We will first consider the physical setting where two immiscible fluids of different properties move through a porous media: the Muskat problem. We will start discussing recent results in the regime where the fluids are of different viscosities and different densities: global existence and uniqueness in the critical space $\dot{\mathcal{F}}^{1,1}$, instant gain of analyticity for the interface, large-time decay of the interface and ill-posedness in the unstable regime. We will take advantage of these results to later prove analogous results for star-shape bubbles of a fluid surrounded by another one, when surface tension is included in the evolution equation for the free boundary.

On a second part, we will show that sharp fronts of temperature modeled by the Boussinesq system propagate their initial interface regularity globally in time. The results also hold in 3D and for piecewise H\"older initial temperature.