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 .

In this talk we will survey some results regarding the possibility of a self-similar blow-up for the Euler equations. We will also prove that under a mild Lp -growth assumption on the self-similar profile we obtain that the solution carries a positive amount of energy up to the time of blow-up. As a consequence, we will recovery and extend several previously known exclusion criteria. Also, we will present some preliminary studies on the fractal dimension of the energy measure, which roughly speaking is the limit of the measures on the space induced by the velocity squared as time approaches the time of blow-up. We will explore the relation between the fractal dimension of the energy measure and the growth of the velocity as time approaches the time of singularity formation. This is joint work with Roman Shvydkoy .