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 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 B positive definite everywhere.

Automated summary

This paper studies the partial differential equations governing the two-dimensional flows of a robust class of viscoelastic rate-type fluids with stress diffusion. The incompressible Navier-Stokes equations are generalized by introducing an additional term in the constitutive equation for the Cauchy stress expressed in terms of a positive definite tensor B. The evolution of tensor B follows a diffusive variant of a combination of Oldroyd-B and Giesekus models. The study proves the existence of a unique globally defined weak solution for arbitrary initial data and appropriate forcing in spatially periodic problems, and more regular initial data and forcing result in a solution with B positive definite everywhere.