  We consider the system of partial differential equations governing
two-dimensional flows of a robust class of viscoelastic rate-type fluids with
stress diffusion, involving a general objective derivative. The studied system
generalizes the incompressible Navier--Stokes equations for the fluid velocity
$v$ and pressure $p$ by the presence of an additional term in the constitutive
equation for the Cauchy stress expressed in terms of a positive definite tensor
$B$. The tensor $B$ evolves according to a diffusive variant of an equation
that can be viewed as a combination of corresponding counterparts of Oldroyd-B
and Giesekus models. Considering spatially periodic problem, we prove that for
arbitrary initial data and forcing in appropriate $L^2$ spaces, there exists a
unique globally defined weak solution to the equations of motion, and more
regular initial data and forcing launch a more regular solution with $\bs B$
positive definite everywhere.
