The Reconstruction Problem for the Euler-Poisson System in Cosmology

The motion of a continuum of matter subject to gravitational interaction is classically described by the Euler-Poisson system. Prescribing the density of matter at initial and final times, we are able to obtain weak solutions for this equation by minimizing the action of the Lagrangian which is a convex functional. Through this variational formulation, the reconstruction problem becomes very similar to an optimal transportation problem. Then we see that such minimizing solutions are consistent with smooth solutions of the Euler-Poisson system and enjoy some special regularity properties.