Eventual Regularity of the Two-Dimensional Boussinesq Equations with Supercritical Dissipation

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 suchweak 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 establishedby 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. Communicated by Peter Constantin. Q. Jiu School of Mathematics, Capital Normal University, Beijing 100037, People’s Republic of China e-mail: [email protected] J. Wu (B) Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078, USA e-mail: [email protected] W. Yang Department of Mathematics, Beifang University of Nationalities, Ningxia 750021, People’s Republic of China e-mail: [email protected]