  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.

Analysis of PDEs