  In this paper, we mainly study global existence and optimal $L^2$ decay rate
of weak solutions to the co-rotation Hooke dumbbell model. This micro-macro
model is a coupling of the Navier-Stokes equation with a nonlinear
Fokker-Planck equation. Based on the defect measure propagation method, we
prove that the co-rotation Hooke dumbbell model admits a global weak solution
provided the initial data under different integrable conditions. Moreover, we
obtain optimal time decay rate in $L^2$ for the weak solutions obtained by the
Fourier splitting method.
