Global Well-posedness for 3D Generalized Navier-Stokes-Boussinesq Equations

In this paper, we study the Cauchy problem for the 3D generalized Navier-Stokes-Boussinesq equations with fractional diffusion: { u(t) + (u . del)u + nu A(2 alpha)u = -del p + theta e(3), e(3) = (0,0,1)(T), theta(t) + (u . del)theta = 0, Divu = 0 With the help of the smoothing effect of the fractional diffusion operator and a logarithmic estimate, we prove the global well-posedness for this system with alpha >= 5/4. Moreover, the uniqueness and continuity of the solution with weaker initial data is based on Fourier localization technique. Our results extend ones on the 3D Navier-Stokes equations with fractional diffusion.