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Fractional regularity, global persistence, and asymptotic properties of the Boussinesq equations on bounded domains

Mustafa Sencer Aydın, Pranava Chaitanya Jayanti

TL;DR

This work analyzes the 2D Boussinesq equations on bounded domains without thermal diffusion, proving global persistence of strong solutions for a spectrum of initial data regularities using parabolic maximal regularity and initial-time compatibility. The authors establish persistence for $V\times H^1$, $D(A)\times H^2$, and $H^k\times H^k$ data, extend these results to fractional regularity, and derive detailed long-time behavior including sharp asymptotics and a necessary-and-sufficient condition for convergence to stratified steady states. The approach blends Galerkin approximations, maximal regularity estimates, and commutator techniques to propagate regularity across the coupled velocity-density system. These results advance understanding of the long-time dynamics of viscous, non-diffusive Boussinesq flows on bounded domains, with precise regularity and asymptotic characterizations that were previously open in this setting.

Abstract

We address the long-time behavior of the 2D Boussinesq system, which consists of the incompressible Navier-Stokes equations driven by a non-diffusive density. We construct globally persistent solutions on a smooth bounded domain, when the initial data belongs to $(H^k\cap V)\times H^k$ for $k\in\mathbb{N}$ and $H^s\times H^s$ for $0<s<2$. The proofs use parabolic maximal regularity and specific compatibility conditions at the initial time. Additionally, we also deduce various asymptotic properties of the velocity and density in the long-time limit and present a necessary and sufficient condition for the convergence to a steady state.

Fractional regularity, global persistence, and asymptotic properties of the Boussinesq equations on bounded domains

TL;DR

This work analyzes the 2D Boussinesq equations on bounded domains without thermal diffusion, proving global persistence of strong solutions for a spectrum of initial data regularities using parabolic maximal regularity and initial-time compatibility. The authors establish persistence for , , and data, extend these results to fractional regularity, and derive detailed long-time behavior including sharp asymptotics and a necessary-and-sufficient condition for convergence to stratified steady states. The approach blends Galerkin approximations, maximal regularity estimates, and commutator techniques to propagate regularity across the coupled velocity-density system. These results advance understanding of the long-time dynamics of viscous, non-diffusive Boussinesq flows on bounded domains, with precise regularity and asymptotic characterizations that were previously open in this setting.

Abstract

We address the long-time behavior of the 2D Boussinesq system, which consists of the incompressible Navier-Stokes equations driven by a non-diffusive density. We construct globally persistent solutions on a smooth bounded domain, when the initial data belongs to for and for . The proofs use parabolic maximal regularity and specific compatibility conditions at the initial time. Additionally, we also deduce various asymptotic properties of the velocity and density in the long-time limit and present a necessary and sufficient condition for the convergence to a steady state.
Paper Structure (19 sections, 8 theorems, 99 equations)

This paper contains 19 sections, 8 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and the main results
  3. $V\times H^1$ persistence --- Proof of Theorem \ref{['T01']}$(i)$
  4. Approximate equations and a priori estimates
  5. Density in $L^p$
  6. Energy estimates
  7. Gradient of the density
  8. Higher order energy estimates
  9. Passing to the limit
  10. Uniqueness of solutions
  11. Time continuity of density
  12. Improved velocity regularity and asymptotic properties
  13. Time derivatives
  14. $D(A)\times H^2$ persistence --- Proof of Theorem \ref{['T01']}$(ii)$
  15. $H^k\times H^k$ persistence --- Proof of Theorem \ref{['T01']}$(iii)$
  16. ...and 4 more sections

Key Result

Theorem 2.1

Assume that $(u_0,\rho_0) \in D(A) \times H^{1}$. Then, for any $T\in (0,\infty)$,

Theorems & Definitions (15)

  • Theorem 2.1: AKZ Global persistence with $D(A)\times H^1$ data
  • Theorem 2.2: Global persistence with $(H^k\cap V)\times H^k$ data
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Theorem 2.8: Global persistence with fractional regularity data
  • Theorem 2.9: Asymptotic properties
  • Lemma 2.1
  • ...and 5 more