Iterates Actual Oscillatory Five Parts #
Support for the Armstrong–Vicol anomalous-diffusion formalization.
theorem
AVenhance.Infra.Section4.iterate_increment_oscillatory_flux_five_parts
{β : ℝ}
(I : Ingredients β)
{Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ}
(hΦ : IsStreamSeq I Φ)
{m : ℕ}
{κm κprev : ℝ}
{θ₀ : Homogenization.Vec 2 → ℝ}
{θprev : ℝ → Homogenization.Vec 2 → ℝ}
{T : ℕ → ℝ → Homogenization.Vec 2 → ℝ}
(hT : I.IsTIterates hΦ m κm κprev θ₀ θprev T)
(hθ : IsClassicalSol (streamVel (Φ (m - 1))) κprev (fun (x : ℝ) (x_1 : Homogenization.Vec 2) => 0) θ₀ θprev)
(hm : 1 ≤ m)
(hκm : 0 < κm)
(i : ℕ)
(hi : i + 1 ≤ Nstar β)
(w : List (Fin 2))
{s : ℝ}
(hs : 0 ≤ s)
(hs1 : s ≤ 1)
:
have u := fun (t : ℝ) => iterateSpatialWord w (iterateIncrement T (i + 1) t);
have v := fun (t : ℝ) => iterateSpatialWord w (iterateIncrement T i t);
have b := streamVel (Φ (m - 1));
have Q := iterateKmatPrimitive I κm m;
∫ (z : AmnrSpace) in iterateTruncatedCell s, Homogenization.vecDot (spaceGrad (u z.1) z.2)
((I.Kmat κm m z.1 - timeAvgMat (I.Kmat κm m)).mulVec (spaceGrad (v z.1) z.2)) = ((((∫ (x : Homogenization.Vec 2) in unitCube, Homogenization.vecDot (spaceGrad (u s) x) ((Q s).mulVec (spaceGrad (v s) x))) - ∫ (z : AmnrSpace) in iterateTruncatedCell s, Homogenization.vecDot (spaceGrad (u z.1) z.2) ((Q z.1).mulVec (spaceGrad (amnrMaterial b v z.1) z.2))) - ∫ (z : AmnrSpace) in iterateTruncatedCell s, Homogenization.vecDot (spaceGrad (amnrMaterial b u z.1) z.2) ((Q z.1).mulVec (spaceGrad (v z.1) z.2))) + ∫ (z : AmnrSpace) in iterateTruncatedCell s, Homogenization.vecDot (spaceGrad (u z.1) z.2)
((Q z.1).mulVec ((gradMatrix (b z.1) z.2).mulVec (spaceGrad (v z.1) z.2)))) + ∫ (z : AmnrSpace) in iterateTruncatedCell s, Homogenization.vecDot ((gradMatrix (b z.1) z.2).mulVec (spaceGrad (u z.1) z.2))
((Q z.1).mulVec (spaceGrad (v z.1) z.2))
Exact five-term oscillatory flux decomposition for the actual increment sequence: terminal boundary, previous material, current material, and two flow commutators. All endpoint and integrability issues have been discharged.