Uniqueness Of Weak Solutions To The N-dimensional Non-diffusive Boussinesq System

In this paper,we mainly investigate the global existence and uniqueness of weak solutions of the Boussinesq system of the n-dimensional(n?3)without thermal diffusion in the periodic domain Tn with a periodic boundary conditions for the initial data(u0,?0)? L2(Tn)× L2(Tn)and ?0 ? L4/n+2(Tn).Firstly,we use Fourier series and spectral theory to establish Bernstein-type inequalities and related norm estimates of equations,Here we avoid complex harmonic methods.And then we use these estimates and stream function methods to prove the uniqueness of Boussinesq equations.The method of verifying the uniqueness of the Boussinesq equations is used to study the vanishing thermal diffusion limit of the equation in the periodic domain.The strong convergence of the solution of the equations and the better convergence rate of the equations are obtained.Ultimately,when ??1/2+n+4,we use the Brezis-Wainger type inequality established by spectral theory to study the equations in the periodic domain Tn regularity problems and relevant conclusions.