Properties of the solutions of delocalised coagulation and inception problems with outflow boundaries

TL;DR

This paper proves well-posedness and regularity of solutions for a class of particle coagulation models with advection and outflow in bounded domains, extending results to one-dimensional cases and connecting to simulation convergence.

Contribution

It establishes well-posedness, uniqueness, and regularity of solutions for delocalised coagulation equations with outflow boundaries, including one-dimensional cases.

Findings

Proved well-posedness and uniqueness of solutions.

Established differentiability and bounded variation regularity.

Connected theoretical results to simulation convergence in 1D.

Abstract

Well posedness is established for a family of equations modelling particle populations undergoing delocalised coagulation, advection, inflow and outflow in a externally specified velocity field. Very general particle types are allowed while the spatial domain is a bounded region of $d$-dimensional space for which every point lies on exactly one streamline associated with the velocity field. The problem is formulated as a semi-linear ODE in the Banach space of bounded measures on particle position and type space. A local Lipschitz property is established in total variation norm for the propagators (generalised semi-groups) associated with the problem and used to construct a Picard iteration that establishes local existence and global uniqueness for any initial condition. The unique weak solution is shown further to be a differentiable or at least bounded variation strong solution under smoothness assumptions on the parameters of the coagulation interaction. In the case of one spatial dimension strong differentiability is established even for coagulation parameters with a particular bounded variation structure in space. This one dimensional extension establishes the convergence of the simulation processes studied in [Patterson, Stoch. Anal. Appl. 31, 2013] to a unique and differentiable limit.