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

## TL;DR

This paper provides a direct proof that solutions of the 3D Boussinesq system exhibit large-time energy growth in certain norms, with kinetic energy increasing as t^{1/2}, contrasting with Navier-Stokes behavior.

## Contribution

It offers the first direct proof of large-time energy growth for the Boussinesq system with explicit estimates, highlighting differences from Navier-Stokes dynamics.

## Key findings

- L^{p}-norms grow large for 1<p<3 as t→∞
- L^{p}-norms decay for 3<p≤∞ as t→∞
- Kinetic energy grows like c t^{1/2} for large t

## Abstract

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.