Boussinesq Equation Group Of Space-time Decay Nature Of The Study

We consider the following Cauchy problem of the Boussinesq equations:Here u(x,t) = (u₁,u₂,u₃) is the velocity field of the flow; 9{x,t) is the active scalar function(i.e. temperture); p(x, t) is the scalar pressure of the flow.And f(x,t) = (f₁,f₂,f₃) is the external potential; u₀(x),θ₀(x) is the initial velociy and temperature respectively; γ ≥ 0 and ε≥ 0 is the viscosity coefficient and the thermal expansion coefficient of the flow respectively.The contents of the paper include the following three parts:1. Construction of the approximation solutions of the Boussinesq equations andtheir integral representations .Applying the linearized approximation equations of the Boussinesq equations and the fundamental solution of the Stokes equations and the heat equation, we derive the integral representations of the approximate solutions.2. Uniform moment estimates for the approximation solutions .In this part, under some conditions of the initial data u₀,θ₀ and the given function ∫, we obtain the uniform weighted L²-estimates for the approximate solutions .3. Uniform weighted L^p -estimates(p > 3) for the approximation solutions .