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

Amjad Saef, Wilhelm Stannat

TL;DR

This work develops a rigorous analytical framework for stochastic phase-field models with singular diffusion in moving-boundary problems, motivated by cell motility and chemotaxis. The authors establish global existence of probabilistically weak (martingale) solutions in weighted $L^2$ spaces, addressing both an uncoupled phase-field and a fully coupled system where the phase-field evolves via a viscous Hamilton–Jacobi type equation. Central to the approach are weighted derivatives with respect to the phase-field, truncation/taming of nonlinearities, and compactness arguments (including Jakubowski–Skorokhod) to handle singular diffusion near $\phi=0$. With additional entropy-type bounds, the results yield classical martingale solutions and positivity of the phase-field under suitable initial data, providing a solid mathematical foundation for stochastic diffuse-interface models of cell motility and chemotaxis that incorporate fluctuations. The framework has potential implications for accurately simulating biologically realistic boundary dynamics under stochastic perturbations, while highlighting the importance of weighted spaces to manage singular diffusion terms.

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.

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

TL;DR

This work develops a rigorous analytical framework for stochastic phase-field models with singular diffusion in moving-boundary problems, motivated by cell motility and chemotaxis. The authors establish global existence of probabilistically weak (martingale) solutions in weighted spaces, addressing both an uncoupled phase-field and a fully coupled system where the phase-field evolves via a viscous Hamilton–Jacobi type equation. Central to the approach are weighted derivatives with respect to the phase-field, truncation/taming of nonlinearities, and compactness arguments (including Jakubowski–Skorokhod) to handle singular diffusion near . With additional entropy-type bounds, the results yield classical martingale solutions and positivity of the phase-field under suitable initial data, providing a solid mathematical foundation for stochastic diffuse-interface models of cell motility and chemotaxis that incorporate fluctuations. The framework has potential implications for accurately simulating biologically realistic boundary dynamics under stochastic perturbations, while highlighting the importance of weighted spaces to manage singular diffusion terms.

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 -spaces with weights which compensate for the singularities.
Paper Structure (6 sections, 17 theorems, 220 equations)

This paper contains 6 sections, 17 theorems, 220 equations.

Table of Contents

  1. Introduction
  2. Mathematical Setting, Notations and Main Assumptions
  3. Solution theory for uncoupled phase-fields
  4. Solution theory of the coupled system
  5. Applications
  6. Appendix

Key Result

Theorem 3.6

Let $\phi$ be as described in the introduction to this section. Then there exists a filtered probability space $(\Omega', \mathcal{F}', (\mathcal{F}'_t)_{t\geq 0}, \mathbb P')$ and a solution $(\phi', \bm c)$ of the weighted martingale problem (Definition MartSolStationary) associated with equation

Theorems & Definitions (54)

  • Definition 2.1: Strong variational solution
  • Definition 2.2: Martingale solution
  • Definition 3.1: Weighted weak derivative
  • Remark 3.2
  • Remark 3.3
  • Definition 3.4: Weighted martingale solution, uncoupled case
  • Remark 3.5
  • Theorem 3.6
  • Proposition 3.7
  • proof
  • ...and 44 more