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May/June 2009 Weak-renormalized solution for a nonlinear Boussinesq system
Abdelatif Attaoui, Dominique Blanchard, Olivier Guibé
Differential Integral Equations 22(5/6): 465-494 (May/June 2009).

Abstract

We give a few existence results of a weak-renormalized solution for a class of nonlinear Boussinesq systems: \begin{eqnarray*} & \dfrac{\partial u}{\partial t}+(u\cdot\nabla)u- 2 \textrm{ div } (\mu(\theta) D u)+\nabla p= F(\theta) & \textrm{ in } \Omega\times(0,T),\\ & \dfrac{\partial b(\theta)}{\partial t}+u\cdot\nabla b(\theta)-\Delta \theta = 2 \mu(\theta) |D u |^2 & \textrm{ in } \Omega\times(0,T),\\ & \textrm{div }u = 0 & \textrm{ in } \Omega\times(0,T), \end{eqnarray*} where $u$ is the velocity field of the fluid, $p$ is the pressure and $\theta$ is the temperature. The function $\mu(\theta)$ is the viscosity of the fluid and the function $F(\theta)$ is the buoyancy force which satisfies a growth assumption in dimension $2$ and is bounded in dimension $3$. The usual techniques for Navier-Stokes equations are mixed with the tools involved for renormalized solutions.

Citation

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Abdelatif Attaoui. Dominique Blanchard. Olivier Guibé. "Weak-renormalized solution for a nonlinear Boussinesq system." Differential Integral Equations 22 (5/6) 465 - 494, May/June 2009.

Information

Published: May/June 2009
First available in Project Euclid: 20 December 2012

zbMATH: 1240.35401
MathSciNet: MR2501680

Subjects:
Primary: 35Q35
Secondary: 35D30, 76D03

Rights: Copyright © 2009 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.22 • No. 5/6 • May/June 2009
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