Global Well-Posedness for Two-Dimensional Flows of Viscoelastic Rate-Type Fluids with Stress Diffusion

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