  The three-dimensional incompressible Boussinesq system is one of the
important equations in fluid dynamics. The system describes the motion of
temperature-dependent incompressible flows. And the temperature naturally has
diffusion. Recently, Elgindi, Ghoul and Masmoudi constructed a $C^{1,\alpha}$
finite time blow-up solutions for Euler systems with finite energy. Inspired by
their works, we constructed $C^{1,\alpha}$ finite time blow-up solution for
Boussinesq equations where the temperature has diffusion and finite energy.
Generally speaking, the diffusion of temperature smooths the solution of the
system which is against the formations of singularity. The main difficulty is
that the Laplace operator of the temperature equation is not coercive under the
Sobolev weighted norm introduced by Elgindi. We introduced a new time depending
scaling formulation and new weighted Sobolev norms, under which we obtain the
nonlinear estimate. The new norm is well-coupled with the original norm, which
enables us to finish the proof.
