A short proof of the large time energy growth for the Boussinesq system

We give a direct proof of the fact that the $L^{p)$-norms of global solutions of the Boussinesq system in $R^{3}$ grow large as $ t \rightarrow + \infty $ for $ 1 < p < 3 $ and decay to zero for $ 3 < p \leq \infty $, providing exact estimates from below and above using a suitable decomposition of the space-time space $ R^{+} \times R^{3} $. In particular, the kinetic energy blows up as $ \| u(t) \|_{2}^{2} \sim c t^{1/2} $ for large time. This contrasts with the case of the Navier-Stokes equations.