Global existence and uniqueness of solutions to the three-dimensional Boussinesq equations

where p = p(x, t) is the scalar pressure, u = u(x, t) is the velocity vector field, and θ = θ (x, t) is a scalar quantity such as the concentration of a chemical substance or the temperature variation in a gravity field, in which case θe represents the buoyancy force. The nonnegative parameters μ and ν denote the molecular diffusion and the viscosity, respectively; u · ∇u := ua∂au, ∂au := ∂u ∂xa := u;a, u · ∇θ := ua∂aθ , ∇ · u := ua;a = , ∂u  ∂xa := (u );a, u = ubub. Over the repeated indices a and b summation is understood, ≤ a, b≤ . The Boussinesq system is one of themost commonly used fluidmodels since it has a vortex stretching effect similar to that in the D incompressible flow. The Boussinesq system has an important roles in the atmospheric sciences [] and is a model in many geophysical applications []. For this reason, this system is studied systematically by scientists from different domains.