The intention of this talk is to show two different approaches to obtain global in time results for fluid mechanics problems modeled through parabolic equations, without assuming small enough initial data.
The first part of the talk will be about the Muskat problem. This problem models the movement of two incompressible and immiscible fluids through porous media. We will introduce the equations coming from fluid mechanics that describe this process and deduce the contour dynamic formulation in terms of the interface. We will also comment on the physical implications of considering different viscosities and surface tension, as well as its mathematical consequences. The main goal is to show global in time results in different scenarios (whole line and closed curves) for critical initial data of medium size.
We will later talk about temperature sharp fronts governed by the Boussinesq equations. We will show how a different approach can be used in this problem, obtaining global in time results for arbitrary size initial data.