  In this paper, we derive the optimal time-decay estimates for 2-D
inhomogeneous Navier-Stokes equations. In particular, we prove that
$\|u(t)\|_{\dot{B}^{\theta}_{p,1}({\mathop{\mathbb R\kern
0pt}\nolimits}^2)}={\mathcal O} (t^{\frac1p-\frac32-\frac{\theta}2})$ as
$t\rightarrow\infty$ for any $p\in[2,\infty[,~\theta\in [0,2]$ if initially
$\rho_0u_0\in \dot{B}^{-2}_{2,\infty}({\mathop{\mathbb R\kern
0pt}\nolimits}^2)$. This is optimal even for the classical homogeneous
Navier-Stokes equations. Different with Schonbek and Wiegner's Fourier
splitting device, our method here seems more direct, and can adapt to many
other equations as well. Moreover, our method allows us to work in the
$L^p$-based spaces.
