Sharp Interface Limits of Some Models of Cell Migration

Abstract

This dissertation is concerned with the analysis of some variants of the classical Patlak-Keller-Segel systems of partial differential equations. These systems are common models for the population density of a group of organisms subject to the simultaneous antagonistic effects of aggregation and a mechanism that prevents overcrowding. The aggregation is the result of an attraction to a chemical signal (chemotaxis) that is produced by the organisms themselves. Overcrowding is modeled by a population pressure resulting in a degenerate, nonlinear diffusion, e.g., of porous medium-type. Our aim is understanding the dynamics of these systems in regimes where the organism population is large and when either the timescale for organism migration is instantaneous relative to the relaxation of the chemical signal (elliptic-parabolic systems) or when the organism motion is only somewhat faster than the chemical relaxation (parabolic-parabolic systems). We show, in the limit of a large population, that centered energies for which these systems are gradient flows Γ-converge toward (multiples of) perimeter functionals. Then, we study the singular limit of the gradient flows themselves. For each system, when the initial state of the system is well-prepared, phase separation occurs in the limit of a large population, resulting in the formation of a sharp interface partitioning the environment into a region with no organisms and a region containing the organisms arranged at a uniform density. We then derive, conditional on the satisfaction of an energy convergence hypothesis, evolution equations for these sharp interfaces. For the elliptic-parabolic system, the sharp interface evolves by volume-preserving mean-curvature flow while parabolic-parabolic system's emergent interface evolves by a Hele-Shaw free boundary problem with surface tension and kinetic undercooling.