Entropy and shear-parabolic admissibility of algebraic closure models for rarefied gas dynamics beyond one-dimensional calibration

One-dimensional normal-shock structure is the standard benchmark for higher-order rarefied-gas closures. Uniaxial kinematics confines the sampled deviatoric stress to the prolate (ξ=+1) and oblate (ξ=−1) vertices of the two-dimensional kinematic envelope indexed by the Lode parameter ξ, and the compression shock itself exercises only the prolate vertex. We show that, at any axisymmetric background of an algebraic closure, the linearized viscosity tensor decomposes under axial rotation symmetry into three scalar eigenvalues. These eigenvalues yield three 1D-visible admissibility tests on the scalar constitutive curve h(P) (where P is the uniaxial stress amplitude): the sign condition h(P)P≥0, monotonicity dh/dP≥0, and the branch-ratio bound 13≤h(P)/|h(−P)|≤3. Under the standard requirement that the closure reduce to Navier–Stokes–Fourier at equilibrium, monotonicity subsumes the sign condition, leaving monotonicity and branch-ratio as the two logically independent requirements. If any eigenvalue is negative, the linearized frozen-coefficient Cauchy problem at that background is ill-posed in L2. The strictly stronger nonlinear Hadamard statement via Lax's linearization principle remains conditional on a Kawashima–Shizuta rank condition, verified here only at transverse wavevectors near the parabolic threshold. Restoring heat-flux coupling tightens the admissible thresholds at every non-zero heat-flux magnitude whenever the stress–heat-flux cross-coefficient is positive. Finally, we show that two three-dimensional closures can share the same one-dimensional curve yet disagree on admissibility inside a physically reachable band of states, so one-dimensional shock data alone cannot determine the three-dimensional extension.