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

This paper studies the one dimensional Navier-Stokes-Fourier-Korteweg system of equations describing the evolution of a heat-conducting compressible fluid that exhibits viscosity and capillarity. The main goal of the present analysis is to examine the dissipative structure of the system and to prove the global existence and the asymptotic decay of perturbations of equilibrium states. For that purpose, a novel nonlinear change of perturbed state variables, which takes into account that the conserved quantities contain density gradients, is introduced. These new perturbation variables satisfy a partially symmetric system whose linearization fulfills the generalized genuine coupling condition of Humpherys (J. Hyperbolic Differ. Equ. 2, 2005, no. 4, 963-974) for higher order systems. It is shown that the linearized system is symbol symmetrizable and an appropriate compensating matrix is constructed. This procedure allows to obtain linear decay rates which underlie a dissipative mechanism of regularity-gain type. This linear dissipative structure implies, in turn, the global decay of small perturbations to constant equilibrium states as solutions to the full nonlinear system.