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 ${\lambda }_1\le {\lambda }_2\le 1$, and non-integrable fragment daughter distributions. In particular, global existence of mass-conserving weak solutions is shown when $1\le \lambda :={\lambda }_1+{\lambda }_2\le 2$ 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.