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The two-dimensional incompressible Boussinesq equations with general critical dissipation. (English) Zbl 1319.35193
From authors’ summary: The paper studies the global (in time) regularity of solutions to the two-dimensional incompressible Boussinesq equations with a general critical dissipation. The novelty of the paper is the reduction of the critical Boussinesq system to a critical active scalar equation or, more precisely, the generalized critical surface quasi-geostrophic equation. When the parameter \(\alpha\) of the Zigmund operator is restricted to a suitable range, the global regularity of the critical Boussinesq system is obtained by exploiting the global regularity of this scalar equation and the global bound for a combined quantity of the vorticity and the temperature.
Some parts of the authors’ summary were used.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B35 Stability in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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