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

2025/07/22 by Nagayama, Masaharu, Sakakibara, Koya, Takasao, Keisuke
#35K57 #53E10 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.16447

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 ε 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.

Citations

• Weak-strong uniqueness for volume-preserving mean curvature flow
• Convergence of perturbed Allen-Cahn equations to forced mean curvature flow
• The Allen-Cahn Action functional in higher dimensions
• A theoretical and experimental study on the unidirectional motion of a camphor disk
• Self-Running Droplet: Emergence of Regular Motion from Nonequilibrium Noise

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