  Following Arnold's geometric interpretation, the Euler equations of an
incompressible fluid moving in a domain D are known to be the optimality
equation of the minimizing geodesic problem along the group of orientation and
volume preserving diffeomorphisms of D. This problem admits a well-established
convex relaxation which generates a set of "relaxed", "multi-stream", version
of the Euler equations. However, it is unclear that such relaxed equations are
appropriate for the initial value problem and the theory of turbulence, due to
their lack of well-posedness for most initial data. As an attempt to get a more
relevant set of relaxed Euler equations, we address the multi-stream
pressure-less gravitational Euler-Poisson system as an approximate model, for
which we show that the initial value problem can be stated as a concave
maximization problem from which we can at least recover a large class of smooth
solutions for short enough times.
