We consider a 2d fluid in the domain defined by $b(x)<y<f(t,x)$, $x\in\mathbb{R}^d$ where $f(t,x)$ is the free interface, and $b(x)$ the fixed bottom. We assume that the fluid is incompressible and is governed by the Darcy law. In the case of constant densities, Nguyen–Pausader proved that the interface maintains Sobolev regularity for all time (provided the Sobolev regularity is large enough). Our long-term goal is to investigate stability of the constant-density case. The purpose of this talk is twofold: first I will explain how we can rewrite our problem as a quasilinear problem (a usual procedure in this context) and spend some time explaining why such problems pose non-trivial issues such as derivative losses. Then I will use the rest of my time to explain some specific difficulties linked to non-constant densities and explain how one can achieve the first part of our program, that is short-time propagation of the initial Sobolev regularity of the free boundary. This is based on a joint work with Huy Q. Nguyen (University of Maryland).