Weak Solution Of The Boussinesq Equations In The Half Space L ~ 2 Decay Estimate

We consider the following Cauchy problem for Boussinesq equations:Here n is space dimension u = u(x,t),is the velocity field of the flow, θ is the active scalar (i.e. temperature), p(x, t) is the scalar pressure of the flow , f(x, t) is the exteral potential, u₀,θ₀ is the intial velocity and temperature respectively, γ≥0 and e ε≥ 0 is viscosity coefficient of the flow respectively.This paper is mainly concerned with the L² decay and its improvement of the weak solutions of the Boussinesq equations under the appropriate assumption of the external force f.Also, the decay rate of the difference between the solutions the Boussinesq equations and those of Stokes equation and heat equation is also discussed.