Coarse-Grained Resolution and Pressure-Flux Work Depletion for Navier-Stokes CKN Badness

Technical Summary: Coarse-Grained Resolution and Pressure–Flux Work Depletion for Navier–Stokes CKN Badness

Problem Statement
The paper addresses the local regularity problem for the three-dimensional incompressible Navier–Stokes equations, specifically within the framework of the Caffarelli–Kohn–Nirenberg (CKN) theory. The central challenge is to understand the behavior of "CKN-bad" scales—spacetime regions where the scale-critical quantity $\Psi(r)$ (involving $L^3$ velocity and $L^{3/2}$ pressure norms) is large. The author seeks to decompose this local regularity problem into two distinct components: a coarse-grained resolution (what is visible at a filtered scale) and a subfilter residual (what is hidden by the filter). Furthermore, the paper investigates the energy balance of the resolved fields, specifically the signed "combined pressure–flux work" distribution $G_\ell$, to determine how energy is transferred, dissipated, or backscattered across a finite chain of scales.

Methodology
The paper employs a rigorous mathematical analysis based on spatial coarse graining (mollification) and exact energy identities, avoiding probabilistic or numerical heuristics.

1. Spatial Coarse Graining: The author defines a spatial filter $S_\ell$ with length $\ell > 0$, producing resolved velocity $U_\ell$ and pressure $P_\ell$, and a Reynolds stress tensor $R_\ell = S_\ell(u \otimes u) - U_\ell \otimes U_\ell$.
2. Decomposition of CKN Quantities: The full CKN quantity $\Psi(r)$ is decomposed into a resolved part $\Psi_\ell(r)$ and a residual part $\Omega_\ell(r)$ representing the subfilter oscillations.
3. Finite-Chain Telescoping: The analysis is conducted on a fixed geometric chain of adjacent parabolic slabs $Q_k$ with radii $r_k = \theta^k r_0$. The author constructs a "telescoping" sum of localized energy identities across these slabs.
4. Active Work Extraction: Instead of assuming the work distribution $G_\ell$ is non-negative or has a fixed sign, the paper uses a finite-dimensional family of test profiles to constructively extract a non-negative weight. This allows the author to identify whether the work is "forward" (dissipative) or "backward" (backscatter) on each slab.
5. Pressure Decomposition: The pressure term is split into a local singular-integral part (via Riesz transforms) and a harmonic tail, allowing for precise control of gauge invariance and harmonic contributions to work.

Key Contributions and Results

1. The Resolution Lemma (Unconditional)
The first main result is an algebraic inequality relating the full CKN badness to its coarse-grained and residual components:
$$\Psi(r) \leq 4\Psi_\ell(r) + 4\Omega_\ell(r)$$

• Significance: This lemma is unconditional; it does not rely on the Navier–Stokes equations, local energy inequalities, or smallness conditions. It establishes that if a scale is "bad" (large $\Psi$), it must be either visible in the resolved fields ($\Psi_\ell$) or carried by a large subfilter residual ($\Omega_\ell$).
• Implication: A detector based solely on resolved fields cannot miss a singularity unless the subfilter residual is quantitatively large. This prevents "false negatives" in regularity detection due to filtering artifacts.

2. The Finite-Chain Work Depletion Theorem
The second main result is an exact identity for the combined pressure–flux work $G_\ell = \Pi_\ell + \nabla \cdot (P_\ell U_\ell)$ over a finite chain of slabs.

• Constructive Extraction: For any finite chain, the author proves the existence of a specific test weight $\phi_k$ on each slab such that the magnitude of the work $|W_k|$ is bounded below by a coefficient norm of the work distribution.
• Telescoping Inequality: Summing the local energy identities yields a weighted inequality:
  $$\sum_{k} w_k (W_k^+ + D_k) \leq E_0^- + \sum_{k} w_k |L_k| + \sum_{k} w_k W_k^-$$
  where $W_k^+$ is forward work, $W_k^-$ is backscatter (negative work), $D_k$ is resolved dissipation, and $L_k$ represents localization leakage.
• Significance: This theorem rigorously accounts for "backscatter" (energy transfer from small to large scales) and "leakage" (energy lost at the boundaries of the test functions). It shows that forward work and dissipation are "paid for" by initial kinetic energy, leakage, and negative work.

3. The Conditional Detector Interface
The paper synthesizes these results into a conditional implication. If an independent "coarse observability" estimate exists (i.e., if large resolved badness $\Psi_\ell$ implies a large active work coefficient $A_k(G_\ell)$), then the depletion theorem applies to the detected work.

• The Gap: The paper explicitly states that the implication $\Psi_\ell \geq \epsilon \implies A_k(G_\ell) \geq c$ is not proved here. It is identified as a separate, difficult compactness-rigidity problem that may fail due to pressure–flux cancellation, harmonic pressure tails, or coherent low-frequency profiles.

4. Diagnostic Examples and Limitations
The author provides examples demonstrating that:

• Harmonic pressure (which is a gauge in the global sense) performs real local work and cannot be ignored.
• Forward transfer and backscatter are both algebraically possible for the coarse fields.
• Pressure–flux cancellation can hide large individual components within a small combined work $G_\ell$.

Significance and Claims
The paper claims to provide a finite-scale coarse-grained obstruction calculus for the Navier–Stokes regularity problem. Its primary contribution is structural and algebraic:

1. Separation of Concerns: It cleanly separates the problem of "detecting" a singularity (which requires proving that resolved badness implies large work) from the problem of "accounting" for energy (which is handled by the exact depletion theorem).
2. Unconditional Resolution: It removes the ambiguity of subfilter scales in the CKN criterion via the resolution lemma.
3. Exact Accounting: It provides an exact ledger for energy flux, dissipation, and backscatter on finite chains, explicitly tracking leakage and harmonic tails.

The author modestly frames the work as isolating the "unconditional resolution and fixed-chain work identities" needed for a detector interface. The paper does not claim to solve the Navier–Stokes regularity problem or prove that singularities cannot exist. Instead, it establishes the precise mathematical conditions under which a detector based on resolved work would succeed or fail, highlighting that the remaining difficulty lies in the "coarse observability" step (ruling out silent profiles like harmonic tails or cancellation).