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Eventual regularity of the two-dimensional Boussinesq equations with supercritical dissipation. (English) Zbl 1311.35221
Summary: This paper studies solutions of the two-dimensional incompressible Boussinesq equations with fractional dissipation. The spatial domain is a periodic box. The Boussinesq equations concerned here govern the coupled evolution of the fluid velocity and the temperature and have applications in fluid mechanics and geophysics. When the dissipation is in the supercritical regime (the sum of the fractional powers of the Laplacians in the velocity and the temperature equations is less than 1), the classical solutions of the Boussinesq equations are not known to be global in time. Leray-Hopf type weak solutions do exist. This paper proves that such weak solutions become eventually regular (smooth after some time $$T>0$$) when the fractional Laplacian powers are in a suitable supercritical range. This eventual regularity is established by exploiting the regularity of a combined quantity of the vorticity and the temperature as well as the eventual regularity of a generalized supercritical surface quasi-geostrophic equation.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 35B65 Smoothness and regularity of solutions to PDEs 35D30 Weak solutions to PDEs 26A33 Fractional derivatives and integrals
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