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

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

References in corpus (2)

• Fine properties of self-similar solutions of the Navier-Stokes equations
• Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system

Cited by in corpus (1)

• Norm inflation for the Boussinesq system