We show the existence of H\"{ o}lder continuous periodic weak solutions of the 2d Boussinesq equation with diffusive temperature which satisfy the prescribed kinetic energy. More precisely, for any smooth $e(t):[0,1]\rightarrow R_+$ and $\varepsilon\in (0, \frac{ 1}{ 10})$, there exist $v\in C^{ \frac{ 1}{ 10}-\varepsilon}([0,1]\times { \rm T}^2), \theta\in C_t^{ 1,\frac{ 1}{ 20}-\frac{ \varepsilon}{ 2}}C_x^{ 2,\frac{ 1}{ 10}-\varepsilon}([0,1]\times { \rm T}^2)$ which solve boussinesq equation in the sense of distribution and satisfy e(t)=\int_{ { \rm T}^2}|v(t,x)|^2dx, \quad \forall t\in [0,1].
Categories: math.AP