A Bowel Movement In The Global Three-dimensional Boussinesq Equations L ~ 2 Stability

We consider the following Boussinesq equationsΩ is a domain of R³, 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 global L² stability for large solutions to the Boussinesq equations in three-dimensional bounded or unbounded domains. Under suitable conditions of the large solutions, it was shown that the large solutions were stable. And we got the equivalent condition of this main stability condition. Moreover, the global existence and the stability of two-dimensional Boussinesq equations under three-dimensional perturbations were also obtained.