THE 3D INCOMPRESSIBLE BOUSSINESQ EQUATIONS WITH FRACTIONAL PARTIAL DISSIPATION

The system of the 3D Boussinesq equations is one of the most important models for geophysical fluids. The fundamental problem of whether or not reasonably smooth solutions to the 3D Boussinesq equations with the standard Laplacian dissipation can blow up in a finite time is an outstanding open problem. The Boussinesq equations with partial or fractional dissipation not only naturally generalize the classical Boussinesq equations, but also are physically relevant and mathematically important. This paper focuses on a system of the 3D Boussinesq equations with fractional partial dissipation and proves that any H-1-initial data always leads to a unique and global-in-time solution. The result of this paper is part of our efforts devoted to the global well-posedness problem on the Boussinesq equations with minimal dissipation.