A concentrated capacity model for diffusion-advection: advection localized to a moving curve

TL;DR

This paper develops a new PDE framework for modeling diffusion with advection localized to a moving curve, inspired by biological processes involving proteins and solvent interactions.

Contribution

It introduces a concentrated capacity approach to rigorously model advection confined to a moving curve within a diffusion-advection equation.

Findings

Established the form of the concentrated capacity limit.

Proved small time existence of solutions.

Applied the model to biological protein-solvent interactions.

Abstract

In this work I show how a diffusion-advection equation in three space-dimensions may have its advection term weakly limited to a velocity field localized to a moving curve. This is rigorously accomplished through the technique of concentrated capacity, and the form of the concentrated capacity limit along with small time existence of solutions is determined. This problem is motivated by mathematical biology and the study of proteins in solvent where the latter is modeled as a diffusing quantity and the protein is taken to be a 1d object which advects the solvent by contact and its own motion. This work introduces a novel PDE's framework for that interaction.