Regular Solutions to the Coagulation Equations with Singular Kernels

TL;DR

This paper proves the existence of regular solutions to coagulation equations with singular kernels, including the Smoluchowski kernel, using weighted L^1-spaces and compactness methods.

Contribution

It introduces a novel approach employing weighted L^1-spaces to handle singularities in coagulation equations, extending the class of kernels with proven solutions.

Findings

Existence of solutions for coagulation equations with singular kernels.

Application of weighted L^1-space techniques to singular problems.

Coverage of the Smoluchowski kernel within the proof framework.

Abstract

In this article we prove the existence of solutions to the coagulation equation with singular kernels. We use weighted L^1-spaces to deal with the singularities in order to obtain regular solutions. The Smoluchowski kernel is covered by our proof. The weak L^1 compactness methods are applied to suitably chosen approximating equations as a base of our proof. A more restrictive uniqueness result is also mentioned.