Global existence of viscous heat conducting fluids

The aim of this work is to extend to compressible and heat conducting flows the well known concept of weak solutions to the incompressible Navier-Stokes equations due to J. Leray in 1933 and extended by P.-L. Lions in 1995 and E. Feireisl in 2001 to barotropic flows. Global existence and stability properties are obained in dimension 2 and 3 for general equation of state including polytropic gas law except close to vacuum. The main idea is to use a new mathematical entropy expressed as some additional Lyapunov function on the whole system which arises when the viscosity coefficients depend in a suitable way on the density. We will compare the results to the one by E. Feireisl where an inequality is obtained on the temperature but viscosity coefficients may depend on temperature. We will also explain why our result help to provide existence of global weak solutions for viscous shallow water systems. This is a joint work with Benoit Desjardins.