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.
