On the asymptotic behavior of the quasi-static problem for a linear viscoelastic fluid

TL;DR

This paper investigates the long-term behavior of a quasi-static viscoelastic fluid model using the minimal state concept, establishing existence, uniqueness, and decay results for various memory kernels.

Contribution

It introduces a new free energy framework in a broader data space to analyze the quasi-static problem for viscoelastic fluids, proving key mathematical properties.

Findings

Existence and uniqueness of solutions in the new framework

Asymptotic decay established for exponential and polynomial kernels

Applicable to a wide class of memory kernels

Abstract

In this paper we study the quasi-static problem for a viscoelastic fluid by means of the concept of minimal state. This implies the use of a different free energy defined in a wider space of data. The existence and uniqueness is proved in this new space and the asymptotic decay for the problem with non vanishing supplies is obtained for a large class of memory kernels, including those presenting an exponential or polynomial decay.