Global well-posedness for two-dimensional flows of viscoelastic rate-type fluids with stress diffusion

TL;DR

This paper proves the global existence and uniqueness of solutions for a class of two-dimensional viscoelastic fluid models with stress diffusion, extending classical fluid equations with additional stress tensor dynamics.

Contribution

It establishes the global well-posedness for a generalized viscoelastic fluid model with stress diffusion, including cases with arbitrary initial data and forcing.

Findings

Existence of unique global weak solutions for the model.

More regular initial data lead to more regular solutions.

Solutions maintain positive definiteness of the stress tensor B.

Abstract

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.