On the large time behavior of the 2D inhomogeneous incompressible viscous flows

TL;DR

This paper analyzes the long-term behavior of 2D inhomogeneous incompressible viscous flows, characterizing steady states, their stability, and conditions for convergence to hydrostatic equilibrium, with improved regularity results.

Contribution

It provides a rigorous characterization of steady states, stability analysis around hydrostatic profiles, and conditions for convergence to hydrostatic density profiles in 2D inhomogeneous Navier-Stokes flows.

Findings

All admissible equilibria are hydrostatic under Dirichlet boundary conditions.

The system relaxes to a hydrostatic equilibrium despite transient Rayleigh--Taylor growth.

Conditions on initial density perturbations determine convergence to linear hydrostatic density.

Abstract

This paper studies the two-dimensional inhomogeneous Navier--Stokes equations governing stratified flows in a bounded domain under a gravitational potential \(f\). Our main results are as follows. First, we provide a rigorous characterization of steady states, proving that under the Dirichlet condition \(\mathbf{u}|_{\partial \Omega} = \mathbf{0}\), all admissible equilibria are hydrostatic and satisfy \(\nabla p_s = -\rho_s \nabla f\). Second, through a perturbative analysis around arbitrary hydrostatic profiles, we show that despite possible transient growth induced by the Rayleigh--Taylor mechanism, the system always relaxes to a hydrostatic equilibrium. Third, we identify a necessary and sufficient condition on the initial density perturbation for convergence to a linear hydrostatic density profile of the form \(\rho_s = -\gamma f + \beta\), with \(\gamma > 0\) and \(\beta > 0\). Finally, we establish improved regularity estimates for strong solutions corresponding to initial data in the Sobolev space \(H^3(\Omega)\).