Keywords: Incompressible Euler equations, Arnold distance, Relaxation Abstract: According to Arnold's interpretation, the Euler equations for incompressible fluids can be seen as a geodesic equation in the space of measure preserving diffeomorphism. Hence, one can look for solutions by minimizing the geodesic action functional with fixed endpoints. Since this problem is in general ill-posed, Brenier introduced at the end of the 80's a relaxed version of Arnold's approach. The aim of the talk is to describe some recent results on this problem.