  In \cite{GW12} (Y. Guo, Y. Wang, Decay of dissipative equations and negative
Sobolev spaces, Commun. Partial Differ. Equ. 37 (2012) 2165--2208), Y. Guo and
Y. Wang developed a general new energy method for proving the optimal time
decay rates of the solutions to dissipative equations. In this paper, we
generalize this method in the framework of homogeneous Besov spaces. Moreover,
we apply this method to a model arising from electro-hydrodynamics, which is a
coupled system of the Navier-Stokes equations and the Poisson-Nernst-Planck
equations through charge transport and external forcing terms. We show that the
negative Besov norms are preserved along time evolution, and obtain the optimal
temporal decay rates of the higher-order spatial derivatives of solutions by
the Fourier splitting approach and the interpolation techniques.
