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A global regularity result for the 2D Boussinesq equations with critical dissipation. (English) Zbl 1420.35263
Summary: This paper examines the global regularity problem on the two-dimensional incompressible Boussinesq equations with fractional dissipation, given by $$\Lambda^\alpha u$$ in the velocity equation and by $$\Lambda^\beta \theta$$ in the temperature equation, where $$\Lambda - \sqrt { - \Delta }$$ denotes the Zygmund operator. We establish the global existence and smoothness of classical solutions when ($$\alpha, \beta$$) is in the critical range: $$\alpha > (\sqrt {1777} - 23)/24 = 0.789103 \ldots$$, $${\beta} > 0$$, and $$\alpha +\beta = 1$$. This result improves previous work which obtained the global regularity for $$\alpha > (23-\sqrt {145})/12 \approx 0.9132,\;\beta>0$$, and $$\alpha + \beta= 1$$.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35R11 Fractional partial differential equations 35Q86 PDEs in connection with geophysics 86A10 Meteorology and atmospheric physics 35B65 Smoothness and regularity of solutions to PDEs 35A01 Existence problems for PDEs: global existence, local existence, non-existence 76D05 Navier-Stokes equations for incompressible viscous fluids 35A09 Classical solutions to PDEs
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