  We consider $L^2$ minimizing geodesics along the group of volume preserving
maps $SDiff(D)$ of a given 3-dimensional domain $D$. The corresponding curves
describe the motion of an ideal incompressible fluid inside $D$ and are
(formally) solutions of the Euler equations. It is known that there is a unique
possible pressure gradient for these curves whenever their end points are
fixed. In addition, this pressure field has a limited but unconditional
(internal) regularity. The present paper completes these results by showing: 1)
the uniqueness property can be viewed as an infinite dimensional phenomenon
(related to the possibility of relaxing the corresponding minimization problem
by convex optimization), which is false for finite dimensional configuration
spaces such as O(3) for the motion of rigid bodies; 2) the unconditional
partial regularity is necessarily limited.
