Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models

TL;DR

This paper investigates a viscoelastoplastic fluid model, establishing short-time existence, uniqueness, and the behavior of solutions as stress diffusion vanishes, introducing energy-variational solutions for the non-diffusive case.

Contribution

It introduces the concept of energy-variational solutions for the non-diffusive limit and analyzes their properties and existence, extending previous work on generalized solutions.

Findings

Short-time existence and uniqueness of strong solutions.

Existence of energy-variational solutions in the non-diffusive limit.

Convergence of solutions as stress diffusion approaches zero.

Abstract

We study a model for a fluid showing viscoelastic and viscoplastic behavior, which describes the flow in terms of the fluid velocity and an internal stress. This stress tensor is transported via the Zaremba--Jaumann rate, and it is subject to two dissipation processes: one induced by a nonsmooth convex potential and one by stress diffusion. We show short-time existence of strong solutions as well as their uniqueness in a class of Leray--Hopf type weak solutions satisfying the tensorial component in the sense of an evolutionary variational inequality. The global-in-time existence of such generalized solutions has been established in a previous work. We further study the limit when stress diffusion vanishes. In this case, the above notion of generalized solutions is no longer suitable, and we introduce the concept of energy-variational solutions, which is based on an inequality for the relative energy. We derive general properties of energy-variational solutions and show their existence by passing to the non-diffusive limit in the relative energy inequality satisfied by generalized solutions for non-zero stress diffusion.