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d5b562f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingCompactCommutator
import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingCompactLowProduct
import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingCommutatorCoefficient
import OAI.MathematicalPhysics.DefocusingNLS.Certificates.ApproximateNullLimit
namespace OAI
/-! # Compact action for coefficients approximated by fixed compact profiles -/
open Filter Topology
open scoped SchwartzMap
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
/-- Uniform eventual approximation in the actual expanding norm by sampled compact functions. -/
def ExpandingCompactApproximation (a k : ℝ) (ha1 : a < 1) (hk : 8 < k)
(L : ℕ → ℝ) (hL : ∀ n, 1 ≤ L n) (q : ℕ → FourierL2) : Prop :=
∀ δ : ℝ, 0 < δ → ∃ (R : ℝ) (K : 𝓢(E, ℂ)),
(∀ y : E, R < ‖y‖ → K y = 0) ∧ ∀ᶠ n in atTop,
‖q n - schwartzTorusSample a k (L n) ha1 hk (hL n) (radianFourierKernel K)‖ < δ
theorem tendsto_expandingApproximate_commutator (a M : ℝ) (N : ℕ)
(ha : 0 < a) (ha1 : a < 1) (hN : 8 < ((N + 1 : ℕ) : ℝ))
(L : ℕ → ℝ) (hL : ∀ n, 1 ≤ L n) (hLinf : Tendsto L atTop atTop)
(q : ℕ → FourierL2) (hq : ExpandingCompactApproximation a (N + 1 : ℕ) ha1 hN L hL q)
(f : ℕ → FourierL2) (hf : ∀ n, ‖f n‖ ≤ M)
(hlocal : ∀ R ε : ℝ, 0 < ε → ∀ᶠ n in atTop, ∀ y : E, ‖y‖ ≤ R →
‖expandingTorusFunction a (N + 1 : ℕ) (L n) (f n)
(euclideanToTorus ((L n)⁻¹ • y))‖ < ε)
(j : Fin (N + 1) → Fin 12) :
Tendsto (fun n => expandingProductCommutator a (L n) (N + 1) ha ha1 hN (hL n) j (q n) (f n))
atTop (𝓝 0) := by
let C : ℝ := (2 * Real.sqrt (4 ^ ((N + 1 : ℕ) : ℝ)) + 1) * expandingEmbeddingBound a (N + 1 : ℕ)
have hC : 0 ≤ C := by dsimp [C]; unfold expandingEmbeddingBound; positivity
have hM : 0 ≤ M := (norm_nonneg (f 0)).trans (hf 0)
apply tendsto_zero_of_null_approximations
intro ε hε
let δ := ε / (C * M + 1)
have hδ : 0 < δ := div_pos hε (by positivity)
obtain ⟨R, K, hK, herr⟩ := hq δ hδ
refine ⟨fun n => expandingProductCommutator a (L n) (N + 1) ha ha1 hN (hL n) j
(schwartzTorusSample a (N + 1 : ℕ) (L n) ha1 hN (hL n) (radianFourierKernel K)) (f n),
tendsto_expandingSample_commutator a M R N ha ha1 hN L hL hLinf f hf hlocal K hK j, ?_⟩
have hδbound : C * δ * M ≤ ε := by
have he : δ * (C * M + 1) = ε := div_mul_cancel₀ _ (by positivity)
nlinarith [hδ.le]
filter_upwards [herr] with n hn
exact (expandingProductCommutator_coefficient_difference a (L n) (N + 1) ha ha1 hN (hL n)
j (q n) _ (f n)).trans ((mul_le_mul
(mul_le_mul_of_nonneg_left hn.le hC) (hf n) (norm_nonneg _)
(mul_nonneg hC hδ.le)).trans hδbound)
theorem tendsto_expandingApproximate_lowEnergy_product (a M : ℝ) (N : ℕ)
(ha : 0 < a) (ha1 : a < 1) (hN : 8 < (N : ℝ))
(L : ℕ → ℝ) (hL : ∀ n, 1 ≤ L n) (hLinf : Tendsto L atTop atTop)
(q : ℕ → FourierL2) (hq : ExpandingCompactApproximation a N ha1 hN L hL q)
(f : ℕ → FourierL2) (hf : ∀ n, ‖f n‖ ≤ M)
(hlocal : ∀ R ε : ℝ, 0 < ε → ∀ᶠ n in atTop, ∀ y : E, ‖y‖ ≤ R →
‖expandingTorusFunction a N (L n) (f n) (euclideanToTorus ((L n)⁻¹ • y))‖ < ε) :
Tendsto (fun n => expandingLowEnergy a N (L n) (hL n)
(expandingProduct a N (L n) ha ha1 hN (hL n) (q n) (f n))) atTop (𝓝 0) := by
let C : ℝ := 2 * Real.sqrt (4 ^ (N : ℝ)) * expandingEmbeddingBound a N
have hC : 0 ≤ C := by dsimp [C]; unfold expandingEmbeddingBound; positivity
have hM : 0 ≤ M := (norm_nonneg (f 0)).trans (hf 0)
apply tendsto_zero_of_null_approximations
intro ε hε
let δ := ε / (C * M + 1)
have hδ : 0 < δ := div_pos hε (by positivity)
obtain ⟨R, K, hK, herr⟩ := hq δ hδ
let r := fun n => schwartzTorusSample a N (L n) ha1 hN (hL n) (radianFourierKernel K)
refine ⟨fun n => expandingLowEnergy a N (L n) (hL n)
(expandingProduct a N (L n) ha ha1 hN (hL n) (r n) (f n)),
tendsto_expandingSample_lowEnergy_product a M R N ha ha1 hN L hL hLinf f hf hlocal K hK, ?_⟩
have hδbound : C * δ * M ≤ ε := by
have he : δ * (C * M + 1) = ε := div_mul_cancel₀ _ (by positivity)
nlinarith [hδ.le]
filter_upwards [herr] with n hn
have he : expandingProduct a N (L n) ha ha1 hN (hL n) (q n) (f n) -
expandingProduct a N (L n) ha ha1 hN (hL n) (r n) (f n) =
expandingProduct a N (L n) ha ha1 hN (hL n) (q n - r n) (f n) := by
exact (congrArg (fun B : FourierL2 →L[ℂ] FourierL2 => B (f n))
((expandingBilinearProduct a N (L n) ha ha1 hN (hL n)).map_sub (q n) (r n))).symm
rw [← map_sub, he]
exact (expandingLowEnergy_norm_le a N (L n) (hL n) _).trans
((expandingProduct_norm_le a N (L n) ha ha1 hN (hL n) _ _).trans
((mul_le_mul (mul_le_mul_of_nonneg_left hn.le hC) (hf n) (norm_nonneg _)
(mul_nonneg hC hδ.le)).trans hδbound))
end DefocusingNLS
end OAI
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