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 .

Two proofs of the persistence of regularity of the boundary of a two-dimensional vortex patch were published in 1993, one by Chemin the other by Bertozzi and Constantin. A third, by Ph. Serfati, was published in 1994. His proof, while sharing elements of Chemin's proof, differs quite substantially, and is more elementary. We describe the main features of his proof and discuss directions in which it might be extended. This is a report on work in progress, joint with Hantaek Bae of UC Davis .