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

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

2010-03-25

arxiv.org/abs/1003.4921v2

Mathematics

Analysis of PDEs

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

Scientific paper

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.

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