Global existence of smooth solutions and stability of the constant state for dissipative hyperbolic systems with applications to extended thermodynamics

The entropy principle plays an important role in hyperbolic systems of balance laws: symmetrization, principal subsystems and nesting theories, equilibrium manifold. After a brief survey on these questions we present recent results concerning the local and global well-posedness of the Cauchy problem for smooth solutions with particular attention to the genuine coupling Kawashima condition. These results are applied to the case of extended thermodynamics and we prove that the K-condition is satisfied in the case of the 13-moment Grad theory with the consequence that there exist global smooth solutions for small initial data and the solutions converge to constant equilibrium states.