Large time decay and growth for solutions of a viscous Boussinesq system

Abstract

In this paper we analyze the decay and the growth for large time of weak and strong solutions to the three-dimensional viscous Boussinesq system. We show that generic solutions blow up as $t\to \infty$ in the sense that the energy and the $L^p$-norms of the velocity field grow to infinity for large time, for $1\le p<3$. In the case of strong solutions we provide sharp estimates both from above and from below and explicit asymptotic profiles. We also show that solutions arising from $(u_0,{\theta }_0)$ with zero-mean for the initial temperature ${\theta }_0$ have a special behavior as $|x|$ or $t$ tends to infinity: contrarily to the generic case, their energy dissipates to zero for large time.

arXiv comment35 pages. 2nd version revised according to referee's remarks. to appear in Trans. Amer. Math. Soc

References in corpus (2)

• Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces
• Fine properties of self-similar solutions of the Navier-Stokes equations