PDE analysis of a class of thermodynamically compatible viscoelastic rate-type fluids with stress-diffusion

TL;DR

This paper proves the long-time existence of weak solutions for a simplified thermodynamically consistent viscoelastic fluid model with stress diffusion, providing insights into more complex non-Newtonian fluid models.

Contribution

It establishes the mathematical existence of solutions for a simplified viscoelastic rate-type fluid model with stress diffusion, linking thermodynamics and PDE analysis.

Findings

Proved long-time existence of weak solutions for the model.

Showed thermodynamic consistency of the simplified model.

Provided insights into complex non-Newtonian fluid models.

Abstract

We establish the long-time existence of large-data weak solutions to a system of nonlinear partial differential equations. The system of interest governs the motion of non-Newtonian fluids described by a simplified viscoelastic rate-type model with a stress-diffusion term. The simplified model shares many qualitative features with more complex viscoelastic rate-type models that are frequently used in the modeling of fluids with complicated microstructure. As such, the simplified model provides important preliminary insight into the mathematical properties of these more complex and practically relevant models of non-Newtonian fluids. The simplified model that is analyzed from the mathematical perspective is shown to be thermodynamically consistent, and we extensively comment on the interplay between the thermodynamical background of the model and the mathematical analysis of the corresponding initial-boundary-value problem.