Long-time existence of classical solutions to a 1-D swelling gel

TL;DR

This paper models the swelling dynamics of a gel in one dimension with a free boundary, transforming the problem into a fixed boundary and proving long-time existence of smooth solutions.

Contribution

It introduces a new model for gel swelling, employs a mass-Lagrangian transformation, and proves long-time existence of classical solutions in a one-dimensional setting.

Findings

Existence of long-time, smooth solutions established.

Transformation to fixed boundary simplifies analysis.

Model captures hyperbolic swelling dynamics with dissipation.

Abstract

In this paper we derived a model which describes the swelling dynamics of a gel and study the system in one-dimensional geometry with a free boundary. The governing equations are hyperbolic with a weakly dissipative source. Using a mass-Lagrangian formulation, the free-boundary is transformed into a fixed-boundary. We prove the existence of long time, continuously differentiable solutions to the transformed fixed boundary problem.