Hyperbolic-parabolic systems have spatially homogeneous equilibria. Whenathe dissipation is weak, one can derive weakly nonlinear-dissipativeaapproximations that govern perturbations of these equilibria. Theseaapproximations are quadratically nonlinear. Up to a linearatransformation, they are independent of the dependent variables used toaexpress the original system. When the original system has an entropy,athe approximation is formally dissipative in a natural Hilbert space. Weashow that under a mild structural hypothesis, this approximation hasaglobal weak solutions for all initial data in that Hilbert space. Thisatheory applies to the compressible Navier-Stokes system. The resultingaapproximate system is an incompressible Navier-Stokes system coupled toaequations that govern the acoustic modes. The solution of thisaapproximate system is unique if the incompressible modes are uniquelyadetermined.a