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fac6777 | 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 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.ContinuousTorusDiscreteData
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.StableSmallRemainderContinuity
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffRemainderContinuity
namespace OAI
/-! # Continuous stable endpoint sequences for the expanding torus -/
open scoped NNReal
universe u
namespace DefocusingNLS
local notation "Radius" => {L : ℝ // 1 ≤ L}
theorem exists_continuous_cutoffStable_sequence_threshold
{F : Type*}
[NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F]
(a δ B : ℝ) (ha : 0 < a) (hδ : 0 < δ) (hB : 0 ≤ B)
(T : ℝ≥0) (hratio : 2 * Real.exp (-(2 + a) * T / 2) ≤ 1)
(A : Radius → FourierL2 →L[ℝ] FourierL2)
(hAc : Continuous (fun q : Radius × FourierL2 => A q.1 q.2))
(hA : HasContinuousTorusLinearStep (F := F) T A) :
∃ C : ℝ, 0 < C ∧ ∃ ρ : ℝ, 0 < ρ ∧ 2 * ρ ≤ δ ∧ ∃ L₀ : ℝ,
∃ ζ : Radius → ℕ → F →L[ℝ] FourierL2,
∃ π : Radius → ℕ → FourierL2 →L[ℝ] F,
(∀ n, ContinuousOn (fun L => ζ L n) {L : Radius | L₀ ≤ L.1}) ∧
(∀ n, Continuous (fun p : Radius × FourierL2 => π p.1 n p.2)) ∧
(∀ L : Radius, L₀ ≤ L.1 → ∀ n, ‖ζ L n‖ ≤ 1 ∧ ‖π L n‖ ≤ C) ∧
(∀ L : Radius, L₀ ≤ L.1 → ∀ n v, π L n (ζ L n v) = v) ∧
∀ (P : Type u) [TopologicalSpace P], ∀ L : P → Radius,
Continuous L → (∀ p, L₀ ≤ (L p).1) →
∀ h : P → ℕ → {v : FourierL2 // ‖v‖ ≤ δ} → FourierL2,
(∀ n, Continuous (fun q : P × {v : FourierL2 // ‖v‖ ≤ δ} => h q.1 n q.2)) →
(∀ p n d, 0 < d → d ≤ δ → ∀ v u, ‖v.1‖ ≤ d → ‖u.1‖ ≤ d →
‖h p n v - h p n u‖ ≤
B * (d + (expandingDiscreteRadius (L p) T n).1 ^ (-2 - a)) * ‖v.1 - u.1‖) →
(∀ p n v, v.1 = 0 → ‖h p n v‖ ≤
B * (expandingDiscreteRadius (L p) T n).1 ^ (-2 - a)) →
∀ w : P → FourierL2, Continuous w → (∀ p, ‖w p‖ ≤ ρ / 4) →
(∀ p, π (L p) 0 (w p) = 0) →
∃ z : P → ℕ → FourierL2,
(∀ n, Continuous (fun p => z p n)) ∧
(∀ p, stableFrameProjection (ζ (L p) 0) (π (L p) 0) (z p 0) = w p) ∧
(∀ p n, z p (n + 1) = A (expandingDiscreteRadius (L p) T n) (z p n) +
cutoffRemainderExtension δ (h p) n (z p n)) ∧
(∀ p n, ‖z p n‖ ≤ (2 * ρ) * (1 / 2 : ℝ) ^ n) := by
obtain ⟨C, hC, hlinear⟩ := continuous_torusLinearStep_discrete T A hA
obtain ⟨σ, Llip, hσ, hσδ, hLlip, hlip⟩ := exists_cutoffRemainder_small_lipschitz
a B δ (1 / (128 * (C + 1))) ha hB hδ (by positivity)
let ρ := σ / 2
have hρ : 0 < ρ := by dsimp [ρ]; positivity
have h2ρ : 2 * ρ = σ := by dsimp [ρ]; ring
obtain ⟨Llin, ζ, π, D, R, hinv, hR, hζc, hπc, hdata⟩ := hlinear (1 / 128) (by norm_num)
obtain ⟨Lforce, _, hforce⟩ := exists_cutoffResidual_small_scale a (C * B) (ρ / 8) ha (by positivity)
let L₀ := max Llin (max Llip Lforce)
have hL₀lip : Llip ≤ L₀ := (le_max_left _ _).trans (le_max_right _ _)
have hL₀lin : Llin ≤ L₀ := le_max_left _ _
have hL₀force : Lforce ≤ L₀ := (le_max_right _ _).trans (le_max_right _ _)
have hL₀pos : 0 < L₀ := lt_of_lt_of_le zero_lt_one (hLlip.trans hL₀lip)
let ε := B * L₀ ^ (-2 - a)
have hε : 0 ≤ ε := mul_nonneg hB (Real.rpow_nonneg hL₀pos.le _)
have hεsmall : C * ε ≤ ρ / 8 := by
simpa only [ε, ← mul_assoc] using hforce L₀ hL₀force
refine ⟨C, hC, ρ, hρ, by rw [h2ρ]; exact hσδ, L₀, ζ, π, ?_, hπc,
(fun L hL n => ⟨(hdata L (hL₀lin.trans hL)).2.1 n,
(hdata L (hL₀lin.trans hL)).2.2.1 n⟩),
(fun L hL => (hdata L (hL₀lin.trans hL)).1), ?_⟩
· intro n
apply (hζc n).mono
intro L hL
exact hL₀lin.trans hL
intro P _ L hLc hL h hhc hl hz w hwc hwb hker
have hd (p : P) := hdata (L p) (hL₀lin.trans (hL p))
have hZc (n : ℕ) : Continuous (fun p => ζ (L p) n) :=
(hζc n).comp_continuous hLc (fun p => hL₀lin.trans (hL p))
have hZjoint (n : ℕ) : Continuous (fun q : P × F => ζ (L q.1) n q.2) :=
((hZc n).comp continuous_fst).clm_apply continuous_snd
have hPjoint (n : ℕ) : Continuous (fun q : P × FourierL2 => π (L q.1) n q.2) :=
(hπc n).comp ((hLc.comp continuous_fst).prodMk continuous_snd)
have hAjoint (n : ℕ) : Continuous
(fun q : P × FourierL2 => A (expandingDiscreteRadius (L q.1) T n) q.2) :=
hAc.comp ((((continuous_expandingDiscreteRadius T n).comp hLc).comp
continuous_fst).prodMk continuous_snd)
have hpow (p : P) : (L p).1 ^ (-2 - a) ≤ L₀ ^ (-2 - a) :=
Real.rpow_le_rpow_of_nonpos hL₀pos (hL p) (by linarith)
have hsmall (p : P) : ‖w p‖ + 2 * (C * ε) ≤ ρ / 2 := by
linarith [hwb p]
have hLip : ∀ p n v u, ‖v‖ ≤ 2 * ρ → ‖u‖ ≤ 2 * ρ →
‖cutoffRemainderExtension δ (h p) n v - cutoffRemainderExtension δ (h p) n u‖ ≤
(1 / (128 * (C + 1))) * ‖v - u‖ := by
intro p n v u hv hu
have hh := cutoffRemainderExtension_lipschitz a (L p).1 T δ σ B
(lt_of_lt_of_le zero_lt_one (L p).2) T.2 ha hσ hσδ hB (h p) (hl p)
n v u (by rwa [← h2ρ]) (by rwa [← h2ρ])
exact hh.trans (mul_le_mul_of_nonneg_right
(hlip (L p).1 (hL₀lip.trans (hL p))) (norm_nonneg _))
have hZero : ∀ p n, ‖cutoffRemainderExtension δ (h p) n 0‖ ≤
ε * (Real.exp (-(2 + a) * T / 2)) ^ n := by
intro p n
apply (cutoffRemainderExtension_zero_bound a (L p).1 T δ B
(lt_of_lt_of_le zero_lt_one (L p).2) hδ.le (h p) (hz p) n).trans
exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left (hpow p) hB) (by positivity)
obtain ⟨z, hzc, hz0, hzstep, hzn⟩ := exists_continuous_stableGraph_small_remainder
(fun p => ζ (L p)) (fun p => π (L p))
(fun p n => A (expandingDiscreteRadius (L p) T n)) D R hR hinv
(fun p => (hd p).1) (fun p => (hd p).2.2.2.2.1) (fun p => (hd p).2.2.2.2.2.1)
hZjoint hPjoint hAjoint C ρ ε (Real.exp (-(2 + a) * T / 2)) hC hρ.le hε
(Real.exp_nonneg _) hratio (fun p => (hd p).2.1) (fun p => (hd p).2.2.1)
(fun p => (hd p).2.2.2.1) (fun p => (hd p).2.2.2.2.2.2)
(fun p => cutoffRemainderExtension δ (h p))
(continuous_cutoffRemainderExtension_on_ball δ (2 * ρ) (h2ρ ▸ hσδ) h hhc)
hLip hZero w hwc hker hsmall
refine ⟨z, hzc, hz0, hzstep, fun p n => (hzn p n).trans ?_⟩
apply mul_le_mul_of_nonneg_right _ (by positivity)
linarith [hsmall p]
end DefocusingNLS
end OAI
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