Global regularity of the two-dimensional fractional Boussinesq equations with critical dissipation

AMS Subject Classification 2020: 35Q35; 35B65; 76D03.

Keywords: Boussinesq equations; Global regularity; Fractional dissipation.

[email protected] (Qian Li), [email protected] (Jiahong Wu), [email protected] (Xiaojing Xu), [email protected] (Zhuan Ye)

Introduction

We consider the Cauchy problem for the two-dimensional (2D) incompressible Boussinesq equations with fractional dissipation

where $u=(u_1,u_2)$ denotes the velocity field, $p$ is the scalar pressure, $\theta$ represents the temperature (or buoyancy), and $e_2=(0,1)$ is the vertical unit vector. The fractional dissipation operator $\Lambda=(-\Delta)^{\frac{1}{2}}$ is defined via the Fourier transform $\widehat{\Lambda^\gamma f}(\xi)=|\xi|^\gamma\hat{f}(\xi)$, with $\alpha$ and $\beta$ characterizing the intensity of the non-local dissipation in the fluid and the thermal diffusion, respectively.

The Boussinesq system (1.1) plays a fundamental role in geophysical fluid dynamics, modeling atmospheric and oceanic flows driven by density differences [27, 29]. Mathematically, it also serves as a critical lower-dimensional toy model for the 3D incompressible Euler and Navier-Stokes equations, owing to the structural similarities in their vortex stretching mechanisms. Beyond the classical viscous and thermal diffusion models ($\alpha=\beta=2$), the incorporation of fractional dissipation operators carries profound physical significance in modern fluid dynamics. In large-scale geophysical scenarios, oceanic and atmospheric flows are inherently turbulent, rendering standard Brownian molecular diffusion inadequate to capture the complex, multi-scale mixing processes. Instead, the fractional Laplacian serves as a rigorous macroscopic parameterization for anomalous diffusion driven by Lévy flights of fluid parcels in fully developed turbulence [30]. Furthermore, specific fractional orders directly correspond to observable geophysical boundary effects; for instance, the case $\alpha=1$ mathematically encapsulates the Ekman pumping friction originating from the bottom boundary layers of rotating fluids [10, 29]. Therefore, identifying the exact analytical threshold for global regularity in the fractional regime not only resolves a longstanding mathematical challenge but also helps determine the minimum intensity of turbulent mixing required to prevent catastrophic atmospheric frontogenesis or finite-time singularity formation [10].

The principal mathematical challenge in establishing global regularity for (1.1) lies in the derivative forcing present in the vorticity equation. Denoting the scalar vorticity by $\omega=\partial_1u_2-\partial_2u_1$, applying the curl operator to (1.1) yields:

While the temperature equation obeys a robust maximum principle ($\|\theta(t)\|_{\infty}\leq\|\theta_0\|_{\infty}$), it provides no immediate bound for the derivative term $\partial_1\theta$ on the right-hand side of (1.2). This mismatch induces a severe “loss of derivatives.” Overcoming this loss heavily depends on the subtle interplay between the fractional dissipation regimes $\alpha$ and $\beta$. The regime where $\alpha+\beta=1$ is universally recognized as the critical line, whereas $\alpha+\beta>1$ and $\alpha+\beta<1$ are referred to as the subcritical and supercritical regimes, respectively.

Historical context

Due to its profound physical relevance and structural analogy to the 3D Euler equations—where the temperature gradient $\nabla^\perp\theta$ acts as a potent driving mechanism analogous to vortex stretching—the 2D Boussinesq system has inspired an extensive body of mathematical and computational literature [2, 16, 26]. The fundamental local well-posedness and blow-up criteria for the Boussinesq equations were pioneered by Chae and Nam [5]. Conversely, the quest for finite-time singularity formation in the purely inviscid equations has witnessed spectacular breakthroughs recently. Triggered by the striking numerical discovery of Luo and Hou [25], Elgindi and Jeong rigorously demonstrated finite-time singularity formation for strong solutions to the Boussinesq system [15]. Subsequently, Chen and Hou [7] provided a monumental proof of stable, finite-time blow-up for the 2D inviscid Boussinesq equations with smooth data in the presence of a boundary. These groundbreaking results, alongside Elgindi’s finite-time singularity formation for the 3D Euler equations with $C^{1,\alpha}$ data [14], profoundly underscore the virulent nature of the quadratic buoyancy forcing. In light of these finite-time singularity formations in the absence of dissipation, establishing global regularity necessitates an exquisite balance of partial or fractional dissipation. For global regularity, early breakthroughs were achieved in the partial dissipation regimes. Hou and Li [22], and independently Chae [4], established global well-posedness for the viscous but nondiffusive case ($\alpha= 2, \beta= 0$). The inviscid but strictly diffusive case ($\alpha= 0, \beta= 2$) was subsequently resolved by Chae [4] and further refined by Danchin and Paicu [13], and Hmidi and Keraani [18]. Concurrently, remarkable progress was made regarding anisotropic partial dissipation; for instance, Cao and Wu [3], as well as Larios, Lunasin, and Titi [24], established global regularity for systems lacking dissipation in certain spatial directions. Hmidi and Zerguine [21] further considered Euler–Boussinesq systems with fractional thermal diffusion. These works systematically demonstrate how dissipation in one equation can control the coupled system through sophisticated transport estimates and vorticity bounds.

An essential step toward weaker dissipation was made by Hmidi, Keraani, and Rousset. They proved global well-posedness for the Boussinesq–Navier–Stokes system with only first-order velocity dissipation [19] and for the Euler–Boussinesq system with only first-order thermal diffusion [20]. In these two models, the other dissipative term is absent. Their approach uses a combined quantity involving the vorticity and a Riesz transform of the temperature, replacing the derivative forcing by a commutator that can be estimated in suitable function spaces. This change of unknowns is a basic component of many subsequent arguments for fractional Boussinesq equations. Based on regularity estimates for the associated drift–diffusion equation, Ye [39] (see also [40]) gave an alternative proof for the Euler–Boussinesq system with critical thermal diffusion without the use of the combined quantity introduced in [20].

The global well-posedness in the subcritical regime has also attracted extensive attention and has been investigated in depth in recent years. It is worth emphasizing that the global regularity of (1) in the subcritical ranges is not a trivial problem. In fact, many subcritical cases have not been resolved. In particular, we do not know the global regularity for the case when $\alpha$ and $\beta$ are close to one half and $\alpha+ \beta> 1$. In the regimes with stronger velocity dissipation, Miao and Xue [28] obtained global well-posedness for a class of such systems, namely, $\alpha> \frac{6-\sqrt{6}}{4}$. Ye [38] enlarged the range of velocity dissipation to $\alpha> \frac{10-2\sqrt{10}}{5}$. Zhou, Li, Shang, Wu, Yuan, and Zhao [42] proved global well-posedness for $\frac{2}{3} < \alpha< 1$ and $\beta> \frac{1-\alpha}{\alpha}$. These results illustrate that the inequality $\alpha+ \beta> 1$ alone does not capture all the conditions needed by a given proof. Recently, Stefanov, Wu, Xu, and Ye [34], Theorem 1.1 made interesting progress in the subcritical regime, establishing global well-posedness for by pushing the threshold down to $\alpha> \frac{2}{3}$. In regimes with stronger thermal diffusion, Constantin and Vicol [11 Theorem 6.1] proved global well-posedness for smooth solutions of the fractional Boussinesq system with $0 < \alpha,\beta< 1$ under the condition $\beta> \frac{2}{2+\alpha}$, which was further improved by Ye and Xu [41]. The very recent preprint of Stefanov, Wu, Xu, and Ye [35] addresses subcritical dissipation for $\alpha\le\frac{2}{3}$, and their result holds under the condition

Compared with [19, 20], when both dissipative terms are present and $0 < \alpha,\beta< 1$, the general critical line $\alpha+\beta=1$ requires a more delicate balance. Jiu, Miao, Wu, and Zhang [23] established global regularity for

Stefanov and Wu [33] obtained the threshold

and Wu, Xu, Xue, and Ye [36] proved global regularity for $\alpha> \frac{10}{13}$. The proofs combine estimates for the corrected vorticity with regularity mechanisms for a critical active scalar. On the critical line $\alpha+\beta=1$, Stefanov, Wu, Xu, and Ye [34 Theorem 1.2] recently managed to lower the regularity threshold to $\alpha> \frac{2}{3}$. However, to control the explosive growth of the nonlinear commutators in this low-dissipation regime, a strict smallness condition on the initial temperature amplitude ($\lVert\theta_0\rVert_{L^\infty}\ll1$) was inevitably required. This smallness assumption has remained a major bottleneck in the literature, as the velocity dissipation alone proves fundamentally insufficient to close the energy estimates when $\alpha$ drops below $\frac{10}{13}$.

1.2. Main contributions and methodological innovations. In this paper, we completely remove the aforementioned smallness assumption and establish unconditional global well-posedness for arbitrary, large initial data on the critical line, pushing the regularity threshold further down to an algebraic root $\alpha_0=\frac{\sqrt{57}-5}{4}\approx0.637$ (the positive root of $2\alpha^2+5\alpha-4=0$). The present paper treats the critical interval $\alpha_0<\alpha\le\frac{10}{13}$ with arbitrary $H^s$ data. The main results are stated as follows.

To breach the barrier of the smallness assumption and transcend the $\alpha=\frac{10}{13}$ threshold, this paper introduces a highly refined Quadratic Normal Form (QNF) method coupled with exact scaling analysis. Our strategy unfolds in the following novel steps:

1. Bilinear normal form and frequency splitting: The standard linear correction $G = \omega- R_{\alpha}\theta$ leaves behind a virulent purely quadratic temperature term $C_2(\theta,\theta)$. We introduce a bilinear Fourier multiplier $\mathcal{B}(\theta,\theta)$ designed to algebraically cancel the non-resonant portion of $C_2$. A sophisticated high-output cutoff $\chi_{\mathrm{out}}$ is deployed to isolate the interactions into strictly separated non-resonant, resonant, and low-output regimes.

2. Eradicating the smallness assumption via exact exponent tracking: In the $L^m$ energy estimates of the corrected vorticity $H = \omega- R_{\alpha}\theta- \mathcal{B}(\theta,\theta)$, past works relied on loose Hölder interpolations, yielding a severe time-integrability failure unless $\lVert\theta\rVert_{L^\infty}$ was artificially small. By introducing a highly precise fractional chain rule and rigorously avoiding any over-estimation in the conjugate exponents (see Proposition 5.2), we ensure that the nonlinear growth is universally bounded by the natural temperature dissipation energy $\int_0^T \lVert\Lambda^{\frac{\beta}{2}}\theta\rVert_2^2\,dt < \infty$.

3. Positive Hölder gain via commutator cancellation: We demonstrate that as long as the Lebesgue exponent satisfies $m > \frac{2}{\alpha}$, the fractional dissipation $\Lambda^\alpha H$ strictly overtakes the transport commutators. This directly bootstraps the corrected vorticity to a Besov regularity gain $H \in B^{3\alpha-2}_{r,\infty}$, which in turn guarantees the velocity field $u \in C^{1-\beta}$.

4. The Silvestre-Schauder bootstrap: The regularity $u \in C^{1-\beta}$ perfectly aligns with the critical-drift threshold of Silvestre’s theorem [31] for advection fractional-diffusion equations. This triggers a spontaneous generation of Hölder regularity for the temperature field, immediately elevating the velocity drift regularity strictly beyond the critical threshold.

More detailed innovations can be summarized as follows:

Innovation 1: Tripartite phase-space decomposition and resonant geometry. Dividing by $D(k,l)$ inevitably encounters small-divisor singularities near the resonant variety $D(k,l) = 0$. We overcome this by constructing a smooth partition of unity in phase space: