Optimal transport by omni-potential flow and cosmological reconstruction

One of the simplest models used in studying the dynamics of large-scale structure in cosmology, known as the Zeldovich approximation, is equivalent to the three-dimensional inviscid Burgers equation for potential flow. For smooth initial data and sufficiently short times it has the property that the mapping of the positions of fluid particles at any time $t_1$ to their positions at any time $t_2\ge t_1$ is the gradient of a convex potential, a property we call omni-potentiality. Are there other flows with this property, that are not straightforward generalizations of Zeldovich flows? This is answered in the affirmative in both two and three dimensions. How general are such flows? Using a WKB technique we show that in two dimensions, for sufficiently short times, there are omni-potential flows with arbitrary smooth initial velocity. Mappings with a convex potential are known to be associated with the quadratic-cost optimal transport problem. This has important implications for the problem of reconstructing the dynamical history of the Universe from the knowledge of the present mass distribution.