arXiv subjects

Mashkoor Ali

Publications and source records attributed to Mashkoor Ali.

8 recordsLinked to original sources

arXiv2026

Gelation and Positivity of Solutions to the Discrete Oort--Hulst--Safronov Coagulation Equation

Motivated by the recent deterministic approach of Fournier~\cite{F2025} to gelation for the continuous Smoluchowski coagulation equation, we adapt his method to the discrete Oort--Hulst--Safronov (OHS) coagulation system. We show that under a suitable condition on the coagulation kernel, every solution with finite initial mass loses mass in finite time, and we give an explicit bound on the gelation time. We also prove gelation in the critical logarithmic case and provide a sufficient condition for mass conservation. Finally, we study the positivity of solutions and show that, for any positive time, a cluster size has positive concentration if and only if it is at least as large as the smallest cluster present initially.

math.CA↗

arXiv2026

On a Class of Continuous Collision-Induced Breakage Equation

In this work, we establish the existence of mass-conserving weak solutions to a nonlinear collision-induced breakage equation in which binary collisions may trigger particle breakup. The result is proved for a class of product-type collision kernels whose small-size behavior is controlled by a power-law function of the form $\omega_0(x)\le A_1\,x^\ell$, while no growth restriction is imposed on the large-size factor $\omega_\infty$. The qualitative behavior of the solutions depends crucially on the exponent $\ell$ near the origin. Sublinear growth corresponding to $\ell<\tfrac12$ yields existence only on finite time intervals, whereas superlinear growth corresponding to $\ell>\tfrac12$ ensures global-in-time existence.

math.AP↗

arXiv2025

Global Solutions to the Discrete Nonlinear Breakage Equations without Mass Transfer

Global existence of mild solutions to the discrete collisional breakage equations is established for a broad class of collision kernels, without imposing any growth assumptions. In addition, classical solutions are constructed, and uniqueness is proved for an appropriate class of kinetic coefficients and initial data. The large time behavior of solutions is also discussed, and numerical simulations are presented to support the theoretical results.

math.CA↗

arXiv2023

The discrete collision-induced breakage equation with mass transfer: well-posedness and stationary solutions

The discrete collisional breakage equation, which captures the dynamics of cluster growth when clusters encounter binary collisions with possible matter transfer, is discussed in this article. The existence of global mass-conserving solutions is investigated for the collision kernels $a_{i,j}=A(i^α j^β + i^βj^α)$, $i, j \ge 1$, with $α\in (-\infty,1)$, $β\in [α,1]\cap (0,1]$, and $A>0$ and for a large class of possibly unbounded daughter distribution functions. All algebraic superlinear moments of these solutions are bounded on time intervals $[T,\infty)$ for any $T>0$. The uniqueness issue is further handled under additional restrictions on the initial data. Finally, non-trivial stationary solutions are constructed by a dynamical approach.

math.CA↗

arXiv2023

On the discrete Safronov-Dubovskii coagulation equation: well-posedness, mass-conservation and asymptotic behaviour

The global existence of mass-conserving weak solutions to the Safronov-Dubovskii coagulation equation is shown for the coagulation kernels satisfying the at most linear growth for large sizes. In contrast to previous works, the proof mainly relies on the de la Vallee-Poussin theorem [8, Theorem 7.1.6], which only requires the finiteness of the first moment of the initial condition. By showing the necessary regularity of solutions, it is shown that the weak solutions con-structed herein are indeed classical solutions. Under additional restrictions on the initial data, the uniqueness of solutions is also shown. Finally, the continuous dependence on the initial data and the large-time behaviour of solutions are also addressed.

math.AP↗

arXiv2023

Well-posedness to the discrete collision-induced breakage equation and various properties of solutions

A discrete version of the nonlinear collision-induced breakage equation is studied. Existence of solutions is investigated for a broad class of unbounded collision kernels and daughter distribution functions, the collision kernel $a_{i,j}$ satisfiying $a_{i,j} \leq A i j$ for some $A>0$. More precisely, it is proved that given suitable conditions, there exists at least one mass-conserving solution for all times. A result on the uniqueness of solutions is also demonstrated under reasonably general conditions. Furthermore, the propagation of moments, differentiability, and the continuous dependence of solutions are established, along with some invariance properties and the large-time behaviour of solutions.

math.CA↗

arXiv2022

On Classical solution to the Discrete Coagulation Equations with Collisional Breakage

In this article, the existence of global classical solutions to the discrete coagulation equations with collisional breakage is established for collisional kernel having linear growth whereas the uniqueness is shown under additional restrictions on collisional kernel. Moreover, mass conservation property and propagation of moments of solutions are also discussed.

math.AP↗

arXiv2022

Global existence to the discrete Safronov-Dubovskiǐ coagulation equations and failure of mass-conservation

This paper presents the existence of global solutions to the discrete Safronov-Dubvoskiǐ coagulation equations for a large class of coagulation kernels satisfying $Λ_{i,j} = θ_i θ_j + κ_{i,j}$ with $κ_{i,j} \leq Aθ_i θ_j, \ \ \forall \ \ i,j\ge 1$ where the sequence $(θ_i)_{i\geq 1}$ grows linearly or superlinearly with respect to $i$. Moreover, the failure of mass-conservation of the solution is also addressed which confirms the occurrence of the gelation phenomenon.

math.AP↗