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This paper studies the one dimensional Navier-Stokes-Fourier-Korteweg systemof equations describing the evolution of a heat-conducting compressible fluidthat exhibits viscosity and capillarity. The main goal of the present analysisis to examine the dissipative structure of the system and to prove the globalexistence and the asymptotic decay of perturbations of equilibrium states. Forthat purpose, a novel nonlinear change of perturbed state variables, whichtakes into account that the conserved quantities contain density gradients, isintroduced. These new perturbation variables satisfy a partially symmetricsystem whose linearization fulfills the generalized genuine coupling conditionof Humpherys (J. Hyperbolic Differ. Equ. 2, 2005, no. 4, 963-974) for higherorder systems. It is shown that the linearized system is symbol symmetrizableand an appropriate compensating matrix is constructed. This procedure allows toobtain linear decay rates which underlie a dissipative mechanism ofregularity-gain type. This linear dissipative structure implies, in turn, theglobal decay of small perturbations to constant equilibrium states as solutionsto the full nonlinear system.

Global decay of perturbations of equilibrium states for one-dimensional heat conducting compressible fluids of Korteweg type