In this talk we consider a minimizing movements type scheme for the incompressible Navier-Stokes equations, combining the Lagrangian and Eulerian viewpoints. Our scheme is an improved version of the split scheme introduced in Ebin-Marsden. An essential ingredient is the H^1 projection problem, which is a viscous analogue of the L^2 projection introduced by Brenier for evolving the incompressible Euler equation. We will discuss the scheme and the regularity of solutions it produces. The talk is based on a joint work with Wilfrid Gangbo and Matt Jacobs.