THERMODYNAMICS AND STABILITY OF NON-EQUILIBRIUM STEADY STATES IN OPEN SYSTEMS – CASE STUDY FOR COMPRESSIBLE HEAT CONDUCTING FLUID

The paper defines the isolated-system Lyapunov functional V_meq, shows d/dt V_meq = -θ̂ ∫Ω ξ ≤ 0 using the entropy balance with no-entropy-flux boundary and conservation of mass/energy (equations (6)–(9)) . It rewrites V_meq via the Helmholtz free energy and, under the thermodynamic stability conditions cV>0 and ∂ρ pth>0, proves a lower bound with a unique minimizer at the rest state (formulas (28)–(29), (43)–(44)) . For general steady states, it constructs the nonequilibrium functional V_neq by an affine (Bregman) correction (63), uses momentum variables to obtain the kinetic remainder ∫ 1/2 ρ|v−v̂|^2 (80), and identifies V_neq with Feireisl’s relative energy (86) based on the ballistic free energy (87), via identity (90) . The candidate solution reproduces these steps and conclusions. Minor presentational differences: it (i) attributes the −(p̂_th/ρ̂)(ρ−ρ̂) term in V_neq to “keeping” the affine mass term rather than showing it arises after the Bregman subtraction of the core terms (the paper’s derivation makes this clear), and (ii) states V_neq equals the relative energy up to an affine mass term vanishing by mass conservation, whereas the paper shows exact identity after algebraic cancellation. These do not affect the correctness of the main claims.