Boussinesq Equations Solution Regular Problem

We consider the following Cauchy problem for Boussinesq equations:The unknown functions u = u(x,t), θ= θ(x,t) and p = p(x,t) are the velocity field , the scalar temperature and the scalar pressure of the flow respectively. f = f(x,t) is the known external potential, u0 = u0(x) and θ0 = θ0(x) are the initial velocity and temperature respectively. The constants v ≥ 0 and μ ≥ 0 are the viscosity coefficient and the thermal expansion coefficient of the flow respectively.In this paper, we mainly study the regularity of the weak solutions and the partial regularity for the suitable weak solutions to the three-dimensional incompressible Boussinesq equations.The contents of the paper include two parts:1. We consider the " Serrin class " of the weak solutions of the three-dimensional incompressible Boussinesq equations. We discuss two class sufficient conditions to guarantee that the weak solutions are regular, which are similar to the corresponding results of the Navier-Stokes equations.2. We consider the partial regularity for the suitable weak solutions of (*). Firstly, based on the generalized energy inequality , we get estimates of some scaled nondimensional quantities . Secondly, we employ the iterative technique to obtain the smallness of some scaled quantities of temperature field. Finally, by scaling arguments we get the partial regularity for the suitable weak solutions.