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Regularity results for the 2D Boussinesq equations with critical or supercritical dissipation. (English) Zbl 1358.35136
Authors’ abstract: The incompressible Boussinesq equations serve as an important model in geophysics as well as in the study of Rayleigh-Bénard convection. One generalization is to replace the standard Laplacian operator by a fractional Laplacian operator, namely $$(-\Delta)^{\alpha/2}$$ in the velocity equation and $$(-\Delta)^{\beta/2}$$ in the temperature equation. This paper is concerned with the two-dimensional (2D) incompressible Boussinesq equations with critical dissipation $$(\alpha + \beta = 1)$$ or supercritical dissipation $$(\alpha + \beta < 1)$$. We prove two main results. This first one establishes the global-in-time existence of classical solutions to the critical Boussinesq equations with $$(\alpha + \beta = 1)$$ and $$0.7692 \approx \frac{10}{13} < \alpha < 1$$. The second one proves the eventual regularity of Leray-Hopf type weak solutions to the Boussinesq equations with supercritical dissipation $$(\alpha + \beta < 1)$$ and $$0.7692 \approx \frac{10}{13} < \alpha < 1$$.

##### 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
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