  Hyperbolic-parabolic systems have spatially homogenous stationary states.
When the dissipation is weak, one can derive weakly nonlinear-dissipative
approximations that govern perturbations of these constant states. These
approximations are quadratically nonlinear. When the original system has an
entropy, the approximation is formally dissipative in a natural Hilbert space.
We show that when the approximation is strictly dissipative it has global weak
solutions for all initial data in that Hilbert space. We also prove a
weak-strong uniqueness theorem for it. In addition, we give a Kawashima type
criterion for this approximation to be strictly dissipative. We apply the
theory to the compressible Navier-Stokes system.
