  This paper is concerned with regular flows of incompressible weakly
viscoelastic fluids which obey a differential constitutive law of Oldroyd type.
  We study the newtonian limit for weakly viscoelastic fluid flows in $\R^N$ or
$\T^N$ for $N=2, 3$, when the Weissenberg number (relaxation time measuring the
elasticity effect in the fluid) tends to zero. More precisely, we prove that
the velocity field and the extra-stress tensor converge in their existence
spaces (we examine the Sobolev-$H^s$ theory and the Besov-$B^{s,1}_2$ theory to
reach the critical case $s= N/2$) to the corresponding newtonian quantities.
  These convergence results are established in the case of "ill-prepared"'
data.We deduce, in the two-dimensional case, a new result concerning the global
existence of weakly viscoelastic fluids flow. Our approach makes use of
essentially two ingredients : the stability of the null solution of the
viscoelastic fluids flow and the damping effect,on the difference between the
extra-stress tensor and the tensor of rate of deformation, induced by the
constitutive law of the fluid.
