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 $Φ$ satisfying $Φ(x_1,x_2)={x_1}^{λ_1} {x_2}^{λ_2}+{x_2}^{λ_1} {x_1}^{λ_2}$, $(x_1,x_2)\in(0,\infty)^2$, with ${λ_1} \leq {λ_2}\leq 1$, and non-integrable fragment daughter distributions. In particular, global existence of mass-conserving weak solutions is shown when $1\leλ:={λ_1}+{λ_2}\le2$ with $λ_1\ge k_0$, 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 $2k_0 \le λ< 1$ with $λ_1\ge k_0$ 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 $λ_1<k_0$. It is worth mentioning that the previous literature on the nonlinear fragmentation equation does not treat non-integrable fragment daughter distribution functions.

Theorems & Definitions (28)

• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Lemma 3.3
• proof