  We prove global existence and uniqueness of solutions of Oldroyd-B systems
with relatively small data in $\Rr^d$, in a large functional setting
($C^{\alpha}\cap L^1$). This is a stability result, solutions select an
equilibrium and converge exponentially to it. Large spatial derivatives of the
initial density and stress are allowed, provided the $L^{\infty}$ norm of the
density and stress are small enough. We prove global regularity for large data
for a model in which the potential responds to high rates of strain in the
fluid. We also prove global existence for a class of large data for a didactic
scalar model which attempts to capture in the simplest way the essence of the
dissipative nature of the coupling to fluid. This latter model has an
unexpected cone invariance in function space that is crucial for the global
existence.
