  In this paper, we deal with some Oldroyd type models, which describe
incompressible viscoelastic fluids. There are 3 parameters in these models: the
viscous coefficient of fluid $\nu_{1}$, the viscous coefficient of the elastic
part of the stress tensor $\nu_{2}$, and the damping coefficient of the elastic
part of the stress tensor $\alpha$. In this paper, we assume at least one of
the parameters is zero: $(\nu_{1}, \nu_{2},\alpha)=(+,0,+), (+,0,0), (0,+,+),
(0,+,0), (0,0,+)$ and prove the global existence of unique solutions to all
these 5 cases in the framework of hybrid Besov spaces. We also derive decay
rates of the solutions except for the case $(\nu_{1}, \nu_{2},\alpha)=(+,0,0)$.
To the best of our knowledge, decay rates in this paper are the first results
in this framework, and can improve some previous works.
