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Journal of Nonlinear Science

, Volume 27, Issue 5, pp 1589–1608 | Cite as

A Short Proof of the Large Time Energy Growth for the Boussinesq System

  • Lorenzo BrandoleseEmail author
  • Charafeddine Mouzouni
Article

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 \(t\rightarrow \infty \) for \(1<p<3\) and decay to zero for \(3<p\le \infty \), 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 \(\Vert u(t)\Vert _2^2\sim ct^{1/2}\) for large time. This contrasts with the case of the Navier–Stokes equations.

Keywords

Kato spaces The Boussinesq equation and asymptotic behavior 

Mathematics Subject Classification

Primary 76D05 Secondary 35B40 

Notes

Acknowledgements

The authors would like to thank the referees for their careful reading and useful suggestions that have been incorporated in this revised version of the manuscript.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Univ Lyon, Université Claude Bernard Lyon 1CNRS UMR 5208, Institut Camille JordanVilleurbanne CedexFrance
  2. 2.Univ Lyon, École centrale de Lyon, CNRS UMR 5208Institut Camille JordanÉcully CedexFrance

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