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Hele-Shaw limit for phenotype-structured pressure models with phenotype-dependent mobility

Prove a rigorous asymptotic derivation of a Hele-Shaw type free-boundary problem from the phenotype-structured pressure-based movement model (equation (structuredPM)) in the incompressible limit when the cell mobility coefficient depends on the phenotypic state, and clarify the corresponding results for related systems with cross-diffusion terms that model binary phenotypic states.

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Background

Pressure-based movement models describe cell motion via advection down pressure gradients coupled to growth inhibition by pressure, and have connections to porous-medium type equations and Hele-Shaw free-boundary limits under stiff pressure laws.

While incompressible-limit results to Hele-Shaw problems are known in settings with phenotype-independent mobility, the authors highlight the absence of rigorous asymptotic results for the phenotype-structured pressure model when mobility varies with phenotype. They further note this difficulty relates to open issues in cross-diffusion systems for binary phenotypic states, underscoring the challenge and importance of establishing a rigorous limit in the heterogeneous case.

References

Furthermore, there are no rigorous asymptotic results, of the type presented by, on the derivation of free-boundary problems of Hele-Shaw type from PS-PDE models of form~structuredPM, when the mobility parameter is a function of the phenotypic state. This problem appears to be far from being closed given that, as of today, related problems remain open even for systems of PDEs with cross-diffusion terms corresponding to the case where the phenotypic state is binary~\citep{lorenzi2016interfaces,david2024degenerate}.

structuredPM:

{∂tn=∇x⋅[μ(y)n∇xP]+Dˉ∇y2n+nR(y,P) ,x∈X ,  y∈Y ,P(t,x):=Π[ρ](t,x),    ρ(t,x):=∫Yn(t,x,y) dy \begin{cases} \displaystyle{\partial_t n = \nabla_{\bm x} \cdot \left[\mu({\bm y}) n \nabla_{\bm x} P\right] + \bar{D} \nabla_{\bm y}^2 n + n R({\bm y},P)\, , \quad {\bm x} \in \mathcal{X} \ , \; {\bm y} \in \mathcal{Y} \ , }\\[5pt] \displaystyle{P(t,{\bm x}) := \Pi[\rho](t,{\bm x}), \;\; \rho(t,{\bm x}) := \int_{{\cal {Y}}} n(t,{\bm x},{\bm y}) \, {\rm d}{\bm y} \ } \end{cases}

— Phenotype structuring in collective cell migration:a tutorial of mathematical models and methods (2410.13629 - Lorenzi et al., 17 Oct 2024) in Section 6 (Challenges and perspectives), item VI