Gradient-flow interpretation of the coupled cell–ECM system

Determine whether and in which sense the coupled PDE–ODE system for cell invasion into extracellular matrix—comprising the non-mass-conserving reaction–cross-diffusion equation for cell density u in Eq. (1a) and the non-diffusive ODE for ECM density m in Eq. (1b)—admits a gradient-flow formulation. Specifically, ascertain a precise gradient-flow framework (if any) under which the full system can be interpreted, despite the reaction term in Eq. (1a) and the ODE coupling in Eq. (1b).

Background

The paper studies a model of cell invasion into extracellular matrix given by a PDE for cell density u and an ODE for ECM density m. The PDE’s flux structure resembles that of volume-filling cross-diffusion systems where boundedness-by-entropy methods and Otto–Wasserstein gradient-flow frameworks are commonly used to obtain existence and qualitative properties of solutions.

However, the presence of a non-mass-conserving reaction term in the PDE and the coupling to an ODE without diffusion complicate the identification of a gradient-flow structure for the entire coupled system. The authors therefore develop existence proofs without relying on a standard gradient-flow framework, noting the lack of clarity regarding any such interpretation for the full system.