proposed a new kinetic model which is known as the continuous coagulation and collisional breakage model, i.e. (1.1)–(1.2), see [22] which has been further studied by Wilkins in [27]. The Eq. (1.1) becomes the continuous nonlinear fragmentation model if \(E\equiv 0\). In 2001, Laurençot and Wrzosek [20] have discussed the existence and uniqueness of weak solutions to the discrete coagulation and collisional breakage model. The existence proof was based on a weak \(L^1\) compactness argument. They have also studied mass conservation, gelation and large time behaviour of solutions. Recently, in [5], Barik and Giri have shown the existence of weak solutions for a particular classes of nonsingular unbounded collision kernels. The main novelty of the present work is to include the singularity for small volume particles in the collision kernel in the existence and uniqueness results to the continuous coagulation and collisional breakage models. Here, the proof of the uniqueness result is motivated by [24].
In order to prove the existence result, we first consider the following basic assumptions on the collision kernel \(\Psi \), initial data \(g_{\text {in}}\), probability E and the distribution function P. Assume that the collision kernel \(\Psi \) satisfies
where \(\sigma \in (0, {1}/{2}), 0 \le \omega < 1\) and some constant \(k \ge 0\).
Next, we assume that the probability E enjoys the following relation for small volume particles
where \( \eta (r) \ge N \ge 2\), for \( r=0, \sigma , 2\sigma \) and N total number of fragments is obtained after the collision between a pair of particles given in (1.4).
We further state the following four assumptions on the distribution function P: recalling \(\eta (r)\) and r from (1.6), we have
Next, for each \(\lambda >0\) and \(\nu \in (0, \lambda )\) and any small measurable subset A of \((0, \lambda )\) with \(|A| \le \delta \), there exists \(\theta _1\) \(\in (0, 2\sigma ]\) such that
Here, |A| denotes the Lebesgue measure of A and \(\chi _{A}\) is the characteristic function on a set A. For \(\nu +\tau > \lambda \) and \(\mu \in (0, \lambda )\) for some \(\tau _2 \in [0,1-\sigma )\), such that P satisfies
For the proof of the uniqueness of weak solutions, we need the following further restriction on collision kernel \(\Psi \) given in (1.5)
Finally, let us suppose that the initial data \(g_{\text {in}}\) satisfies
where \({\mathcal {S}}^{+}\) is the positive cone of the Banach space
endowed with the norm
It can easily be seen that \({\mathcal {S}}\) is a Banach space with respect to norm \(\Vert \cdot \Vert _{{\mathcal {S}}}\), see [8].
This paper is organized as follows: In Sect. 2, some assumptions on the \(\Psi \) and \(g_{\text {in}}\), and the definition of solution together with the main results of this article are stated. By using a weak \(L^1\) compactness method, the existence of weak solutions to (1.1)–(1.2) is established in Sect. 3. In the last section, a uniqueness result of weak solutions to (1.1)–(1.2) is demonstrated for a class of further restricted collision kernels.
2 Preliminaries and results
Let us start this section by formulating the notion of weak solutions to (1.1)–(1.2) by means of the following definition.
Definition 2.1
A solution g of (1.1)–(1.2) is a non-negative continuous function \(g: [0,T)\rightarrow {\mathcal {S}}^+\) such that, for a.e. \(\mu \in {\mathbb {R}}_{+}\) and all \(t\in [0, T)\),
-
(i)
the following integrals are finite
$$\begin{aligned} \int \nolimits _{0}^{t}\int \nolimits _{0}^{\infty }\Psi (\mu , \nu )g(\nu , s)d\nu ds<\infty ,\ \ \text{ and } \ \ \int \nolimits _{0}^{t} {\mathcal {B}}_b(g)(\mu , s)ds < \infty , \end{aligned}$$ -
(ii)
the function g satisfies the following weak formulation of (1.1)–(1.2)
$$\begin{aligned} g(\mu , t)=g_{\text {in}}(\mu )+ \int \nolimits _{0}^{t} \{ {\mathcal {B}}_c(g)(\mu , s)-{\mathcal {D}}_{cb}(g)(\mu , s)+{\mathcal {B}}_b(g)(\mu , s)\} ds, \end{aligned}$$where \(T \in (0, \infty ]\).
Let us consider the following example of collision kernel which satisfies (1.5), see [1]:
Next, we consider the following distribution function
By inserting the above value of P into (1.4), it can easily be observed that we get the finite number of daughter particles, only if \(-1< \theta \le 0\), which is denoted by N and written as \(N = \frac{\theta +2}{\theta +1}\). In particular, for \(\theta =0\), we have \(P(\mu |\nu ;\tau )=\frac{2}{\nu }\), which gives the case of binary breakage, i.e. \(N=2\), once substituted in (1.4). On the other hand, for \(\theta \in (-2, -1]\), we get an infinite number of daughter particles. The result presented in this article deals with the case of finitely many daughter particles. Therefore, it is meaningful to consider \(\theta \in (2\sigma -1, 0]\) for our settings. Now, let us verify (1.7)–(1.9) by considering the above example of distribution function P. First, we check (1.7) as
where \(\eta (r)= \frac{(\theta +2)}{(\theta -r +1)} \).
Next, applying Hölder’s inequality, for \(p>1\), to verify (1.8) as
where \(\theta _1 :=1+\sigma -\frac{1}{p} \in (0, 2\sigma ]\).
In order to check (1.9), we have
where \(\tau _2=-\theta \in [0,1-\sigma )\) and \(k'(\lambda )\ge \frac{\theta +2}{\lambda ^{1+\theta }}\).
Now, we are in a position to state the following existence and uniqueness results:
Theorem 2.2
Assume that (1.5)–(1.9) hold. Then, for \(g_{\text {in}}\in {\mathcal {S}}^+\), there exists a weak solution g to (1.1)–(1.2) on \([0, \infty )\).
Theorem 2.3
Suppose \(g_{\text {in}}\in {\mathcal {S}}^+\). Assume that the collision kernel \(\Psi \) satisfies (1.10) and (1.6)–(1.9) hold. Then, (1.1)–(1.2) has a unique weak solution on \([0, \infty )\).
3 Existence of weak solutions
In this section, we first, construct a sequence of functions \((\Psi _n)\) with compact support for each \(1 < n \in {\mathbb {N}}\), such that
Next, we may argue as in [5, Proposition 1], by using the upper bound of \(\Psi _n ~~ (\le 4kn^{2+\sigma })\) from (3.1) or [26], to show that the truncated equation
with initial data
where
has a unique non-negative solution \(g^n\) for each \(n> 1\). These family of solutions \((g^n)_{n>1}\) lie in \( {\mathcal {C}}^1( [0, \infty );L^1((0,n), d\mu ))\). Additionally, it enjoys the mass conserving property for all \(t > 0\), i.e.
Furthermore, we extend the domain of truncated unique solution \(g^n\) by zero in \({\mathbb {R}}_{+} \times [0, \infty )\) as
for \(n> 1\) and \(n \in {\mathbb {N}}\). For notational convenient, we drop the \(\hat{}\) of \(g^n\).
In the coming section, we wish to apply a classical weak compactness technique for the family of solutions \((g^n)\) to obtain required weak solutions.
3.1 Weak compactness
Here we first show the equi-boundedness of the family \((g^n) \subset {\mathcal {S}}^+ \) by means of the following lemma.
Lemma 3.1
Given any \(T>0\). Then the following holds
where \({\mathcal {P}}(T)\) is a positive constant depending on T, \(\eta (2 \sigma )\) and \(g_{\text {in}}\).
Proof
Let \(t\in [0, T]\) and \(1 < n \in {\mathbb {N}}\). Now, using (3.4), (3.3) and \(g_{\text {in}} \in {\mathcal {S}}^+\), we estimate the following integral as
Integrating (3.2) with respect to the volume variable \(\mu \) from 0 to 1, after multiplying \(\mu ^{-2\sigma }\), we find
Next, we estimate each integral on the right-hand side to (3.7), individually. Using Fubini’s theorem, the transformation \(\mu -\nu ={\mu }'\) and \(\nu ={\nu }'\), then the first term on the right-hand side of (3.7) can be written as
Again, using Fubini’s theorem, the third term on the right-hand side of (3.7) can be simplified as
Applying the repeated applications of Fubini’s theorem, (1.7), the transformation \(\nu -\tau =\nu '\) and \(\tau =\tau '\) and finally replacing \(\nu \rightarrow \mu \) and \(\tau \rightarrow \nu \), the integral \(J_1^n\) can be estimated as
Next, by applying (1.7), replacing \(\nu \), \(\tau \) by \(\mu \), \(\nu \) respectively, using Fubini’s theorem, and the transformation \(\mu -\nu =\mu '\) and \(\nu =\nu '\), one can simplify the second integral \(J_2^n\) as
Substituting the above estimates on \(J_1^n \) and \(J_2^n \) into (3.9), we obtain
Further, inserting (3.8) and (3.10) into (3.7), then applying the symmetry of E, F and \(\Psi _n\), and using Fubini’s theorem, (3.7) can be rewritten as
Using the non-negativity of the first and second integrals on the right-hand side to (3.11) guaranteed from (1.6) and then applying (1.5), (3.4), (3.3) and \( g_{\text {in}} \in {\mathcal {S}}^+\), we obtain
Then, an application of Gronwall’s inequality to (3.12) gives
where
\(a:= k \eta (2\sigma ) 2^{2\omega } \Vert g_{\text {in}}\Vert _{{\mathcal {S}}} \) and \(b:= k \frac{\eta (2\sigma )}{2} 2^{2\omega } \Vert g_{\text {in}}\Vert _{{\mathcal {S}}}^2\). Finally, inserting (3.13) into (3.6), we thus have
This completes the proof of Lemma 3.1. \(\square \)
Next, the equi-integrability of the family of functions \((g^n)_{n > 1} \subset {\mathcal {S}}^+ \) is shown in the following lemma for applying the Dunford-Pettis theorem.
Lemma 3.2
Given any \(T>0\). Then, followings hold true:
-
(i)
for all \(t\in [0,T]\) and for any given \(\epsilon > 0\), there exists a positive constant \(\lambda > \max \bigg \{ \bigg ( \frac{2 \Vert g_{\text {in}} \Vert _{{\mathcal {S}}} }{\epsilon } \bigg )^{1/(1+\sigma )},\frac{ 2 \Vert g_{\text {in}} \Vert _{{\mathcal {S}}} }{\epsilon } \bigg \}\) such that
$$\begin{aligned} \sup _{n> 1} \left\{ \int \nolimits _{\lambda }^{\infty }(1+\mu ^{-\sigma })g^n(\mu , t)d\mu \right\} < \epsilon , \end{aligned}$$ -
(ii)
for a given \(\epsilon > 0\), there exists \(\delta _\epsilon >0\) (depending on \(\epsilon \)) such that, for every small Lebesgue measurable set \(A \subset {\mathbb {R}}_{+}\) with \(|A|\le \delta _\epsilon \), \(n > 1\) and \(t\in [0,T]\),
$$\begin{aligned} \int \nolimits _{A}(1+{\mu }^{-\sigma })g^n(\mu , t)d\mu < \epsilon . \end{aligned}$$
Proof
(i) Let \(\epsilon > 0\) be given. Then, by (3.4), for each \(n > 1 \), \(g_{\text {in}} \in {\mathcal {S}}^+ \) and for all \(t\in [0,T]\), we have
which completes the proof of first part of Lemma 3.2.
(ii) Let \( \epsilon > 0\) be given. For \(A \subset {\mathbb {R}}_{+}\), we can choose \(\lambda \) such that \( \lambda < n\) for all \(n >1\) and \(t \in [0,T]\), and using Lemma 3.2 (i), we have
For \(n >1\), \(\delta \in (0, 1)\) and \(t\in [0,T]\), we define
For \(n>1\) and \(t\in [0,T]\), we estimate the following term, by employing (3.2), Leibniz’s rule, the non-negativity of \(g^n\), Fubini’s theorem and the transformation \(\mu -\nu =\mu '\) and \(\nu =\nu '\), as
We denote the first, second and third terms on the right-hand side of (3.15) by \(J_3^n(t)\), \(J_4^n(t)\) and \(J_5^n(t)\), respectively. Then, we estimate each \(J_i^n(t)\), for \(i=3,4,5\) separately. Let us first consider \(J_3^n(t)\) and evaluate it as
Since \((-\nu +A)\cap (0, \lambda -\nu ) \subset (0, \lambda )\) and \(|(-\nu +A)\cap (0, \lambda -\nu )| \le |-\nu +A| =|A| \le \delta \), then from (3.16), we obtain
Next, by utilizing (1.8), \(J_4^n(t)\) is evaluated, as
Applying Fubini’s theorem, then using the transformation \(\nu -\tau =\nu '\) and \(\tau =\tau '\), (1.5) and Lemma 3.1 into (3.17), we have
Similarly, from repeated applications of Fubini’s theorem, (1.9), \(\nu -\tau =\nu '\) and \(\tau =\tau '\), (1.5) and Lemma 3.1, \(J_5^n(t)\) can be estimated as
Choose \(u>1\) such that \( u\tau _2<1\) and for \( \frac{1+\sigma +\tau _2}{1-\sigma -\tau _2}>1\), then, applying Hölder’s inequality to above estimate, we get
Inserting the estimates on \(J_{3}^n(t)\), \(J_{4}^n(t)\) and \(J_{5}^n(t)\) into (3.15), we obtain
Next, integrating (3.18) with respect to time from 0 to t and taking supremum over all A such that \(A \subset {\mathbb {R}}_{+}\) with |A| \(\le \delta \) and using \( g_{\text {in}} \in {\mathcal {S}}^+\), we estimate
Now, after applying Gronwall’s inequality, it is obtained that
where
From (3.19), we thus have
Finally, adding (3.14) and (3.20), we obtain the desired result. This completes the proof to the second part of Lemma 3.2. \(\square \)
Next, we turn to show the time equicontinuity of sequences \((g^n)_{n >1 }\) and \( (\mu ^{-\sigma }g^n)_{n>1 }\).
3.2 Equicontinuity in time
Set \(\Upsilon ^n(\mu , t):=\mu ^{-\zeta }g^n(\mu , t)\) for \(\zeta =\{0, \sigma \}\), \(\mu \in {\mathbb {R}}_{+}\) and \(t \in [0, T]\). At \( \zeta =0\), this gives \(\Upsilon ^n(\mu , t)=g^n(\mu , t)\) and when \(\zeta =\sigma \), \(\Upsilon ^n(\mu , t):=\mu ^{-\sigma }g^n(\mu , t)\). Let \(T>0\). For any given \(\epsilon > 0\) and \(\phi \in L^{\infty }({\mathbb {R}}_{+})\) there exists \(\lambda =\lambda (\epsilon )>1\) in such way that
where the constant \({\mathcal {P}}(T)\) is defined in Lemma 3.1. Consider \(s, t \in [0, T]\) with \(t \ge s\). Then, for each \(n >1\), by Lemma 3.1 and (3.21), we have
Multiplying by \(\mu ^{-\zeta } \phi (\mu )\) on both sides into (3.2), integrating with respect to \(\mu \) from 0 to \(\lambda \), then using Leibniz’s rule and non-negativity of \(g^n\), we simplify the following term as
Now, using Fubini’s theorem, (1.7) and applying the transformation \(\mu -\nu =\mu '\) and \(\nu =\nu '\) to (3.23), we estimate
By employing Fubini’s theorem, (1.5) and Lemma 3.1 to (3.24), we evaluate as
After combining the estimates in (3.22) and (3.25), finally we have
Fix \(\delta >0\) and take s and t such that \(t-s<\delta \). Then the estimate (3.26) implies the equicontinuity of the family \(\{g^n(t), t\in [0,T]\}\) with respect to time variable t, in the topology \(L^1({\mathbb {R}}_{+}, d\mu )\). Then according to a refined version of the Arzelà-Ascoli Theorem, see [23, Theorem 2.1] or Arzelà-Ascoli Theorem [2, Appendix A8.5], we conclude that there exist a subsequence (\({\Upsilon ^{n}}\)) (not relabeled) and a non-negative function \(\Upsilon \in L^\infty ((0,T);L^1({\mathbb {R}}_{+}, d\mu ))\) such that
for all \(T>0\) and \(\phi \in L^\infty ({\mathbb {R}}_{+})\). This implies that
uniformly for all \(t \in [0,T]\) to some \(\Upsilon \in {\mathcal {C}}([0,T]; w-L^1({\mathbb {R}}_{+}, d\mu ))\), where \({\mathcal {C}}([0, T]; w-L^1 ({\mathbb {R}}_{+}, d\mu ))\) is the space of all weakly continuous functions from [0, T] to \(L^1 ({\mathbb {R}}_{+}, d\mu )\). Applying the weak convergence of \(g^n(t) - g^n(s)\) to \(g(t) - g(s)\) in \(L^1({\mathbb {R}}_{+}, d\mu )\) from (3.27), Lemma 3.1, and setting \(\phi (\mu ) = \text {sgn}(\Upsilon ^n(\mu , t)-\Upsilon ^n(\mu , s) ) \) into (3.26), we can easily improve the space from \(\Upsilon \in {\mathcal {C}}([0,T]; w-L^1({\mathbb {R}}_{+}, d\mu ))\) to \(\Upsilon \in {\mathcal {C}}([0,T]; L^1({\mathbb {R}}_{+}, d\mu ))\).
By considering \(\zeta =0\) and \( \zeta =\sigma \), (3.27) implies that there exist two subsequences \((g^{n})\) and \((\mu ^{-\sigma }g^{n})\) such that
and
For any \( a>0, t\in [0,T], \) since \(g^{n} \rightharpoonup g\), we thus obtain
and
Equation (3.4), the non-negativity of each \(g^{n_k}\) and g, and then \( a \rightarrow \infty \) imply that
\(\int \nolimits _0^{\infty }\mu g(\mu , t) d\mu \le \int \nolimits _0^{\infty }\mu g_{\text {in}}(\mu ) d\mu \) and \(g \in {\mathcal {S}}^+ \).
Next, to show that the limit function g constructed in (3.28) is actually a weak solution to (1.1)–(1.2) in an appropriate sense i.e. as given in Definition 2.1.
3.3 Integral convergence
In the following lemma, we wish to show that the truncated integrals on the right-hand side to (3.2) converge weakly to the original integrals on the right-hand to (1.1).
Lemma 3.3
Let \((g^{n})_{ n \ge 1}\) be a bounded sequence in \({\mathcal {S}}^+\) and \(g\in {\mathcal {S}}^{+}\), where \(\Vert g^n\Vert _{{\mathcal {S}}} \le {\mathcal {P}}(T)\) and \(g^n\rightharpoonup g\) in \(L^1({\mathbb {R}}_{+}, d\mu )\) as \(n\rightarrow \infty \). Then, for each \(\lambda > 1\), we have
Proof
Fix \(\lambda \in (1, n)\) and \(\mu \in (0, \lambda )\). Suppose \(\phi \) belongs to \(L^\infty (0, \lambda )\) with compact support included in \((0, \lambda )\). We argue in the similar manner with little modifications as in Camejo and Warnecke [10] to show that the first two terms such that \( {\mathcal {B}}_{c}^n(g^n) \rightharpoonup {\mathcal {B}}_{c}(g)\) and \({\mathcal {D}}_{cb}^n(g^n) \rightharpoonup {\mathcal {D}}_{cb}(g)\) in \(L^1((0, \lambda ), d\mu )\) as \(n\rightarrow \infty \).
Next, in order to show \( {\mathcal {B}}_{b}^n(g^n) \rightharpoonup {\mathcal {B}}_{b}(g)\ \text {in} \ L^1((0, \lambda ), d\mu )\ \text {as}\ n\rightarrow \infty \), it is sufficient to prove that
as \(n \rightarrow \infty \), for \(\phi \in L^\infty (0, \lambda )\). Let us first simplify the following integral, by using triangle inequality, the repeated applications of Fubini’s theorem and \(\nu -\tau =\nu '\) and \(\tau =\tau '\), as
where \({\mathcal {I}}_1^n\), \({\mathcal {I}}_2^n\) and \({\mathcal {I}}_3^n\) denote the first, second and third terms on the right-hand side, respectively, to (3.32), and
One can infer from (3.1), (1.4) and \({F} \le 1\) that
Then using [18, Lemma 2.9] or the repeated application of [19, Lemma A.2], it can easily be obtained that
Next, applying (3.1), (1.4) and Lemma 3.1, we estimate that
Similarly, one can evaluate
Then, employing (3.34) and (3.35), we have
Finally, applying (3.33) and (3.36) into (3.32), we conclude that
Since \(\lambda >1\), thus (3.37) is true for \(\lambda \in (1, \infty )\). Hence, from (3.36) and (3.37), it is clear that (3.31) holds. This completes the Proof of Lemma 3.3. \(\square \)
Now, we turn to the Proof of Theorem 2.2 with the help of above results.
Proof of Theorem 2.2
Fix \(T>0\) and \( \phi \in L^{\infty }( {\mathbb {R}}_{+} )\). Then, for each \( s \in [0, t]\), employing Lemma 3.3, we have
Again, a repeated application of Fubini’s theorem, the transformation \(\mu -\nu =\mu '\), and \(\nu =\nu '\), (1.5), (1.9), (1.4) and Lemma 3.1 confirm that there exists a positive constant C(T) such that
where
Next, one can easily check that the left-hand side of (3.39) is in \(L^{1}((0, t), ds)\). Then from (3.38), (3.39) and the Lebesgue dominated convergence theorem, we obtain
as \(n \rightarrow \infty \). Since \(\phi \) is arbitrary and (3.40) holds for \( \phi \in L^{\infty }( {\mathbb {R}}_{+} )\) as \(n \rightarrow \infty \), hence, by applying Fubini’s theorem, we get
Then, the definition of \(({\mathcal {B}}_c^{n}-{\mathcal {D}}_{cb}^{n}+{\mathcal {B}}_b^{n})\) yields that
Next, using (3.41), (3.28) and (3.42), we thus obtain
for any \(\phi \in L^{\infty }( {\mathbb {R}}_{+} )\). Hence, for all \(\phi \in L^{\infty }( {\mathbb {R}}_{+} )\), we have \(g(\mu , t)\) is a solution to (1.1)–(1.2). This implies that for almost any \(\mu \in {\mathbb {R}}_{+} \), we have
This completes the proof of the existence Theorem 2.2. \(\square \)
In the next section, the uniqueness of weak solutions to (1.1)–(1.2) is investigated under additional growth condition (1.10) on collision kernel \(\Psi \).
4 Uniqueness of weak solutions
Proof of Theorem 2.3
Let g and h be two weak solutions to (1.1)–(1.2) on \([0, \infty )\), with \( g_{in}(\mu )=h_{in}(\mu )\). Set \( Z:=g-h\). For \(n= 1,2,3,\cdots ,\) we define
Substituting the value of \(g(\mu ,t)-h(\mu ,t)\) by using the Definition 2.1 (iii) into (4.1) and simplifying it further by applying the following identity
using Fubini’s theorem, symmetry of \(\Psi \), the transformation \(\mu -\nu =\mu '\) and \(\nu =\nu '\) and \(\nu -\tau =\nu '\) and \(\tau =\tau '\), we have
Next, let us define the term Q as
Using the definition of Q and the properties of signum function, i.e. \(|\Theta |=\Theta \text{ sgn }(\Theta ) \) into (4.2), we obtain
Due to the non-negativity of the third integral on the right-hand side to (4.3), (1.4) and \(r=\sigma \) into (1.7), (4.3) can be further estimated as
where \(S_i^n(t)\), for \(i = 1, 2,\cdots , 6\) are the corresponding integrals in the preceding line. We now solve each \(S_i^n(t)\) individually. Using properties of the signum function, i.e. \(\text{ sgn }(\Theta _1) \Theta _1 = | \Theta _1|\), we consider following two bounds as
and
Let us first estimate \(S_1^n(t)\), by using (4.5), (1.10), Young’s inequality and the definition of norm, as
where
Similarly, (1.10), (4.6), Young’s inequality and the definition of norm help to evaluate \(S_2^n(t)\) as
where
Again, employing the same argument as before, one can show the finiteness of the integral \(S_3^n(t)\) as
Thus, we obtain
Similarly, one can easily show that \(S_6^n(t) \rightarrow 0\) as \(n \rightarrow \infty \).
Next, \(S_4^n(t)\) can be evaluated, by applying (1.10), (1.7), Young’s inequality and the definition of norm, as
where
Analogous to \(S_4^n(t)\), \(S_5^n(t)\) can be calculated as
where
Now, taking \(n \rightarrow \infty \) to (4.4) and then inserting (4.7), (4.8), (4.10) and (4.11), we have
The inequality (4.12) implies that
Then applying Gronwall’s inequality to (4.13), we obtain
This implies \(g(\mu , t)=h(\mu , t)\) a.e. \(\mu \in {\mathbb {R}}_{+}\). This completes the Proof of the Theorem 2.3. \(\square \)
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Acknowledgements
This work was mainly supported by Science and Engineering Research Board (SERB), Department of Science and Technology (DST), India through the project YSS/2015/001306. In addition, authors would like to thank University Grant Commission (UGC) India for providing the PhD fellowship to PKB through the Grant 6405/11/44.
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Barik, P.K., Giri, A.K. Existence and uniqueness of weak solutions to the singular kernels coagulation equation with collisional breakage. Nonlinear Differ. Equ. Appl. 28, 34 (2021). https://doi.org/10.1007/s00030-021-00696-6
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DOI: https://doi.org/10.1007/s00030-021-00696-6