  We investigate the local wellposedness of incompressible inhomogeneous
Navier-Stokes equations on the Torus $\T^3$, with initial data in the critical
Besov spaces. Under some smallness assumption on the velocity in the critical
space $B^{\frac{1}{2}}\_{2,1}(\T^3)$, the global-in-time existence of the
solution is proved. The initial density is required to belong
to~$B^{\frac{3}{2}}\_{2,1}(\T^3)$ but not supposed to be small.
