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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-BĂ©nard 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 \(0.7692 \approx \frac{10}{13} < \alpha < 1\).

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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