Stability of the 3D Boussinesq equations with partial dissipation near the hydrostatic balance

The Boussinesq equations with partial or fractional dissipation not only naturally generalize the classical Boussinesq equations but also are physically relevant and mathematically important. Unfortunately, it is not often well-understood for many ranges of fractional powers. This paper focuses on a system of the 3D Boussinesq equations with fractional horizontal (-Delta(h))(beta)theta.. u and (-Delta(h))(alpha)u dissipation and proves that if the initial data (u0,..0) in the Sobolev space H3(R3) are close enough to the hydrostatic balance state, respectively, the equations with alpha, beta epsilon(1/2, 1] then always lead to a steady solution.