I will review the theory of minimizing geodesics on the semi-group of
volume preserving maps of the unit cube, with respect to the L2 metric,
which exactly corresponds to the equations introduced by Euler in 1755
to describe incompressible fluid motions inside the cube.
Paradoxically, the "weakness" of the L2 metric simplifies the analysis in the sense that the minimization problem can be convexified, which would be impossible for stronger Sobolev metrics or for finite dimensional analogous models (involving the group SO(3), for instance). In particular one can prove the existence, uniqueness and partial regularity of the pressure gradient driving (possibly multiple) minimizing geodesics between two given points on the semi-group, without any restriction on these points.
More recent investigations will be also discussed, such as the harmonic flow in close connection with Moffatt's magnetic relaxation models.