Hele-Shaw flow with surface tension and kinetic undercooling as a sharp interface limit of a fully parabolic Patlak-Keller-Segel system with nonlinear diffusion

Abstract

A large population limit of the parabolic-parabolic Patlak-Keller-Segel (PKS) system with degenerate, nonlinear diffusion, e.g., of porous medium-type $-\frac{m}{m-1}\mathrm{div}(\rho \nabla \rho^{m-1})$, 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.