Derivation of limit equation for a perturbed 3D periodic Boussinesq system

TL;DR

This paper analyzes the long-term behavior of a 3D density-dependent flow system under gravity, proving global well-posedness for small Froude numbers and convergence to a stratified Navier-Stokes system as the Froude number approaches zero.

Contribution

It establishes the global existence of solutions for small Froude numbers and demonstrates convergence to a stratified Navier-Stokes system without initial data restrictions.

Findings

Global well-posedness for small Froude number

Strong convergence to stratified Navier-Stokes system as Froude number tends to zero

No smallness assumption on initial data

Abstract

We consider a system describing the long-time dynamics of an hydrodynamical, density-dependent flow under the effects of gravitational forces. We prove that if the Froude number is sufficiently small such system is globally well posed with respect to a $ H^s, \ s>1/2 $ Sobolev regularity. Moreover if the Froude number converges to zero we prove that the solutions of the aforementioned system converge (strongly) to a stratified three-dimensional Navier-Stokes system. No smallness assumption is assumed on the initial data.