In this paper, we study the analytical smoothing effect of Cauchy problem for
the incompressible Boussinesq equations. Precisely, we use the Fourier method
to prove that the Sobolev H 1-solution to the incompressible Boussinesq
equations in periodic domain is analytic for any positive time. So the
incompressible Boussinesq equation admet exactly same smoothing effect
properties of incompressible Navier-Stokes equations
