Mass-conserving weak solutions to the continuous nonlinear fragmentation equation in the presence of mass transfer
Ram Gopal Jaiswal, Ankik Kumar Giri
TL;DR
The paper proves the existence and uniqueness of mass-conserving weak solutions to the continuous nonlinear fragmentation equation with mass transfer for collision kernels of the form $Φ(x,y)=κ\left(x^{σ_1} y^{σ_2} + y^{σ_1} x^{σ_2}\right)$ under $0\le σ_1 \le σ_2 \le 1$ and $σ_1\neq 1$, with integrable daughter distributions. It develops a two-pronged existence theory via compactness: (i) for $α\in(0,σ_1]$, with initial data in $\Xi_{0,+} \cap \Xi_1$, global existence up to $T_{γ,σ}$ and moment bounds, and (ii) for $α\in(0,1)$, with $u^{in} \in \Xi_{-α} \cap \Xi_1$ and finite $\int x^{-α}u^{in}$, existence up to $T_{γ,σ}$ with analogous moment control. The approach combines kernel truncation, fixed-point arguments, Dunford-Pettis weak compactness, and time equicontinuity to pass to the limit, augmented by lower-mound moment estimates (via a lower bound on $\mu_{σ_1}$ or $\mu_{-α}$) and a de la Vallée-Poussin convex function to obtain higher-moment and uniform integrability bounds. The results broaden the class of admissible collision kernels and breakage distributions in the presence of mass transfer, ensuring mass conservation and uniqueness while avoiding the need for finite initial superlinear moments, with implications for modeling fragmentation phenomena in clouds and related systems.
Abstract
A mathematical model for the continuous nonlinear fragmentation equation is considered in the presence of mass transfer. In this paper, we demonstrate the existence of mass-conserving weak solutions to the nonlinear fragmentation equation with mass transfer for collision kernels of the form $Φ(x,y) = κ(x^{σ_1} y^{σ_2} + y^{σ_1} x^{σ_2})$, $κ>0$, $0 \leq {σ_1} \leq {σ_2} \leq 1$, and ${σ_1} \neq 1$ for $(x, y) \in \mathbb{R}_+^2$, with integrable daughter distribution functions, thereby extending previous results obtained by Giri \& Lauren\c cot (2021). In particular, the existence of at least one global weak solution is shown when the collision kernel exhibits at least linear growth, and one local weak solution when the collision kernel exhibits sublinear growth. In both cases, finite superlinear moment bounds are obtained for positive times without requiring the finiteness of initial superlinear moments. Additionally, the uniqueness of solutions is confirmed in both cases.