Norm inflation for the Boussinesq system

We prove the norm inflation phenomena for the Boussinesq system on $\mathbb T^3$. For arbitrarily small initial data $(u_0,\rho_0)$ in the negative-order Besov spaces $\dot{B}^{-1}_{\infty, \infty} \times \dot{B}^{-1}_{\infty, \infty}$, the solution can become arbitrarily large in a short time. Such largeness can be detected in $\rho$ in Besov spaces of any negative order: $\dot{B}^{-s}_{\infty, \infty}$ for any $s>0$. Notice that our initial data space is scaling critical for $u$ and is scaling subcritical for $\rho$.