Analysis of a nonisothermal Maxwell--Stefan system with degenerate thermal conductivity

We prove the global-in-time existence of weak solutions for the Maxwell–Stefan–Fourier system with physically motivated degenerate thermal conductivity $\kappa (\theta )={\theta }^{\alpha }$, for $0<\alpha \le 2$. In contrast to existing nonisothermal compressible theories based on nondegenerate conductivities, the entropy inequality no longer gives control of the quantities $\mathrm{\nabla }\theta$ and $\mathrm{\nabla }\log \theta$. The main new ingredient is a renormalized energy estimate, which yields compactness of the temperature and allows the identification of the degenerate heat flux.