# A Note on the Volume Conserving Solution to Simultaneous Aggregation and Collisional Breakage Equation

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*Axioms*: Mathematical Analysis)

## Abstract

**:**

## 1. Introduction

- (i)
- $\mathcal{B}(x,y;z)$ is non negative and symmetric with respect to y and z, that is$$\begin{array}{c}\hfill \mathcal{B}(x,y;z)=\mathcal{B}(x,z;y).\end{array}$$
- (ii)
- Volume conservation law$$\begin{array}{c}\hfill {\int}_{0}^{y}x\mathcal{B}(x,y;z)\mathrm{d}x=y\phantom{\rule{1.em}{0ex}}and\phantom{\rule{1.em}{0ex}}\mathcal{B}(x,y;z)=0\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{all}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{1.em}{0ex}}y\le x;\end{array}$$
- (iii)
- Number of particles after fragmentation$$\begin{array}{c}\hfill {\int}_{0}^{y}\mathcal{B}(x,y;z)\mathrm{d}x=\nu (y,z)\le \overline{N}<\infty \phantom{\rule{1.em}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{all}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{1.em}{0ex}}y>0,z>0.\end{array}$$

- (A1)
- $\mathcal{K}(x,y)$ is a non-negative and continuous function on ${\mathbb{R}}_{+}\times {\mathbb{R}}_{+}$.
- (A2)
- $\mathcal{B}(x,y;z)$ is a non-negative, continuous function satisfying the condition$$\begin{array}{c}\hfill {\displaystyle {\int}_{0}^{y}{x}^{-\theta \sigma}\mathcal{B}(x,y;z)\mathrm{d}x\le \Phi \left(y\right),\phantom{\rule{1.em}{0ex}}\mathrm{where}\phantom{\rule{1.em}{0ex}}\Phi \left(y\right)=\eta {y}^{-\theta \sigma},}\end{array}$$

## 2. Existence of Solutions

**Theorem**

**1.**

**Proof.**

- Nonnegativity of the local solution;
- Global existence of the unique solution to the space ${\Psi}_{r,\sigma}^{+}\left(T\right)$.

- If ${x}_{0}\le R$, then $\phi (x,t)>0$ for all $0<x\le R$ and $0\le t<{t}_{0}$. The positivity of the right hand side of (17) implies ${\left.{\partial}_{t}\phi (x,t)\right|}_{\left({x}_{0},{t}_{0}\right)}>0$.
- If ${x}_{0}>R$, we use the property (3) of the breakage function to obtain$$\begin{array}{cc}\hfill {\int}_{0}^{R}{\int}_{{x}_{0}}^{R}\mathcal{K}(y,z)\mathcal{B}({x}_{0},y;z)\phi (y,t)\phi (z,t)\mathrm{d}y\mathrm{d}z& =-{\int}_{0}^{R}{\int}_{{x}_{0}}^{R}\mathcal{K}(y,z)\mathcal{B}({x}_{0},y;z)\phi (y,t)\phi (z,t)\mathrm{d}y\mathrm{d}z\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =0\hfill \end{array}$$Thus, from the Equation (17), we have ${\left.{\partial}_{t}\phi (x,t)\right|}_{\left({x}_{0},{t}_{0}\right)}>0$.

## 3. Conservation of Volume

## 4. Uniqueness Theory

**Theorem**

**2.**

**Proof.**

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Kharchandy, F.W.V.; Das, A.; Thota, V.; Saha, J.; Singh, M.
A Note on the Volume Conserving Solution to Simultaneous Aggregation and Collisional Breakage Equation. *Axioms* **2023**, *12*, 181.
https://doi.org/10.3390/axioms12020181

**AMA Style**

Kharchandy FWV, Das A, Thota V, Saha J, Singh M.
A Note on the Volume Conserving Solution to Simultaneous Aggregation and Collisional Breakage Equation. *Axioms*. 2023; 12(2):181.
https://doi.org/10.3390/axioms12020181

**Chicago/Turabian Style**

Kharchandy, Farel William Viret, Arijit Das, Vamsinadh Thota, Jitraj Saha, and Mehakpreet Singh.
2023. "A Note on the Volume Conserving Solution to Simultaneous Aggregation and Collisional Breakage Equation" *Axioms* 12, no. 2: 181.
https://doi.org/10.3390/axioms12020181