Active phase-separating systems exhibit striking behaviors that go beyond classical phase separation. In these systems, droplets can spontaneously grow, divide, or maintain a steady size, defying the usual coarsening process where small domains shrink to feed larger ones. Such phenomena have been proposed as minimal models for protocells, offering insight into how self-organized structures might have emerged in prebiotic environments.
In this talk, I will present a mathematical framework for understanding the dynamics of active droplets. We will explore two complementary approaches: the phase-field formulation via the Cahn–Hilliard equation, and the sharp-interface formulation based on the Mullins–Sekerka free boundary problem. I will show how these perspectives connect and allow us to rigorously describe droplet evolution. The talk will cover well-posedness results, the stability of planar and radial solutions, and the emergence of complex behaviors such as droplet division and shell formation. Finally, numerical simulations will be presented to illustrate and validate the theoretical predictions, providing a vivid picture of the rich dynamics in active phase-separating systems.