  It is known that the solutions of pure classical 5D gravity with $AdS_5$
asymptotics can describe strongly coupled large N dynamics in a universal
sector of 4D conformal gauge theories. We show that when the boundary metric is
flat we can uniquely specify the solution by the boundary stress tensor. We
also show that in the Fefferman-Graham coordinates all these solutions have an
integer Taylor series expansion in the radial coordinate (i.e. no $log$ terms).
Specifying an arbitrary stress tensor can lead to two types of pathologies, it
can either destroy the asymptotic AdS boundary condition or it can produce
naked singularities. We show that when solutions have no net angular momentum,
all hydrodynamic stress tensors preserve the asymptotic AdS boundary condition,
though they may produce naked singularities. We construct solutions
corresponding to arbitrary hydrodynamic stress tensors in Fefferman-Graham
coordinates using a derivative expansion. In contrast to Eddington-Finkelstein
coordinates here the constraint equations simplify and at each order it is
manifestly Lorentz covariant. The regularity analysis, becomes more elaborate,
but we can show that there is a unique hydrodynamic stress tensor which gives
us solutions free of naked singularities. In the process we write down explicit
first order solutions in both Fefferman-Graham and Eddington-Finkelstein
coordinates for hydrodynamic stress tensors with arbitrary $\eta/s$. Our
solutions can describe arbitrary (slowly varying) velocity configurations. We
point out some field-theoretic implications of our general results.
