On the existence of a singular limit equation for a model of a self-propelled object motion

TL;DR

This paper introduces a phase-field model for self-propelled deformable objects driven by surface tension, demonstrating its convergence to a sharp-interface limit as the interface thickness approaches zero.

Contribution

It establishes the existence of a singular limit equation for a coupled phase-field and reaction-diffusion model of self-propelled objects.

Findings

The phase-field model converges to a sharp-interface limit as interface thickness tends to zero.

The limit involves a normal velocity determined by mean curvature, surface tension, and volume preservation.

The model effectively describes the evolution of deformable, self-propelled objects with surface-tension effects.

Abstract

In this paper, a phase-field model is introduced to describe the evolution of a deformable, self-propelled object driven by surface-tension effects. The model couples an Allen-Cahn-type equation, which distinguishes the body from the surrounding fluid, with a reaction-diffusion equation for the surfactant concentration. As the interface-thickness parameter $\varepsilon$ tends to zero, it is shown that the phase-field model converges to a sharp-interface limit coupled with a reaction-diffusion equation. In particular, the normal velocity is given by the mean curvature, surface tension, and volume-preserving effect.