Smoluchowski coagulation equation with velocity dependence

TL;DR

This paper introduces a kinetic variant of the Smoluchowski coagulation equation incorporating velocity dependence, analyzing its well-posedness and regularity in a spatially-homogeneous setting.

Contribution

It develops a rigorous mathematical framework for a velocity-dependent coagulation PDE, extending classical models with new existence, uniqueness, and regularity results.

Findings

Proved existence and uniqueness of solutions under various initial conditions.

Analyzed regularity properties of solutions in weighted spaces.

Established the kinetic Smoluchowski equation as a valid limit of microscopic models.

Abstract

In the present article we introduce a variant of Smoluchowski's coagulation equation with both position and velocity variables taking a kinetic viewpoint arising as the scaling limit of a system of second-order (microscopic) coagulating particles. We focus on the rigorous study of the PDE system in the spatially-homogeneous case proving existence and uniqueness under different initial conditions in suitable weighted space, investigating also the regularity of such solutions.