Thermo-Acoustic Stabilization and Continuation Structure in Admissible Compressible Navier–Stokes–Fourier Dynamics

This paper develops a thermo-acoustic continuation framework for physically admissible compressible Navier–Stokes–Fourier evolution. The analysis is formulated under the assumptions of positivity of density and temperature, entropy admissibility, free-energy dissipation, finite acoustic propagation, strict hyperbolicity, uniformly subsonic evolution, constitutive smoothness, and a finite-energy weak solution framework. The admissibility conditions are treated as the physical regime of the theory. The central objective is to determine whether thermo-acoustic dissipative structure suppresses scale-critical concentration compatible with singularity formation. A localized entropy concentration quantity is introduced using the entropy-production density generated by viscous deformation and thermal diffusion. The analysis establishes localized thermo-acoustic coercivity, derives nonlinear subcriticality estimates for transport, thermal, acoustic, pressure, coefficient, and commutator remainders, and obtains higher thermo-acoustic integrability through compactness and Meyers-type arguments. Campanato iteration then yields oscillation decay, localized Hölder regularization, and thermo-acoustic ε-regularity. Within the admissible thermo-acoustic regime, persistent scale-critical concentration is excluded. Consequently, admissible thermo-acoustic evolution admits continuation beyond finite admissible evolution intervals. The continuation mechanism is generated by entropy production, thermal diffusion, free-energy dissipation, and finite-speed acoustic redistribution.The paper also studies incompressible projection of the thermo-acoustic system. Using projection fibers and conditional disintegration theory, it is shown that the entropy-generating thermo-acoustic structure is not generally reconstructible from incompressible projected variables alone. The analysis identifies a structural difference between admissible thermo-acoustic compressible evolution and mechanically projected incompressible evolution.The paper does not prove unconditional global regularity for arbitrary compressible Navier–Stokes–Fourier solutions, unconditional propagation of thermo-acoustic admissibility, or regularity or singularity formation for incompressible Navier–Stokes evolution. The continuation result is conditional on persistence of the admissible thermo-acoustic regime.