Wellposedness for density-dependent incompressible viscous fluids on the torus $\T^3$

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.