Iterates Actual Previous Squared Gain #
Support for the Armstrong–Vicol anomalous-diffusion formalization.
theorem
AVenhance.Infra.Section4.iterate_actual_previous_material_squared_gain_of_A3
{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 : 2 ≤ m)
(hκm : 0 < κm)
(hflow : ∀ (l : ℤ), ContDiff ℝ ↑⊤ fun (z : AmnrSpace) => I.flowGrad hΦ m l z.1 z.2)
(hA3 :
∀ (j : ℕ),
1 ≤ j →
∀ (t : ℝ) (n : ℕ),
2 ≤ n →
barNorm n (2 ^ 8 * (epsilon β I.Λ j)⁻¹) (Φ j t) ≤ ENNReal.ofReal (2 ^ 5 * AVenhance.a β I.Λ j * epsilon β I.Λ j ^ 2 * ((↑n + 2) ^ 2 / (↑n + 1) ^ 3)))
(i : ℕ)
(hi : i + 1 ≤ Nstar β)
(w : List (Fin 2))
{C Gcur Gprev r L : ℝ}
(hL : 0 < L)
(hg : 2 * (r / L) ^ 2 ≤ 1 / 2)
(hvel : 2 * (256 * (epsilon β I.Λ (m - 1))⁻¹ / L) ^ 2 ≤ 1 / 2)
(hcoef :
∀ (t : ℝ) (x : Homogenization.Vec 2) (q : List (Fin 2)) (j k : Fin 2),
|iterateMatrixWord (fun (y : Homogenization.Vec 2) => I.Kmat κm m t - κprev • 1 + I.sMat hΦ m κm t y) q x j k| ≤ κprev * C * ↑q.length.factorial * r ^ q.length)
(hcur :
∀ (q : List (Fin 2)),
q.length ≤ w.length + 2 →
(spaceTimeGradNormSq fun (t : ℝ) => spaceGrad (iterateSpatialWord q (iterateIncrement T (i + 1) t))) ≤ Gcur ^ 2 * (↑(q.length + 2 * (i + 1)).factorial * L ^ q.length) ^ 2)
(hprev :
∀ (q : List (Fin 2)),
q.length ≤ w.length + 2 →
(spaceTimeGradNormSq fun (t : ℝ) => spaceGrad (iterateSpatialWord q (iterateIncrement T i t))) ≤ Gprev ^ 2 * (↑(q.length + 2 * i).factorial * L ^ q.length) ^ 2)
{u : ℝ → Homogenization.Vec 2 → ℝ}
(hu : ContDiffOn ℝ (↑⊤) (fun (z : AmnrSpace) => u z.1 z.2) (Set.Ici 0 ×ˢ Set.univ))
{Q : ℝ → Matrix (Fin 2) (Fin 2) ℝ}
(hQc : ContinuousOn (fun (z : AmnrSpace) => Q z.1) (Set.Ici 0 ×ˢ Set.univ))
{D s : ℝ}
(hQ : ∀ (t : ℝ) (j k : Fin 2), |Q t j k| ≤ D)
(hκ : 0 < κprev)
(hs1 : s ≤ 1)
{Cs Cv q : ℝ}
(hCs : 0 ≤ Cs)
(hCv : 0 ≤ Cv)
(hq : 0 ≤ q)
(hscale : D * L ^ 2 ≤ Cs * q)
(hvelocity : 8192 * a β I.Λ (m - 1) * epsilon β I.Λ (m - 1) * (256 * (epsilon β I.Λ (m - 1))⁻¹) ≤ Cv * κprev * L ^ 2)
:
|∫ (z : AmnrSpace) in iterateTruncatedCell s, Homogenization.vecDot (spaceGrad (iterateSpatialWord w (u z.1)) z.2)
((Q z.1).mulVec
(spaceGrad
(amnrMaterial (streamVel (Φ (m - 1))) (fun (t : ℝ) => iterateSpatialWord w (iterateIncrement T (i + 1) t))
z.1)
z.2))| ≤ (κprev / 24 * spaceTimeGradNormSq fun (t : ℝ) => spaceGrad (iterateSpatialWord w (u t))) + Cs ^ 2 * q ^ 2 * κprev * ((192 + 13056 * Cv ^ 2) * Gcur ^ 2 * (↑(w.length + 2 + 2 * (i + 1)).factorial * L ^ w.length) ^ 2 + 12288 * C ^ 2 * Gprev ^ 2 * (↑(w.length + 2 + 2 * i).factorial * L ^ w.length) ^ 2)
The actual preceding PDE gives both squared-gain amplitude contributions; no material-gradient bound is supplied as an assumption.