In this talk, we focus on the two-phase free boundary problem of the Stokes-transport system, which classically models the sedimentation of particles (e.g. sand) transported by steady Stokes flow. We will discuss several results which quantitatively describe possible unstable behaviors of the interface in both Rayleigh-Taylor stable and unstable regimes. In particular, such instability phenomena are manifested by either infinite-in-time growth of certain geometric quantities of the interface, or by a finite-time interface turnover scenario.

This is based on an upcoming joint work with Francisco Gancedo, Rafael Granero-Belinchón, Elena Salguero, and Yao Yao.