On the Doi-Edwards and K-BKZ rheological models for polymer fluids: an existence result for shear flows

TL;DR

This paper proves the existence of smooth solutions for the Doi-Edwards model in shear flows, demonstrating that solutions remain well-behaved over time for small initial data, and relates it to the K-BKZ equation.

Contribution

It provides the first existence result for smooth solutions of the Doi-Edwards model in shear flows, extending the mathematical understanding of polymer fluid models.

Findings

Existence of smooth solutions for small initial data

Solutions remain in the hyperbolic domain over time

Establishes formal equivalence to K-BKZ equations

Abstract

This paper establishes the existence of smooth solutions for the Doi-Edwards rheological model of viscoelastic polymer fluids in shear flows. The problem turns out to be formally equivalent to a K-BKZ equation but with constitutive functions spanning beyond the usual mathematical framework. We prove, for small enough initial data, that the solution remains in the domain of hyperbolicity of the equation for all $t \geq 0$.