The Fourth Workshop on Fluids and PDE was held at the National Institute of Pure and Applied Mathematics (IMPA) in Rio de Janeiro, Brazil, from Monday 26 May to Friday 30 May 2014. This workshop is held every two to three years in Brazil. The fourth edition of the workshop was the closing event of a Thematic Program on Incompressible Fluids Dynamics, to be held at IMPA next Spring. Hence, the focus of the workshop will be incompressible fluid mechanics .

The Muskat equation governs the motion of an interface separation of two incompressible fluids in a porous media. I will describe recent projects on the global well-posedness and we present several scenarios where the interface is initially smooth and at a later time it develops a singularity .