Some existence results for the Brinkman-Fourier system

Abstract

In this paper, we prove the existence of weak solutions to the Brinkman-Fourier system under suitable conditions on the pressure. The main novelty of this paper lies in the fact that the viscosity and heat conductivity coefficients are functions of the temperature, furthermore, the viscosity coefficients may be degenerate in the sense that it may vanish as the temperature goes to zero. The key points to achieve the main results rely on the Galerkin method, the De Giorgi method and some compactness arguments.