A review of results and open problems on mathematical models of motion of viscoelastic media of Jeffreys' type

TL;DR

This paper reviews recent mathematical results and open problems related to the equations of motion for Jeffreys' type viscoelastic media, highlighting their complexity and importance in modeling various materials.

Contribution

It provides a comprehensive review of recent advances and identifies key open problems in the mathematical modeling of Jeffreys' type viscoelastic fluids.

Findings

Mathematical difficulties exceed those of Navier-Stokes equations.

Recent results on existence and uniqueness are summarized.

Open problems in the analysis of Jeffreys' models are highlighted.

Abstract

The Jeffreys model (also associated with the names of Lethersich and Oldroyd) is one of the crucial conceptions in the theory of viscoelastic fluids. The models of Jeffreys type describe behaviour of bitumens, blood, polymers and their solutions, dough, the earth's crust, concrete, lubricants etc. Study of BVPs corresponding to their statics and dynamics meets a lot of mathematical difficulties, which turn out to be much harder than the ones that are related to the celebrated Navier-Stokes system. In this work, we make an attempt to review the recent results and main unsolved problems for equations of motion for the mediums of Jeffreys' type.