On a Port-Hamiltonian Formulation and Structure-Preserving Numerical Approximations for Thermodynamic Compressible Fluid Flow

The paper’s Appendix B proves that the linear relation D defined in (5) with the boundary maps (4) is a Stokes–Dirac structure (D = D⊥) and that, together with (6), it is equivalent to System Ev_pH with the standard power balance ⟨[fω,fB],[eω,eB]⟩ = 0. The isotropy step uses integration by parts with the explicit J(z) and the non-vanishing boundary terms, matched by the boundary port parameterization; co-isotropy is obtained via carefully chosen test efforts in nine steps to recover the bulk and boundary constraints, and the equivalence/power balance follows by direct substitution (Theorem 7 and its proof) . The candidate solution reproduces the same logical structure: (A) shows isotropy via a Green-type identity for J that converts the symmetric bulk term into the prescribed boundary pairing and then shows maximality by testing against a dense generating family; (B) eliminates auxiliary variables to recover Ev_pH and derives the power balance. Differences are present only in presentation (e.g., an explicit boundary Green identity versus the paper’s Q-matrix route), not in substance. Minor omissions in the candidate solution include not restating the exact function-space regularity [peS3, (e/ρ)eS2, (1/ρ)eS2] ∈ H^1(ω)^3 required for trace/integration-by-parts, and not explicitly noting that boundary terms preclude the direct use of the vanishing-boundary skew-adjointness corollary, both of which the paper addresses .