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On the global regularity of the 2D Boussinesq equations with fractional dissipation. (English) Zbl 1372.35254
Summary: In this paper, we consider the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation and thermal diffusion. Attention is focused on the subcritical case when the velocity dissipation dominates. More precisely, we establish the global regularity result of the 2D Boussinesq equations in a new range of fractional powers of the Laplacian, namely \(1-\alpha<\beta<\min\left\{\frac{\alpha}{2},\;\frac{(3\alpha-2)(\alpha+2)}{10-7\alpha},\;\frac{2-2\alpha}{4\alpha-3}\right\}\) with \(0.783\approx\frac{21-\sqrt{217}}{8}<\alpha<1\). Therefore, this result significantly improves the previous work [C. Miao and the last author, NoDEA, Nonlinear Differ. Equ. Appl. 18, No. 6, 707–735 (2011; Zbl 1235.76020)] which obtained the global regularity result for \(1-\alpha<\beta<f(\alpha)\) with \(0.888\approx\frac{6-\sqrt{6}}{4}<\alpha<1\), where \(f(\alpha)<1\) is an explicit function.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B65 Smoothness and regularity of solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35R11 Fractional partial differential equations
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