Scope limits for general wall and viscous initial-value problems

Assume Countable Choice for the Fourier-based periodic claims cited here. The local periodic incompressible draft gives a conditional smooth-evolution argument on a torus; the positive-density polytropic draft gives a conditional argument for only the stated inviscid barotropic system; and the channel heat theorem gives a compatible shear solution conditional on its Fourier interfaces. These results do not provide a local smooth theorem for arbitrary incompressible data on bounded-wall or exterior domains, nor a general nonuniform-density compressible viscous initial-value theorem. The bounded-wall and exterior domain definition only fixes domain, regularity, trace, and possible far-field conventions; it proves no existence or pressure regularity. No claim is made that the broader theories do not exist in the literature. A theorem for such a broader claim would need its own exact data compatibility, pressure/elliptic estimates, and proof chain, so none is inferred from these periodic or shear special cases.

Local smooth periodic Euler and Navier–Stokes

Local smooth positive-density polytropic Euler

Smooth bounded-wall and exterior solution domains

Global compatible no-slip shear evolution

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Brian F. Morton, Local-in-Time Well-Posedness of the Incompressible Euler and Navier–Stokes Equations, §§7.1–7.3

François Golse, The Boltzmann Equation and Its Hydrodynamic Limits, §7.4