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 $t\to +\mathrm{\infty }$ for $1<p<3$ and decay to zero for $3<p\le \mathrm{\infty }$, 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 $\mathrm{\parallel }u(t){\mathrm{\parallel }}_2^2\sim c{t}^{1\mathrm{/}2}$ for large time. This contrasts with the case of the Navier-Stokes equations.