Diffeomorphic shape evolution coupled with a reaction-diffusion PDE on a growth potential

TL;DR

This paper introduces a mathematical model for shape evolution driven by a dynamic growth potential governed by a reaction-diffusion PDE, extending previous static models and providing theoretical analysis and simple simulations.

Contribution

It develops a coupled PDE framework for shape transformation with a evolving growth potential and proves existence, uniqueness, and long-term behavior of solutions.

Findings

Proved uniqueness and long-time existence of solutions.

Extended shape evolution models to include reaction-diffusion growth potentials.

Provided initial simulations demonstrating model behavior.

Abstract

This paper studies a longitudinal shape transformation model in which shapes are deformed in response to an internal growth potential that evolves according to an advection reaction diffusion process. This model extends prior works that considered a static growth potential, i.e., the initial growth potential is only advected by diffeomorphisms. We focus on the mathematical study of the corresponding system of coupled PDEs describing the joint dynamics of the diffeomorphic transformation together with the growth potential on the moving domain. Specifically, we prove the uniqueness and long time existence of solutions to this system with reasonable initial and boundary conditions as well as regularization on deformation fields. In addition, we provide a few simple simulations of this model in the case of isotropic elastic materials in 2D.