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

Song Jiang, Quan Wang

TL;DR

The paper analyzes the two-dimensional inhomogeneous incompressible Navier–Stokes equations with a conservative gravity potential, proving that all finite-energy steady states are hydrostatic and must satisfy $\nabla p_s=-\rho_s\nabla f$.A perturbative framework around hydrostatic profiles is developed to establish global existence and regularity of strong solutions for data in $H^3$, and to control long-time behavior by overcoming the lack of standard decay in the nonzero-forcing setting.The authors derive a necessary and sufficient condition on the initial density perturbation for convergence to a linear hydrostatic density profile $\rho_s=-\gamma f+\beta$, and they demonstrate relaxation to hydrostatic equilibria despite transient Rayleigh–Taylor‑type growth, with detailed asymptotics for the velocity, pressure, and density.Extensions to linear problems on flat domains are presented, along with an appendix that places the results in the broader context of strong-solution theory and instability analyses under gravity.

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 \(\mathbf{u}|_{\partial Ω} = \mathbf{0}\), all admissible equilibria are hydrostatic and satisfy \(\nabla p_s = -ρ_s \nabla f\). 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 \(ρ_s = -γf + β\), with \(γ> 0\) and \(β> 0\). Finally, we establish improved regularity estimates for strong solutions corresponding to initial data in the Sobolev space \(H^3(Ω)\).

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

TL;DR

The paper analyzes the two-dimensional inhomogeneous incompressible Navier–Stokes equations with a conservative gravity potential, proving that all finite-energy steady states are hydrostatic and must satisfy $\nabla p_s=-\rho_s\nabla f$.A perturbative framework around hydrostatic profiles is developed to establish global existence and regularity of strong solutions for data in $H^3$, and to control long-time behavior by overcoming the lack of standard decay in the nonzero-forcing setting.The authors derive a necessary and sufficient condition on the initial density perturbation for convergence to a linear hydrostatic density profile $\rho_s=-\gamma f+\beta$, and they demonstrate relaxation to hydrostatic equilibria despite transient Rayleigh–Taylor‑type growth, with detailed asymptotics for the velocity, pressure, and density.Extensions to linear problems on flat domains are presented, along with an appendix that places the results in the broader context of strong-solution theory and instability analyses under gravity.

Abstract

This paper studies the two-dimensional inhomogeneous Navier--Stokes equations governing stratified flows in a bounded domain under a gravitational potential . Our main results are as follows. First, we provide a rigorous characterization of steady states, proving that under the Dirichlet condition , all admissible equilibria are hydrostatic and satisfy . 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 , with and . Finally, we establish improved regularity estimates for strong solutions corresponding to initial data in the Sobolev space \(H^3(Ω)\).
Paper Structure (28 sections, 28 theorems, 214 equations)

This paper contains 28 sections, 28 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Problem and Literature Review
  3. Steady-State Solutions
  4. Statement of the Main Theorems
  5. Key Ideas of the Proof
  6. Estimates on General Domains
  7. Estimates for $\varrho+\rho_s$
  8. Estimates for $\norm{\sqrt{\varrho+\rho_s}\mathbf{u}}_{L^{\infty}\left((0,\infty);L^2(\Omega)\right)}$ and $\|\nabla \mathbf{u}\|_{ L^{2}\left((0,\infty);L^2(\Omega)\right)}$
  9. Estimates for $\|\nabla \mathbf{u}\|_{ L^{\infty}\left((0,\infty);L^2(\Omega)\right)}$ and $\norm{\sqrt{\varrho+\rho_s}\mathbf{u}_t}_{L^{2}\left((0,\infty);L^2(\Omega)\right)}$
  10. Estimates for $\|\nabla \mathbf{u}_t\|_{ L^{2}\left((0,\infty);L^2(\Omega)\right)}$ and $\norm{\sqrt{\varrho+\rho_s}\mathbf{u}_t}_{L^{\infty}\left((0,\infty);L^2(\Omega)\right)}$
  11. Estimates for $\|\mathbf{u}\|_{ L^{\infty}\left((0,\infty);L^{\infty}(\Omega)\right)}$ and $\norm{\mathbf{u}}_{ L^{2}\left([0,T];W^{2,p}(\Omega)\right)}$
  12. Estimates for $\norm{\mathbf{u}}_{ L^{2}\left([0,T];W^{2,p}(\Omega)\right)}$
  13. Estimates for $\norm{\nabla \mathbf{u}_t}_{ L^{\infty}\left([0,T];L^{2}(\Omega)\right)}$ and $\norm{\mathbf{u}_{tt}}_{ L^{2}\left([0,T];L^{2}(\Omega)\right)}$
  14. Estimates for $\norm{\mathbf{u}}_{ L^{2}\left([0,T];H^{4}(\Omega)\right)}$
  15. Estimates of Linear Problems on $\Omega={\mathbb T}\times (0,h)$
  16. ...and 13 more sections

Key Result

Lemma 1.1

Any steady-state solution $(\mathbf{u}, \rho, P)$ of the system main-2--main-3 with $\mathbf{g}=\nabla f$ must be a hydrostatic equilibrium of the form: where the pressure $p_s$ and density $\rho_s$ satisfy the hydrostatic balance condition:

Theorems & Definitions (53)

  • Lemma 1.1: Hydrostatic equilibrium
  • proof
  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.3
  • Theorem 1.4
  • Lemma 2.1
  • ...and 43 more