Global existence and well-posedness for the Doi-Edwards polymer model

TL;DR

This paper establishes local and global well-posedness results for the Doi-Edwards polymer model, a complex system coupling Navier-Stokes with deformation and memory equations, using advanced functional analysis techniques.

Contribution

It provides the first rigorous proof of local and global existence for the Doi-Edwards model in Besov spaces, extending understanding of polymer dynamics.

Findings

Local well-posedness in Besov spaces

Global existence under small initial data

Application of Littlewood-Paley theory

Abstract

In this paper we mainly investigate the Cauchy problem of the Doi-Edwards polymer model with dimension $d\geq2$. The model was derived in the late 1970s to describe the dynamics of polymers in melts. The system contain a Navier-Stokes equation with an additional stress tensor which depend on the deformation gradient tensor and the memory function. The deformation gradient tensor satisfies a transport equation and the memory function satisfies a degenerate parabolic equation. We first proved the local well-posedness for the Doi-Edwards polymer model in Besov spaces by using the Littlewood-Paley theory. Moreover, if the initial velocity and the initial memory is small enough, we obtain a global existence result.