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

TL;DR

This paper proves the global existence and decay of small perturbations of equilibrium states for a one-dimensional heat-conducting compressible fluid with capillarity, using a novel variable transformation and analyzing the system's dissipative structure.

Contribution

It introduces a new nonlinear change of variables that accounts for density gradients, enabling the analysis of the system's dissipative structure and decay properties.

Findings

Global decay of small perturbations to equilibrium states.

Linearized system is symbol symmetrizable with decay rates.

Dissipative mechanism of regularity-gain type established.

Abstract

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.