## Differential and Integral Equations

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

### Weak-renormalized solution for a nonlinear Boussinesq system

Abdelatif Attaoui, Dominique Blanchard, and Olivier Guibé

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356019602

**Mathematical Reviews number (MathSciNet)**

MR2501680

**Zentralblatt MATH identifier**

1240.35401

**Subjects**

Primary: 35Q35: PDEs in connection with fluid mechanics

Secondary: 35D30: Weak solutions 76D03: Existence, uniqueness, and regularity theory [See also 35Q30]

#### Citation

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