2025/10/18 by Michael Rozowski, Rozowski, Michael
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Micro and Nano Robotics #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2510.16339
openalex publication_date 2025/10/18 · openalex created_date 2025/10/22 · openalex updated_date 2026/07/28
A large population limit of the parabolic-parabolic Patlak-Keller-Segel (PKS) system with degenerate, nonlinear diffusion, e.g., of porous medium-type -(m)/(m-1)div(ρ∇ ρm-1), is studied. We show, asymptotically, a sharp interface develops separating a region containing organisms arranged in a constant-in-time, uniform density from a region without organisms. Under an energy convergence hypothesis, we prove the emergent interface evolves according to a Hele-Shaw free boundary problem with surface tension and kinetic undercooling, and the free boundary satisfies a contact angle-type condition with the fixed boundary. Further, we show that, for well-prepared initial data, phase separation in these systems is, roughly, the result of some compatibility between an antiderivative for the population pressure and the convex conjugate of an antiderivative of the chemical destruction kinetics. When compatible, an energy for which the parabolic-parabolic PKS system is a gradient flow is a penalized Modica-Mortola functional.