Thermodynamics and stability of non-equilibrium steady states in open systems -- case study for compressible heat conducting fluid

TL;DR

The paper develops a thermodynamics-driven framework to construct Lyapunov-like functionals for nonlinear stability of steady states in compressible, heat-conducting fluids. It derives a constrained entropy-based functional for isolated, spatially homogeneous states and identifies the appropriate Lagrange multipliers to ensure decreasing behavior and nonnegativity. The approach is then extended to open systems via an affine correction that yields a Lyapunov functional for spatially inhomogeneous steady states. Throughout, the relative-energy/ballistic-free-energy perspective is shown to align with Feireisl’s established functionals, clarifying their thermodynamic origin and general applicability. The work connects classical thermodynamics, convexity/stability criteria, and modern PDE stability theory in a unified framework applicable to non-equilibrium steady states.

Abstract

We carefully go through all the calculations necessary for the construction of a Lyapunov like functional for the nonlinear stability analysis of steady-states in thermodynamically closed/open systems composed of compressible heat conducting fluids.