On a stochastic phase-field model of cell motility with singular diffusion

TL;DR

This paper establishes the existence of solutions for a class of stochastic phase-field models used in cell motility, addressing singularities at zero level sets and considering both independent and coupled phase-field evolutions.

Contribution

It proves global existence of probabilistically weak solutions for stochastic reaction-diffusion models with singular diffusion in cell motility applications.

Findings

Proved global existence of solutions for singular stochastic phase-field models.

Addressed technical challenges due to singularities at zero level sets.

Analyzed both independent and coupled phase-field evolutions.

Abstract

We study existence of solutions in the variational sense for a class of stochastic phase-field models describing moving boundary problems. The models consist of stochastic reaction-diffusion equations with singular diffusion forced by a phase-field. We investigate both the case of an independently evolving phase-field and of coupled phase-field evolution driven by a viscous Hamilton-Jacobi equation. Such systems are used in the modelling of single-cell chemotaxis, where the contour of the cell shape corresponds to a level set of the phase-field. The technical challenge lies in the singularities at zero level sets of the phase-field. For large classes of initial data, we establish global existence of probabilistically weak solutions in $L^2$-spaces with weights which compensate for the singularities.