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The Boussinesq system in 3-dimensional bounded rough domains: Well-posedness in critical spaces and long-time behavior

Anatole Gaudin

Abstract

We study the three-dimensional Boussinesq system in bounded rough domains, including bounded Lipschitz and $\mathrm{C}^{1,α}$ domains, within a critical functional framework. We establish existence and uniqueness results that are global in time for small initial data and local in time for arbitrary initial data. Well-posedness in critical $\mathrm{L}^p$-based spaces is obtained in domains with Hölder continuous boundaries, relying on $\mathrm{L}^2$-maximal regularity in time. We also prove well-posedness in critical Besov spaces with third index equal to $1$, using $\mathrm{L}^1$-maximal regularity. In this Besov setting, the analysis applies to arbitrary bounded Lipschitz domains. In any case, we show that the fluid velocity stabilizes exponentially for large times and that the temperature converges to the initial averaged temperature of the fluid. The linear theory -- fitting the adapted product estimates and vice versa -- is properly established prior to the nonlinear analysis. With this fully prepared linear framework in hand, the nonlinear estimates that follow are then handled in the critical framework with a simplified treatment -- especially in the case where the velocity belongs to $\mathrm{L}^2(\mathrm{W}^{1,3})$ and the temperature to $\mathrm{L}^2(\mathrm{L}^{3/2})$ -- when compared with previously known similar results in smooth domains. This approach relies on a robust linear theory and sharp product estimates based on operator-theoretic methods and Besov space techniques, even in the case of the $\mathrm{L}^p$-theory. Finally, as part of the analysis, we establish several new results for the underlying linear operators, including refined characterizations for the domains of fractional powers of the Neumann Laplacian and of the Stokes operator in bounded Lipschitz domains.

The Boussinesq system in 3-dimensional bounded rough domains: Well-posedness in critical spaces and long-time behavior

Abstract

We study the three-dimensional Boussinesq system in bounded rough domains, including bounded Lipschitz and domains, within a critical functional framework. We establish existence and uniqueness results that are global in time for small initial data and local in time for arbitrary initial data. Well-posedness in critical -based spaces is obtained in domains with Hölder continuous boundaries, relying on -maximal regularity in time. We also prove well-posedness in critical Besov spaces with third index equal to , using -maximal regularity. In this Besov setting, the analysis applies to arbitrary bounded Lipschitz domains. In any case, we show that the fluid velocity stabilizes exponentially for large times and that the temperature converges to the initial averaged temperature of the fluid. The linear theory -- fitting the adapted product estimates and vice versa -- is properly established prior to the nonlinear analysis. With this fully prepared linear framework in hand, the nonlinear estimates that follow are then handled in the critical framework with a simplified treatment -- especially in the case where the velocity belongs to and the temperature to -- when compared with previously known similar results in smooth domains. This approach relies on a robust linear theory and sharp product estimates based on operator-theoretic methods and Besov space techniques, even in the case of the -theory. Finally, as part of the analysis, we establish several new results for the underlying linear operators, including refined characterizations for the domains of fractional powers of the Neumann Laplacian and of the Stokes operator in bounded Lipschitz domains.
Paper Structure (17 sections, 33 theorems, 240 equations, 2 figures)

This paper contains 17 sections, 33 theorems, 240 equations, 2 figures.

Table of Contents

  1. Introduction
  2. State of the art, the main theorems, and functional analytic motivations
  3. Some notations and background related to semigroup theory and maximal regularity
  4. Main strategy and road map of the present work
  5. Clarifications on notations and base knowledge for the current work.
  6. Functional Setting: Linear theory and maximal regularity estimates
  7. Linear theory for the Neumann Laplacian and the Stokes operator.
  8. The Neumann Laplacian
  9. The Stokes operator
  10. The parabolic maximal regularity estimates.
  11. (Bi-)Linear estimates for the solutions operators of the Boussinesq system.
  12. Well-posedness and asymptotics for the Boussinesq system
  13. Existence results
  14. Uniqueness results
  15. Exponential stabilization towards the equilibrium.
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{R}^3$ be a bounded $\mathrm{C}^{1,\alpha}$ domain with $\alpha>1/3$. Let $(\mathbf{u}_0,\theta_0)\in\mathrm{B}^{0}_{3,2}(\Omega)^3\times\mathrm{B}^{-1}_{3/2,1,0}(\Omega)$ such that $\mathop{\mathrm{div\,}}\nolimits \mathbf{u}_0 =0$ and $\mathbf{u}_0\cdot\mathbf{n}_{|_{\part the system eq:BoussinesqSystemIntro admits a unique weak solution Furthermore, when $T=\infty$, th

Figures (2)

  • Figure 1: Representation of parameters $(s,1/p)$
  • Figure 2: Representation of $(s,1/p)$.

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Da Prato--Grisvard
  • Remark 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Definition 2.1
  • Proposition 2.2
  • Proof 1
  • Proposition 2.3
  • ...and 52 more