Differential and Integral Equations

Weak-renormalized solution for a nonlinear Boussinesq system

Abdelatif Attaoui, Dominique Blanchard, and Olivier Guibé

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

Differential Integral Equations, Volume 22, Number 5/6 (2009), 465-494.

First available in Project Euclid: 20 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q35: PDEs in connection with fluid mechanics
Secondary: 35D30: Weak solutions 76D03: Existence, uniqueness, and regularity theory [See also 35Q30]


Attaoui, Abdelatif; Blanchard, Dominique; Guibé, Olivier. Weak-renormalized solution for a nonlinear Boussinesq system. Differential Integral Equations 22 (2009), no. 5/6, 465--494. https://projecteuclid.org/euclid.die/1356019602

Export citation