  In this paper we establish the global-in-time well-posedness for an arbitrary
$C^{1+\gamma}$, $0<\gamma<1$, initial internal wave for the free boundary
gravity Stokes system in two dimensions. This classical well-posedness result
is complemented with a weak solvability result in the case of $C^\gamma$ or
Lipschitz interfaces. Furthermore, we also propose and study several
one-dimensional models that capture different features of the full internal
wave problem for the gravity Stokes system and show that all of them present
finite time singularities. This fact evidences the fine structure of the
non-linearity in the full system that allows for the free boundary problem to
be globally well-posed while simplifications of it blow-up in finite time.
