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 $\mathbb{R}^{3}$ grow large as $t \to \infty$ 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 $\mathbb{R}^{+}\times\mathbb{R}^{3}$. In particular, the kinetic energy blows up as $\|u(t)\|_{2}^{2} \sim ct ^{1/2}$ for large time. This constrasts with the case of the Navier-Stokes equations.
Domains
Mathematics [math]
Origin : Files produced by the author(s)