Subsonic Thermo-Acoustic Continuation Framework for the Compressible Navier–Stokes–Fourier System: Fourier–Triadic Concentration Exclusion and Thermodynamic Regularization

This paper studies the continuation problem for the three-dimensional compressible Navier–Stokes–Fourier system inside an admissible thermo-acoustic regime under periodic boundary conditions. The analysis considers strong solutions in the Sobolev class HsΩ, s> 5/2, with positive density and temperature and strict subsonic evolution. Using dyadic Fourier–triadic decomposition together with localized Littlewood–Paley analysis, the nonlinear transfer structure is decomposed into perturbative interaction classes and coherent same-scale High–High interactions. Within the present framework, coherent same-scale High–High persistence is identified as the only currently identified potentially nonperturbative concentration mechanism. A transport–acoustic alternative structure is then derived connecting persistent transport concentration with nonvanishing pressure response. The resulting pressure response is decomposed into thermodynamic and acoustic branches. The transonic acoustic branch is shown to be incompatible with the strict subsonic admissible class. The remaining interaction structure is controlled through entropy-driven thermodynamic dissipation and localized thermo-acoustic regularization. The exclusion of dynamically sustained critical thermo-acoustic concentration yields a localized ε-regularity framework combining thermodynamic dissipation, Campanato decay, and interior parabolic regularization. The resulting estimates provide localized Lipschitz control sufficient for the continuation of admissible strong solutions within the same thermo-acoustic class. The framework further remains compatible with weak–strong stability and irreversible long-time thermodynamic relaxation through the relative entropy structure and free-energy dissipation.