Local and global solutions to continuous fragmentation-coagulation equations with vanishing diffusion and unbounded fragmentation and coagulation rates

TL;DR

This paper analyzes spatial particle systems with vanishing diffusion and unbounded fragmentation and coagulation rates, establishing local well-posedness and global solutions under certain conditions.

Contribution

It proves local well-posedness for a broad class of coefficients and demonstrates global solutions for power rate cases across all spatial dimensions.

Findings

Systems are locally well-posed for a large class of coefficients.

Existence of global solutions in all spatial dimensions for power rate cases.

No restrictions on input data size in the power rate case.

Abstract

In the paper, we study spatially distributed particle systems whose time evolution is governed by vanishing diffusion in space $\mathbb{R}^d$, $d\ge 1$, and by size-continuous fragmentation and coagulation processes with unbounded rates. We show that for a large class of coefficients, such systems are classically locally well-posed, provided the diffusion and the coagulation processes are suitably dominated by the fragmentation. In the special case of power rates, we demonstrate existence of global in time classical solutions in all spatial dimensions $d\ge 1$ and without any restrictions on the size of input data.