Partially dissipative hyperbolic systems: hypocoercivity and hyperbolisation.

We discuss the stability properties and relaxation limits associated with partially dissipative hyperbolic systems. Here, "partially dissipative" refers to systems with dissipative terms that do not affect all the components of the system. First, we show that under an algebraic structural condition (necessary in 1d) the coupling between the hyperbolic and the dissipative parts of the system allows us to recover time-decay properties for the whole solution. To that matter, we revisit the well-known hypocoercivity theory in this hyperbolic context and combine it with harmonic analysis tools such as the Littlewood-Paley theory.
Then, we interpret partially dissipative systems as hyperbolic approximations of parabolic systems and provide an element of answer to the infinite speed of propagation paradox that arises in fluid mechanics. As an application, we analyse a hyperbolic version of the compressible Navier-Stokes-Fourier system.