HAL: hal-00464890, version 2

arXiv: 1003.4921

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v2 (2010-07-28)

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

Lorenzo Brandolese 1, Maria Elena Schonbek 2

(2010-03-18)

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.

1:	Institut Camille Jordan (ICJ)
	CNRS : UMR5208 – Université Claude Bernard - Lyon I – Ecole Centrale de Lyon – Institut National des Sciences Appliquées (INSA) - Lyon
2:	Department of Mathematics, University of California Santa Cruz
	University of California, Santa Cruz

Subject

:

Mathematics/Analysis of PDEs

Keyword(s): Boussinesq – energy – heat convection – fluid – dissipation – Navier-Stokes – long time behaviour – blow up at infinity

hal-00464890, version 2

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From: Lorenzo Brandolese <>

Submitted on: Wednesday, 28 July 2010 09:30:01

Updated on: Thursday, 17 March 2011 20:07:22