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

TL;DR

This paper analyzes the behavior of density-dependent incompressible fluids on a 3D torus as the Froude number approaches zero, establishing the limit system and its well-posedness under various initial conditions.

Contribution

It explicitly determines the limit system for small Froude number and proves its global well-posedness, even for large initial data, considering general initial conditions and resonant phenomena.

Findings

Explicit limit system as Froude number tends to zero

Global well-posedness of the limit system

Well-posedness for large initial data with small Froude number

Abstract

The present work is devoted to the analysis of density-dependent, incompressible fluids in a 3D torus, when the Froude number $\varepsilon$ 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 $\varepsilon \to 0$ and prove its global wellposedness. Finally, we prove that for large initial data, the density-dependent, incompressible fluid system is globally wellposed, provided that $\varepsilon$ is small enough.