Absence of gelation and regularity for coagulation-fragmentation-diffusion equations with critical coagulation rate in low dimension

Abstract

We consider a discrete coagulation-fragmentation-diffusion system on a bounded domain of ${\mathbb{R}}^d$. In dimension $d\le 2$, we prove mass conservation (that is absence of gelation) when the coagulation coefficient satisfies the critical additive bound $a_{i,j}\le C(i+j)$ for some constant $C>0$. We thereby close the remaining gap between the spatially homogeneous setting and the spatially inhomogeneous diffusive setting. In dimension $d\le 4$, we impose a higher decay rate on the coagulation and fragmentation coefficients, but we nonetheless prove regularity and absence of gelation under mild assumptions on the diffusion rates.