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.
Similar content being viewed by others
References
Brandolese, L.: Fine properties of self-similar solutions of the Navier–Stokes equations. Arch. Rational Mech. Anal. 92, 375–401 (2009)
Brandolese, L., Schonbek, M.E.: Large time decay and growth for solutions of a viscous Boussinesq system. Trans. Amer. Math. Soc. 364, 5057–5090 (2012)
Duoandikoetxea, J., Zuazua, E.: Moments, masses de Dirac et décompositions de fonctions. C.R. Aca. Sci. Paris, t. série I 315, 693–698 (1992)
Fujigaki, Y., Miyakawa, T.: Asymptotic profiles of nonstationary incompressible Navier–Stokes flows in the whole space. SIAM J. Math. Anal. 33(3), 523–544 (2001)
Guo, B., Yuan, G.: On the suitable weak solutions for the Cauchy problem of the Boussinesq equations. Nonlinear Anal. 26(8), 1367–1385 (1996)
Kato, T.: Strong \(L^{p}\)-solutions of the Navier–Stokes equation in \(\mathbb{R}^{m}\), with applications to weak solutions. Math. Zeit. 187, 471–480 (1984)
Han, P.: Algebraic \(L^2\) decay for weak solutions of a viscous Boussinesq system in exterior domains. J. Differ. Equ. 252(12), 6306–6323 (2012)
Han, P., Schonbek, M.E.: Large time decay properties of solutions to a viscous Boussinesq system in a half space. Adv. Differ. Equ. 19(1–2), 87–132 (2014)
Hernon, D., Walsh, Ed J.: Enhanced energy dissipation rates in laminar boundary layers subjected to elevated levels of freestream turbulence. Fluid Dyn. Res. 39(4), 305–319 (2007)
Lemarié-Rieusset, P.G.: Recent developments in the Navier–Stokes problem. Chapman & Hall/CRC Research Notes in Mathematics, 431. Chapman & Hall/CRC, Boca Raton, FL (2002)
Meyer, Y.: Wavelets, paraproducts, and Navier–Stokes equations. In: Current Developments in Mathematics, 1997, Cambridge, MA, International Press, Boston, MA, pp. 105–212 (1996)
Miyakawa, T.: On space-time decay properties of nonstationary incompressible Navier–Stokes flows in \(\mathbf{R}^n\). Funkcial. Ekvac. 43(3), 541–557 (2000)
Schonbek, M.E.: \(L^2\) decay for weak solutions of the Navier–Stokes equations. Arch. Rational Mech. Anal. 88(3), 209–222 (1985)
Weng, S.: Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system. Math. Methods Appl. Sci 39(15), 4398–4418 (2016)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-017-9379-0