[Paper Review] Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data

Raphaël Danchin, Marius Paicu|HAL (Le Centre pour la Communication Scientifique Directe)|Jun 25, 2008

Navier-Stokes equation solutionsMathematics17 citations

TL;DR

This paper establishes global well-posedness for the 2D inviscid Boussinesq system with Yudovich-type initial data, proving existence and uniqueness of global solutions for finite energy initial velocity and temperature when the initial vorticity is bounded and the initial temperature lies in $L^2 \cap B^{-1}_{\infty,1}$. The result holds without smallness assumptions, relying on Besov space estimates and energy methods with heat semigroup control.

ABSTRACT

The present paper is dedicated to the study of the global existence for the inviscid two-dimensional Boussinesq system. We focus on finite energy data with bounded vorticity and we find out that, under quite a natural additional assumption on the initial temperature, there exists a global unique solution. None smallness conditions are imposed on the data. The global existence issues for infinite energy initial velocity, and for the Bénard system are also discussed.

Motivation & Objective

• To establish global existence and uniqueness of solutions for the 2D inviscid Boussinesq system with initial data in $L^2$ and bounded vorticity.
• To identify minimal regularity conditions on the initial temperature ensuring bounded vorticity propagation over time.
• To extend Yudovich's global well-posedness result for the 2D Euler equations to the coupled Boussinesq system with zero viscosity but positive thermal diffusivity.
• To analyze the role of the initial temperature's gradient in $B^{-1}_{\infty,1}$ for controlling vorticity growth.
• To extend the analysis to the Bénard system and infinite energy initial velocity cases.

Proposed method

• Utilizes the heat semigroup $e^{\kappa t\Delta}$ to control the regularity of the temperature field via $L^1_{\text{loc}}(\mathbb{R}_+; L^\infty)$ estimates on $\nabla \theta_0$.
• Applies Besov space techniques, particularly $B^{-1}_{\infty,1}$, to characterize the required regularity of $\theta_0$ for bounded vorticity propagation.
• Employs energy estimates combining $L^2$ norms of $\theta$ and $u$, with a Gronwall-type argument to control growth in time.
• Uses dyadic decomposition $\Delta_q$ and frequency localization to derive $L^\infty$ bounds on $\omega$ via interpolation and Bernstein-type inequalities.
• Establishes uniqueness via energy estimates and $L^\infty$ control of velocity and vorticity, leveraging the structure of the transport-diffusion equation for $\theta$.
• Adapts the proof framework to the Bénard system by introducing a lower-order term $u_2$ in the temperature equation and adjusting energy estimates accordingly.

Experimental results

Research questions

• RQ1 Under what minimal regularity assumptions on the initial temperature $\theta_0$ does the vorticity remain bounded for all time in the inviscid Boussinesq system?
• RQ2 Can global well-posedness be established for the inviscid Boussinesq system with $\kappa > 0$, $\nu = 0$, and initial data in $L^2$ with bounded vorticity?
• RQ3 How does the Besov space $B^{-1}_{\infty,1}$ for $\theta_0$ relate to the propagation of bounded vorticity in the absence of viscosity?
• RQ4 What is the role of the heat semigroup in controlling the regularity of $\theta$ and ensuring $L^1_{\text{loc}}(\mathbb{R}_+; L^\infty)$ behavior of $\nabla \theta$?
• RQ5 Can the framework be extended to the Bénard system, where an additional $u_2$ term appears in the temperature equation?

Key findings

• The system $(B_{\kappa,0})$ admits a unique global solution for initial data $\theta_0 \in L^2 \cap B^{-1}_{\infty,1}$ and $u_0 \in L^2$ with $\text{div}\,u_0 = 0$, provided the initial vorticity $\omega_0$ is in $L^r \cap L^\infty$ for some $r \geq 2$.
• The solution satisfies $\theta \in \mathcal{C}(\mathbb{R}_+; L^2 \cap B^{-1}_{\infty,1}) \cap L^2_{\text{loc}}(\mathbb{R}_+; H^1) \cap L^1_{\text{loc}}(\mathbb{R}_+; B^1_{\infty,1})$ and $u \in \mathcal{C}^{0,1}_{\text{loc}}(\mathbb{R}_+; L^2)$ with $\omega \in L^\infty_{\text{loc}}(\mathbb{R}_+; L^r \cap L^\infty)$.
• The condition $\theta_0 \in B^{-1}_{\infty,1}$ is equivalent to $\nabla \theta_0 \in B^{-2}_{\infty,1}$, which ensures $\nabla e^{\kappa t\Delta}\theta_0 \in L^1_{\text{loc}}(\mathbb{R}_+; L^\infty)$, crucial for vorticity boundedness.
• The energy estimate yields $\| (\theta,u)(t) \|_{L^2}^2 + 2\kappa \int_0^t \| \nabla \theta(\tau) \|_{L^2}^2 d\tau \leq \| (\theta_0,u_0) \|_{L^2}^2 e^{2t}$, showing exponential-in-time growth control.
• For the Bénard system, the additional $u_2$ term in the temperature equation is handled via modified energy estimates and the same $L^1_{\text{loc}}(\mathbb{R}_+; B^1_{\infty,1})$ control on $\theta$ is preserved.
• The result extends to infinite energy initial velocity and $\theta_0 \in L^p$ or $u_0 \in B^1_{\infty,1}$, with corresponding $L^\infty(\mathbb{R}_+; L^p)$ or $\mathcal{C}(\mathbb{R}_+; B^1_{\infty,1})$ regularity for $\theta$ and $u$.