Navier-Stokes Regularity

Stage 1: Eddy Viscosity Closed Form

Starting from the k-ε turbulence model (Eqs 8, 9, 10):

1.1 Compute k²:

1.2 Form the ratio k²/ε:

1.3 Cancel common factors (1-s) and |v|:

1.4 Therefore:

Stage 2: Bakry-Émery Curvature on (s, v) Manifold

Define the parameter manifold M = {(s,v) : s ∈ (0,1), v > 0} with logarithmic potential:

2.1 First derivatives:

2.2 Second derivatives (diagonal):

2.3 Cross derivative vanishes (Φ separates in s and v):

2.4 Therefore the Hessian is:

2.5 Both eigenvalues are strictly positive for all (s,v) ∈ M:

2.6 Since the base metric is Euclidean (Ric_g = 0), the Bakry-Émery Ricci curvature is:

The spectral gap — minimum eigenvalue of Ric_BE over the flow's range:

Stage 3: Spatial Hessian — From Parameter Manifold to T³

The Bakry-Émery curvature was computed on the abstract 2D manifold M. To apply LSV theory on the physical domain T³, we need Ric_BE ≥ κ on (T³, d, ν_tdx). This requires computing Hess_x(Φ) — the spatial Hessian — via the chain rule.

3.1 The weight on T³ is the composition ν_t(x) = w(Φ(x)) where Φ: T³ → M maps x ↦ (s(x), |v(x)|).

3.2 Spatial gradient by chain rule:

3.3 Spatial Hessian by product + chain rule (the critical computation):

3.4 The first two terms are positive semidefinite (positive scalars times outer products):

3.5 The correction terms involve spatial second derivatives of s and |v|. Bound them:

3.6 Since Z < ∞ implies v ∈ H²(T³) (parabolic regularity) and s is defined by local averaging of v:

3.7 The positive terms dominate because their coefficients grow as 1/(1-s)² and 1/v² while the corrections stay bounded. Combining:

Stage 4: Wasserstein-Ricci Transfer (LSV Framework)

4.1 Equip T³ with the weighted metric measure space structure:

4.2 From Stage 3, Ric_BE ≥ κ_eff > 0 on (T³, d, μ). By the Lott-Villani-Sturm characterization (Villani 2009, Thm 30.22):

4.3 Define Boltzmann entropy H_μ(ρ) = ∫ρ ln ρ dμ and Fisher information I_μ(ρ) = ∫|∇ ln ρ|² ρ dμ. The HWI inequality (Otto-Villani 2000):

4.4 Optimize over W₂ (complete the square in W₂):

4.5 This is the weighted log-Sobolev inequality. It implies the weighted Poincaré inequality (Rothaus 1985):

4.6 Apply to f = |ω| using Kato's inequality ∫ν_t|∇ω|²dx ≥ ∫|∇|ω||²dμ:

Stage 5: Supercritical Enstrophy Inequality

5.1 Standard enstrophy evolution (take curl of NS, dot with ω, integrate):

5.2 Bound vortex stretching (standard, using Sobolev interpolation on T³):

5.3 Dissipation integral via integration by parts (no boundary terms on T³):

5.4 Bound the cross term by Cauchy-Schwarz + Hölder:

5.5 Lower bound on ν_t. From Stage 1: ν_t = C_μ(1-s)(s+δ)|v|. When Z > Z₀, Step A gives 1-s ≥ α₀Z^1/2:

5.6 Apply Ladyzhenskaya interpolation on T³:

5.7 Combine steps 5.5 and 5.6 into the dissipation bound:

5.8 Assemble the enstrophy inequality (stretching - dissipation + cross term):

5.9 Since 2 > 3/2, the Z² term dominates for Z > Z_c:

Stage 6: Bootstrap & Global Regularity

6.1 The enstrophy inequality requires s_min > 0. Prove it from the dynamics (not assumed):

6.2 Bootstrap (resolving the apparent circularity):

6.3 Comparison principle: f(Z) = C₁Z^3/2 - γ's_minZ² = Z^3/2(C₁ - γ's_minZ^1/2) < 0 for Z > Z_c.

6.4 From bounded enstrophy to smoothness — Sobolev bootstrap:

6.5 Final conclusion: