On the influence of gravity on density-dependent incompressible periodic fluids

The present work is devoted to the analysis of density-dependent, incompressible fluids in a 3D torus, when the Froude number $\epsilon$ goes to zero. We consider the very general case where the initial data do not have a zero horizontal average, where we only have smoothing effect on the velocity but not on the density and where we can have resonant phenomena on the domain. We explicitly determine the limit system when $\epsilon \to 0$ and prove its global wellposedness. Finally, we prove that for large initial data, the density-dependent, incompressible fluid system is globally wellposed, provided that $\epsilon$ is small enough.