Global Existence And Decay Of Weak Solutions For Boussinesq Equations With Damping

In this paper,we mainly study the global existence of the weak solution of the Cauchy problem of Boussinesq equations with nonlinear damping term and the decay estimate of the weak solution.The global existence of the weak solution of Boussinesq equations with damping term in R3 is first proved in this paper.We use the classical Galerkin approximation method to prove that for a given initial value u0?L?2(R3),?0? L2(R3),there exist a weak solution when the nonlinear index of the damping term is ??1.We mainly complete the proof of the global existence of weak solution in three steps.Firstly,we use Galerkin approximation method to construct the sequence of approximate solution of the equations.Then,we use various inequalities flexibly to prove the consistent boundedness of approximate solution sequences.Finaly,by using the weak compactness of Sobolev space and embedding theorem,we can prove that the weak limit of approximate solution sequences is the weak solution of the equations.On the basis of the global existence of weak solution in the first part,we used the method of "Fourier Splitting" to prove the decay estimate of the weak solution.In this part,we get two conclusions:1)When the nonlinear index is ??10/3.the weak solution has algebraic decay in the sense of L2 norm.2)When the nonlinear index of the damping term is ??10/3?the L2 decay estimate of error between the weak solution to the model in this paper and the ones to the classical 3 dimensional N-S equations.