Global regularity of temperature patches for the 3D non-diffusive Boussinesq system with large Prandtl number

Technical Summary: Global Regularity of Temperature Patches for the 3D Non-Diffusive Boussinesq System with Large Prandtl Number

Problem Statement
The paper addresses the global well-posedness of strong solutions for the 3D non-diffusive Boussinesq system (B) with large initial data, a problem that has remained open. Specifically, the authors investigate the evolution of "temperature patches," where the initial temperature $\theta_0$ is the characteristic function of a bounded domain $D_0$ (possibly with non-constant values inside). The central questions are:

1. Does the system admit a unique global strong solution for large initial data when the Prandtl number ($Pr$) is sufficiently large?
2. Does the boundary regularity of the temperature patch (specifically $C^{1,\gamma}$, $W^{2,\infty}$, and $C^{2,\gamma}$) persist globally in time?
3. Does the solution of the Boussinesq system converge to the solution of the 3D Stokes-transport system (ST) as $Pr \to \infty$, and does this limit preserve the patch boundary regularity?

Unlike the 2D case or the 3D case with small data, the 3D viscous Boussinesq system with $Pr=1$ and large data is not known to be globally well-posed. The authors propose that the regime of large Prandtl number serves as an alternative mechanism to ensure global regularity without requiring smallness assumptions on the initial data.

Methodology
The authors employ a combination of energy methods, paradifferential calculus, and striated (conormal) estimates within the framework of Besov spaces.

1. Large Prandtl Number Estimates:

   • The authors establish a priori global-in-time estimates for the velocity field $u$ in the spaces $L^\infty(0, T; H^{1/2})$ and $L^2(0, T; H^{3/2})$.
   • Crucially, they derive a global $L^\infty_T(L^{3,\infty})$ estimate for $u$ using a bootstrap argument. This estimate depends on the condition $Pr \geq Pr^*$, where $Pr^*$ is a threshold determined by the scale-invariant norms of the initial data ($\|u_0\|_{L^{3,\infty}} + \|\theta_0\|_{L^1}$).
   • This allows them to close the energy estimates without the smallness assumptions typically required for 3D Navier-Stokes or Boussinesq systems.

2. Striated Regularity and the "Good Unknown":

   • To handle the transport of the patch boundary, the authors introduce an auxiliary quantity $\Gamma = \Omega - R^{-1}\theta$, where $\Omega = \nabla \wedge u$ is the vorticity and $R^{-1}$ involves Riesz transforms.
   • In the limit $Pr \to \infty$, the Stokes-transport system satisfies $\Omega = R^{-1}\theta$, implying $\Gamma \equiv 0$. For finite but large $Pr$, $\Gamma$ satisfies a transport-diffusion equation (4.5) with a source term involving commutators.
   • The authors prove that $\Gamma$ satisfies uniform-in-$Pr$ estimates in appropriate Besov spaces. This "good unknown" decouples the vorticity dynamics from the temperature forcing in a way that facilitates uniform estimates.

3. Boundary Regularity Persistence:

   • The persistence of boundary regularity is linked to the regularity of an admissible system of conormal vector fields $W = \{W^i\}_{i=1}^5$ tangent to the patch boundary.
   • The authors prove that if $\nabla u \in L^1_T(C^\gamma)$, then the flow map preserves $C^{1,\gamma}$ regularity.
   • For higher regularity ($W^{2,\infty}$ and $C^{2,\gamma}$), they analyze the evolution of $\partial_W \nabla u$ (tangential derivatives of the velocity gradient). They decompose $\nabla u$ using the Biot-Savart law into terms involving $\Gamma$ and $\theta$.
   • New commutator estimates and smoothing estimates for the transport-diffusion equation (specifically Lemma 2.8 and Lemma 2.6) are used to bound the nonlinear terms uniformly in $Pr$.
   • A key innovation is a new approach to proving $W^{2,\infty}$ persistence that avoids the specific cancellations used in previous 2D works, making it adaptable to higher-order $W^{k,\infty}$ regularity.

4. Infinite Prandtl Number Limit:

   • Using the uniform estimates established in the large $Pr$ regime, the authors apply compactness arguments (Aubin-Lions lemma) to show that as $Pr \to \infty$, the sequence of solutions $(u_{Pr}, \theta_{Pr})$ converges to a limit $(u, \theta)$.
   • They rigorously justify that this limit satisfies the 3D Stokes-transport system (ST).
   • They further prove that the geometric structure of the patch and its boundary regularity are preserved in the limit.

Key Contributions and Results

• Theorem 1.1 (Global Well-posedness for Large Pr): The paper proves the global existence and uniqueness of strong solutions for the 3D non-diffusive Boussinesq system with initial data $(u_0, \theta_0) \in H^{1/2}(\mathbb{R}^3) \times (L^1 \cap L^s(\mathbb{R}^3))$ ($s>3$), provided the Prandtl number satisfies $Pr \geq Pr^*$. The threshold $Pr^*$ depends only on the scale-invariant norms of the initial data.
• Uniform Regularity Persistence: For non-constant temperature patches, the authors establish the global persistence of $C^{1,\gamma}$, $W^{2,\infty}$, and $C^{2,\gamma}$ boundary regularity. Crucially, the estimates for these regularities are uniform with respect to the Prandtl number in the regime $Pr \in [Pr^*, \infty)$.
• Theorem 1.4 (Infinite Prandtl Limit): The paper rigorously justifies the convergence of the 3D Boussinesq system to the 3D Stokes-transport system as $Pr \to \infty$. It demonstrates that the unique global solution of the limit system preserves the patch structure and the $C^{1,\gamma}$, $W^{2,\infty}$, and $C^{2,\gamma}$ regularity of the boundary globally in time.
• 3D Analogue of 2D Results: The result for the Stokes-transport system is presented as the 3D analogue of the main result by Grayer II (2023) for the 2D Stokes-transport system.

Significance and Claims
The authors claim that their work resolves the global well-posedness problem for the 3D non-diffusive Boussinesq system in the specific regime of large Prandtl numbers, a setting where the standard small-data assumptions are not required. By introducing the "good unknown" $\Gamma$ and developing uniform-in-$Pr$ estimates, they bridge the gap between the Boussinesq system and the Stokes-transport system.

The paper highlights that the contrast between the difficulty of the 3D Boussinesq system with $Pr=1$ and the global well-posedness of the Stokes-transport system (the $Pr \to \infty$ limit) suggests that a large Prandtl number acts as a regularizing mechanism. The rigorous justification of this limit, including the preservation of high-order boundary regularity for temperature patches, is a primary contribution.

The authors modestly note that while their methods can be extended to higher-order $C^{k,\gamma}$ and $W^{k,\infty}$ regularity (Remark 1.2), the current paper focuses on $k=1, 2$. They also acknowledge that extending the convergence result to the 2D whole space remains an open problem due to the lack of Sobolev embeddings available in 3D (Remark 1.5).