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 .

A number of recent works have been devoted to solving system of PDEs governing the evolution of viscous fluids, in so-called critical spaces (a terminology borrowed from the pioneering work by Fujita and Kato on the incompressible Navier-Stokes equations). All those works are mostly based on estimates for the heat flow, for the transport equation and on nonlinear estimates in functional spaces that are scaling invariant (or almost scaling invariant) for the system under consideration. Owing to the hyperbolic nature of the mass equation however, a loss of one derivative occurs in the stability estimates related to the system. Consequently, as critical regularity solutions are not so regular, some restrictions appear on admissible spaces for initial data regarding the well-posedness issue. In the present talk, we will show that using Lagrangian coordinates transforms the mixed type system of PDEs into a parabolic type one, which allows to solve it directly by means of Banach fixed point theorem, and to avoid the loss of derivative. That method turns out to be quite robust. In the talk, we will show how it may be implemented on the density dependent incompressible Navier-Stokes equations, on the isentropic compressible Navier-Stokes equations, and on the full Navier-Stokes equations in the heat-conductive case .