  The purpose of this article is to show that there are many differential
viscoelastic models for which the global existence of a regular solution is
possible. Although the problem of global existence in the classic Oldroyd model
is still open, we show that by adding a non-linear contribution (proposed by
R.G. Larson in 1984), it is possible to obtain more regular and global
solutions, regardless of the size of the data (in the two-dimensional and
periodic case). Similarly, more complex appearance models such as those related
to "pom-pom" polymers are interesting and mathematically richer: some "natural"
bounds on the stress make it possible to obtain global results. On the other
hand, in the last part, we show that other models clearly do not seem to fit
into this framework, and do not even seem to have a global solution in time.
These kinds of results allow to highlight the advantages and disadvantages of
such or such viscoelastic fluid models. They can thus help rheologists and
numericists to make choices with new arguments.
