The continuous collision-induced nonlinear fragmentation equation with non-integrable fragment daughter distributions

Abstract

Existence, non-existence, and uniqueness of mass-conserving weak solutions to the continuous collision-induced nonlinear fragmentation equations are established for the collision kernels $\Phi$ satisfying $\Phi(x_1,x_2)={x_1}^{\lambda_1} {x_2}^{\lambda_2}+{x_2}^{\lambda_1} {x_1}^{\lambda_2}$, $(x_1,x_2)\in(0,\infty)^2$, with ${\lambda_1} \leq {\lambda_2}\leq 1$, and non-integrable fragment daughter distributions. In particular, global existence of mass-conserving weak solutions is shown when $1\le\lambda:={\lambda_1}+{\lambda_2}\le2$ with $\lambda_1\ge k_0$, the parameter $k_0\in(0,1)$ being related to the non-integrability of the fragment daughter distribution. The existence of at least one mass-conserving weak solution is also demonstrated when $2k_0 \le \lambda < 1$ with $\lambda_1\ge k_0$ but its maximal existence time is shown to be finite. Uniqueness is also established in both cases. The last result deals with the non-existence of mass-conserving weak solutions, even on a small time interval, for power law fragment daughter distribution when $\lambda_1<k_0$. It is worth mentioning that the previous literature on the nonlinear fragmentation equation does not treat non-integrable fragment daughter distribution functions.