Upper bounds of the rates of decay for solutions of the Boussinesq equations

In this paper, upper bounds of the L ²-decay rate for the Boussinesq equations are considered. Using the L ² decay rate of solutions for the heat equation, and assuming that the solutions of the Boussinesq equations are smooth, we obtain the upper bounds of L ² decay rate for the smooth solutions and difference between the solutions of the Boussinesq equations and those of the heat system with the same initial data. The decay results may then be obtained by passing to the limit of approximating sequences of solutions. The main tool is the Fourier splitting method.