We review studies on the singularity structure and asymptotic analysis of a 3-brane (flat or curved) embedded in a five-dimensional bulk filled with a 'perfect fluid' with an equation of state [Formula: see text], where [Formula: see text] is the 'pressure' and [Formula: see text] is the 'density' of the fluid, depending on the fifth space coordinate. Regular solutions satisfying positive energy conditions in the bulk exist only in the cases of a flat brane for [Formula: see text] or of AdS branes for [Formula: see text]. More cases can be found by gluing two regular brunches of solutions at the position of the brane. However, only a flat brane for [Formula: see text] leads to finite Planck mass on the brane and thus localizes gravity. In a more recent work, we showed that a way to rectify the previous findings and obtain a solution for a flat brane and a range of [Formula: see text], which is both free from finite-distance singularities and compatible with the physical conditions of energy and finiteness of four-dimensional Planck mass, is by introducing a bulk fluid component that satisfies a nonlinear equation of state of the form [Formula: see text] with [Formula: see text] and [Formula: see text]. This article is part of the theme issue 'The future of mathematical cosmology, Volume 2'.
Keywords: brane-worlds; singularities.