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, I will present our new results of the Euler -a system as a regularization of the incompressible Euler equations in a smooth, two-dimensional, bounded domain. For the limiting Euler system we consider the usual non-penetration boundary condition, while, for the Euler -a regularization, we use velocity vanishing at the boundary. We also assume that the initial velocities for the Euler -a system approximate, in a suitable sense, as the regularization parameter a-> 0, the initial velocity for the limiting Euler system. For small values of a this situation leads to a boundary layer, which is the main concern of this work. Our main result is that, under appropriate regularity assumptions, and despite the presence of this boundary layer, the solutions of the Euler -a system converge, as a -> 0, to the corresponding solution of the Euler equations, in L2 in space, uniformly in time .