Existence and uniqueness of solutions to the Boussinesq system with nonlinear thermal diffusion

The Boussinesq system arises in Fluid Mechanics when motion is governed by density gradients caused by temperature or concentration differences. In the former case, and when thermodynamical coefficients are regarded as temperature dependent, the system consists of the Navier-Stokes equations and the non linear heat equation coupled through the viscosity, bouyancy and convective terms. According to the balance between specific heat and thermal conductivity the diffusion term in the heat equation may lead to a singular or degenerate parabolic equation. In this paper we prove the existence of solutions of the general problem as well as the uniqueness of solutions when the spatial dimension is two

On a quasilinear degenerated system arising in semiconductors theory

A drift-diffusion model for semiconductors with nonlinear diffusion is considered. The model consists of two quasilinear degenerated parabolic equations for the carrier densities and the Poisson equation for the electric potential. We also assume Lipschitz continuous non linearities in the drift and {em generation-recombination terms. Existence of weak solutions is proven by using a regularization technique. Uniqueness of solutions is proven when either the diffusion term $varphi$ is strictly increasing and solutions have spatial derivatives in $L^1(Q_T)$ or when $varphi$ is non decreasing and a suitable entropy condition is fullfilled by the electric potential