Well posedness of a linearized fractional derivative fluid model

TL;DR

This paper investigates the mathematical well-posedness, including existence, uniqueness, and smoothness of solutions, for a generalized fractional derivative fluid model used in modeling viscoelastic responses.

Contribution

It extends previous work by establishing the existence, uniqueness, and regularity of solutions for the generalized fractional derivative Maxwell model.

Findings

Proved existence of weak solutions.

Established uniqueness of solutions.

Analyzed smoothness and regularity properties.

Abstract

The one-dimensional fractional derivative Maxwell model (e.g. Palade et al. Rheol. Acta 35, 265, 1996), of importance in modeling the linear viscoelastic response in the glass transition region, has been generalized in Palade et al. Int. J. Non-Linear Mech. 37, 315, 1999, to objective three-dimensional constitutive equations (CEs) for both fluids and solids. Regarding the rest state stability of the fluid CE, in Heibig and Palade J. Math. Phys. 49, 043101, 2008, we gave a proof for the existence of weak solutions to the corresponding boundary value problem. The aim of this work is to achieve the study of the existence and uniqueness of the aforementioned solutions and to present smoothness results.