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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 | import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.ContinuousTorusLinearStep
import OAI.MathematicalPhysics.DefocusingNLS.Linear.TorusDiscreteRadius
namespace OAI
/-! # Continuous linear data along the expanding discrete orbit -/
open scoped NNReal
namespace DefocusingNLS
local notation "Radius" => {L : ℝ // 1 ≤ L}
theorem continuous_expandingDiscreteRadius (T : ℝ≥0) (n : ℕ) :
Continuous (fun L : Radius => expandingDiscreteRadius L T n) := by
apply Continuous.subtype_mk
exact continuous_subtype_val.mul continuous_const
theorem continuous_torusLinearStep_discrete {F : Type*}
[NormedAddCommGroup F] [NormedSpace ℝ F]
(T : ℝ≥0) (A : Radius → FourierL2 →L[ℝ] FourierL2)
(h : HasContinuousTorusLinearStep (F := F) T A) :
∃ C : ℝ, 0 < C ∧ ∀ ε : ℝ, 0 < ε → ∃ L₀ : ℝ,
∃ ζ : Radius → ℕ → F →L[ℝ] FourierL2,
∃ π : Radius → ℕ → FourierL2 →L[ℝ] F,
∃ D R : F →L[ℝ] F, (∀ v, D (R v) = v) ∧ ‖R‖ ≤ 1 ∧
(∀ 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 v, π L n (ζ L n v) = v) ∧
(∀ n, ‖ζ L n‖ ≤ 1) ∧ (∀ n, ‖π L n‖ ≤ C) ∧
(∀ n, ‖stableFrameProjection (ζ L n) (π L n)‖ ≤ C) ∧
(∀ n, ‖stableProjectedBlock (ζ L n) (ζ L (n + 1)) (π L n) (π L (n + 1))
(A (expandingDiscreteRadius L T n))‖ ≤ 1 / 8) ∧
(∀ n, ‖stableMixedBlock (ζ L n) (ζ L (n + 1)) (π L (n + 1))
(A (expandingDiscreteRadius L T n))‖ ≤ 1 / 16) ∧
(∀ n, ‖(π L (n + 1)).comp (A (expandingDiscreteRadius L T n)) -
D.comp (π L n)‖ ≤ ε) := by
obtain ⟨C, hC, Lb, ζ, π, D, R, hinv, hR, hframe, hbound, hdefect, hζc, hπc⟩ := h
refine ⟨C, hC, ?_⟩
intro ε hε
obtain ⟨L₀, hLb, hεL⟩ := hdefect ε hε
refine ⟨L₀, (fun L n => ζ (expandingDiscreteRadius L T n)),
(fun L n => π (expandingDiscreteRadius L T n)), D, R, hinv, hR, ?_, ?_, ?_⟩
· intro n
exact hζc.comp (continuous_expandingDiscreteRadius T n).continuousOn
(fun L hL => hLb.trans (hL.trans (expandingDiscreteRadius_ge L T n)))
· intro n
exact hπc.comp (((continuous_expandingDiscreteRadius T n).comp continuous_fst).prodMk
continuous_snd)
· intro L hL
have hn (n : ℕ) : L₀ ≤ (expandingDiscreteRadius L T n).1 :=
hL.trans (expandingDiscreteRadius_ge L T n)
have hb (n : ℕ) := hbound (expandingDiscreteRadius L T n) (hLb.trans (hn n))
refine ⟨(fun n => hframe _ (hLb.trans (hn n))), (fun n => (hb n).1),
(fun n => (hb n).2.1), (fun n => (hb n).2.2.1), ?_, ?_, ?_⟩
· intro n
simpa only [expandingDiscreteRadius_succ] using (hb n).2.2.2.1
· intro n
simpa only [expandingDiscreteRadius_succ] using (hb n).2.2.2.2
· intro n
simpa only [expandingDiscreteRadius_succ] using hεL _ (hn n)
end DefocusingNLS
end OAI
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