Boussinesq Equation Weak Solutions Of The L ~ 2 Decay

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, 9 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 ε ≥ 0 is viscosity coefficient of the flow respectively.The contents of the paper include two parts:(1) L² decay for weak solutions of the Cauchy problem for the Boussinesq equations We first get the uniform L2 decay of the smooth solutions. Then we can actuallyobtain the L2 decay for weak solutions by passing limit of the apporximate sequences of solutions. The main tool used is the Fourier splliting method. We first consider the large-time behavior of the temperature of the Boussinesq equation, baesd on which we can obtain, under some assumption of decay rate of given f, the large time behavior of the velocity vector field.(2) Upper bounds estimates for solutions of the Cauchy problem for the Boussinesq equations.Using the L²-decay rate of solution of heat equation, and assuming that the solution of B is smooth, We obtain the L² decay of the solutins of the Boussinesq equations, by comparing the solutions of Boussinesq equations with the solutions of heat equation. The main tool is also the Fourier splitting method.