Closure and Regularity in Partial Differential Equations IV: Shock Formation, Dissipation, and Closure in Compressible Flow

This paper extends the closure framework developed in the earlier papers of the series to the barotropic compressible Navier–Stokes equations and their inviscid limits, with the explicit goal of testing whether the closure mechanism survives genuine singularity formation.

Compressible flow represents the first regime in which finite-time singularities in the form of shocks are expected even for smooth initial data. Rather than treating shock formation as an external obstruction, the paper formulates it as a stress test for closure: either the analytic mechanism persists with all error channels explicitly accounted for, or it fails in a way that must be structurally identifiable.

A canonized high-frequency tail energy is introduced, combining a momentum carrier adapted to weak solutions with a coarse-grained relative density potential compatible with shock formation. An exact distributional tail drift identity is derived for finite-energy weak solutions, separating nonlinear transfer, viscous persistence, and explicit remainder terms. All transfer and cutoff remainders are reduced to fully explicit commutator ledgers and priced by a single scale-local multiplier depending only on low-frequency transport strength and renormalized density potentials.

This yields a strict margin inequality in which viscous dissipation dominates the priced transfer budget with a quantitative gap. Passing to the inviscid limit, the viscous dissipation measures converge along subsequences to a nonnegative Radon measure supported on shock sets. After packetization, this shock entropy production measure replaces viscous dissipation in the same strict margin inequality, with no change in form.

Within this framework, shocks appear not as a breakdown of closure but as the inviscid persistence limit of viscosity. Singularities are compatible with closure provided their entropy production dominates the explicitly priced transfer channels, yielding a quantitative admissibility condition for inviscid limits.

All results are formulated and proved entirely in standard PDE language and are compatible with weak solutions and renormalized densities. The paper does not propose new entropy conditions or global regularity claims. Its contribution is structural: it demonstrates that the closure architecture developed for incompressible flows extends coherently to compressible flow with shocks, and it makes explicit the precise channels through which closure can succeed or fail in the singular regime.

Related papers in the series Closure and Regularity in Partial Differential Equations:

• Part I — From High-Frequency Surplus to Regularity in 3D Navier–Stokes
  https://doi.org/10.5281/zenodo.18371918

• Part II — Inviscid Closure and Anomalous Dissipation in the Euler Equations
  https://doi.org/10.5281/zenodo.18371954

• Part III — Continuity of Closure in the Vanishing Viscosity Limit
  https://doi.org/10.5281/zenodo.18372004

• Part V — A Universality Test of the Strict Margin Closure Mechanism
  https://doi.org/10.5281/zenodo.18372181

• Part VI — Failure Modes and Obstructions to Analytic Closure
  https://doi.org/10.5281/zenodo.18372250