Dissipative structure of one-dimensional isothermal compressible fluids of Korteweg type

TL;DR

This paper analyzes the dissipative structure of a one-dimensional isothermal compressible fluid of Korteweg type, proving decay of solutions to equilibrium states through the genuine coupling condition and symmetrizability techniques.

Contribution

It extends the genuine coupling condition to higher order systems and demonstrates decay estimates for the nonlinear Korteweg fluid system.

Findings

Genuine coupling condition holds for the system.

Solutions decay to equilibrium states over time.

Linear decay estimates imply global nonlinear decay.

Abstract

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.