Regularity Properties of Solutions of a Model for Morphoelastic Growth in the Presence of Nutrients in One Spatial Dimension

TL;DR

This paper establishes regularity properties of solutions for a one-dimensional model of stress-modulated growth influenced by nutrients, combining differential equations and variational methods to analyze growth and deformation.

Contribution

It provides a rigorous mathematical analysis of the regularity of solutions in a coupled growth-deformation-nutrient model, incorporating elastic and nutrient effects in one spatial dimension.

Findings

Proved regularity properties of solutions in the model.

Analyzed the coupling between growth, elastic deformation, and nutrient concentration.

Established existence and uniqueness results for the model equations.

Abstract

Regularity properties of solutions for a class of quasi-stationary models in one spatial dimension for stress-modulated growth in the presence of a nutrient field are proven. At a given point in time the configuration of a body after pure growth is determined by means of a family of ordinary differential equations in every point in space. Subsequently, an elastic deformation, which is given by the minimizer of a hyperelastic variational integral, is applied in order to restore Dirichlet boundary conditions. While the ordinary differential equations governing the growth process depend on the elastic stress and the pullback of a nutrient concentration in the current configuration, the hyperelastic variational problem is solved on the intermediate configuration after pure growth. Additionally, the coefficients of the reaction-diffusion equation determining the nutrient concentration in the current configuration depend on the elastic deformation and the deformation due to pure growth.