Weak-renormalized solution for a nonlinear Boussinesq system

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.