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.
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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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Communicated by Paul Newton.
This work was performed within the framework of the LABEX MILYON (ANR-10-LABX-0070) of Universit de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). L. Brandolese was also supported by project ANR-13-BS01-0003 - DYFICOLTI - DYnamique des Fluides, Couches Limites, Tourbillons et Interfaces.
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Brandolese, L., Mouzouni, C. A Short Proof of the Large Time Energy Growth for the Boussinesq System. J Nonlinear Sci 27, 1589–1608 (2017). https://doi.org/10.1007/s00332-017-9379-0
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DOI: https://doi.org/10.1007/s00332-017-9379-0