  This paper studies the dissipative structure of the system of equations that
describes the motion of a compressible, isothermal, viscous-capillar fluid of
Korteweg type in one space dimension. It is shown that the system satisfies the
genuine coupling condition of Humpherys (J. Hyperbolic Differ. Equ. 2, 2005,
no. 4, 963-974), which is, in turn, an extension to higher order systems of the
classical condition by Kawashima and Shizuta (Tohoku Math. J. 40, 1988, no. 3,
449-464; Hokkaido Math. J. 14, 1985, no. 2, 249-275) for second order systems.
It is proved that genuine coupling implies the decay of solutions to the
linearized system around a constant equilibrium state. For that purpose, the
symmetrizability of the Fourier symbol is used in order to construct an
appropriate compensating matrix. These linear decay estimates imply the global
decay of perturbations to constant equilibrium states as solutions to the full
nonlinear system, via a standard continuation argument.
