The Two-Dimensional Incompressible Boussinesq Equations with General Critical Dissipation

Abstract

The two-dimensional incompressible Boussinesq equations with partial or fractional dissipation have recently attracted considerable attention and the global regularity issue has been extensively investigated. This paper aims at the global regularity in the case when the dissipation is critical. The critical dissipation refers to $\alpha +\beta=1$ when $\Lambda^\alpha \equiv (-\Delta)^{\frac{\alpha}{2}}$ and $\Lambda^\beta$ represent the fractional Laplacian dissipation in the velocity and the temperature equations, respectively. When $\alpha=1$ and $\beta =0$ or when $\alpha=0$ and $\beta=1$, the global regularity was obtained in [T. Hmidi, S. Keraani, and F. Rousset, J. Differential Equations, 249 (2010), pp. 2147--2174; T. Hmidi, S. Keraani, and F. Rousset, Comm. Partial Differential Equations, 36 (2011), pp. 420--445]. However, the approaches there do not apply to the situation when $\alpha+\beta=1$ with both $\alpha>0$ and $\beta>0$. The novelty here is to reduce the critical Boussinesq system to a critical active scalar equation or, more precisely, the generalized critical surface quasi-geostrophic equation. When $\alpha$ is restricted to a suitable range, the global regularity of the critical Boussinesq system can be obtained by exploiting the global regularity of this scalar equation and the global bound for a combined quantity of the vorticity and the temperature.