Closure and Regularity in Partial Differential Equations II: Inviscid Closure and Anomalous Dissipation in the Euler Equations

This paper analyzes the three-dimensional incompressible Euler equations from the perspective of high-frequency closure, with the aim of identifying the precise analytic mechanism by which inviscid dynamics could enforce regularity, if such a mechanism exists at all.

Unlike the viscous Navier–Stokes equations, the Euler equations possess no intrinsic smoothing and are time-reversible in the smooth regime. Any inviscid closure principle must therefore arise from a fundamentally different persistence object. Working within the Duchon–Robert local energy balance framework, this paper isolates anomalous dissipation (the defect measure in the local energy balance) as the only admissible inviscid mechanism capable of producing irreversible high-frequency decay.

Using a scale-localized tail energy formulation and packetization in time via Fejér averaging, the paper derives an exact packet-center drift identity for the high-frequency energy tail. This identity decomposes the drift into nonlinear transfer across a cutoff minus a defect contribution arising from anomalous dissipation. A strict inviscid closure margin hypothesis is then introduced, asserting that at sufficiently high frequencies the packetized defect contribution strictly dominates nonlinear transfer by a scale-consistent gap.

Under this single quantitative assumption, the paper proves a deterministic contraction of the high-frequency tail on discrete packet time steps. Iteration of the contraction yields exponential decay of high-frequency modes and instantaneous Sobolev smoothing for all positive times, provided the contraction scale grows with frequency. The argument is entirely implication-based and requires no viscosity, no smallness assumptions, no probabilistic hypotheses, and no turbulence modeling.

The results make explicit a sharp structural dichotomy: smooth Euler solutions, for which the defect vanishes, lie in a reversible boundary regime where no strict time-directed closure mechanism is forced; by contrast, any inviscid regime exhibiting strict defect dominance necessarily enforces contraction and regularity. As the second paper in the series Closure and Regularity in Partial Differential Equations, this work isolates anomalous dissipation as the unique inviscid persistence mechanism capable of replacing viscosity in the closure architecture and provides the essential inviscid endpoint for subsequent analysis of the vanishing-viscosity limit.

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

• Part IV — Shock Formation, Dissipation, and Closure in Compressible Flow
  https://doi.org/10.5281/zenodo.18372079

• 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