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


Authors: Lorenzo Brandolese and Maria E. Schonbek
Journal: Trans. Amer. Math. Soc. 364 (2012), 5057-5090
MSC (2010): Primary 76D05; Secondary 35Q35, 35B40
DOI: https://doi.org/10.1090/S0002-9947-2012-05432-8
Published electronically: May 30, 2012
MathSciNet review: 2931322
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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 $ \vert x\vert$ or $ t$ tends to infinity: contrary to the generic case, their energy dissipates to zero for large time.


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Additional Information

Lorenzo Brandolese
Affiliation: Université de Lyon, CNRS UMR 5208 Institut Camille Jordan, Université Lyon 1, 43 bd. du 11 novembre, Villeurbanne Cedex F-69622, France
Email: brandolese{@}math.univ-lyon1.fr

Maria E. Schonbek
Affiliation: Department of Mathematics, University of California Santa Cruz, Santa Cruz, California 95064
Email: schonbek@math.ucsc.edu

DOI: https://doi.org/10.1090/S0002-9947-2012-05432-8
Keywords: Boussinesq, energy, heat convection, fluid, dissipation, Navier–Stokes, long time behaviour, blow up at infinity
Received by editor(s): March 18, 2010
Received by editor(s) in revised form: July 27, 2010
Published electronically: May 30, 2012
Additional Notes: The work of both authors was partially supported by FBF Grant SC-08-34
The work of the second author was partially supported by NSF Grants DMS-0900909 and grant FRG-09523-503114.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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