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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 | import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.ContinuousTorusFrame
import OAI.MathematicalPhysics.DefocusingNLS.Linear.TorusLinearStep
import OAI.MathematicalPhysics.DefocusingNLS.Linear.TorusFrameNormalization
namespace OAI
/-! # Continuous normalized linear data for the discrete stable graph -/
open scoped NNReal
namespace DefocusingNLS
local notation "Radius" => {L : ℝ // 1 ≤ L}
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
def HasContinuousTorusLinearStep (T : ℝ≥0) (A : Radius → FourierL2 →L[ℝ] FourierL2) : Prop :=
∃ C : ℝ, 0 < C ∧ ∃ L₀ : ℝ,
∃ ζ : Radius → F →L[ℝ] FourierL2, ∃ π : Radius → FourierL2 →L[ℝ] F,
∃ D R : F →L[ℝ] F, (∀ v, D (R v) = v) ∧ ‖R‖ ≤ 1 ∧
(∀ L : Radius, L₀ ≤ L.1 → ∀ v, π L (ζ L v) = v) ∧
(∀ L : Radius, L₀ ≤ L.1 →
‖ζ L‖ ≤ 1 ∧ ‖π L‖ ≤ C ∧ ‖stableFrameProjection (ζ L) (π L)‖ ≤ C ∧
‖stableProjectedBlock (ζ L) (ζ (torusEndpointRadius T L))
(π L) (π (torusEndpointRadius T L)) (A L)‖ ≤ 1 / 8 ∧
‖stableMixedBlock (ζ L) (ζ (torusEndpointRadius T L))
(π (torusEndpointRadius T L)) (A L)‖ ≤ 1 / 16) ∧
(∀ ε : ℝ, 0 < ε → ∃ Lε : ℝ, L₀ ≤ Lε ∧ ∀ L : Radius, Lε ≤ L.1 →
‖(π (torusEndpointRadius T L)).comp (A L) - D.comp (π L)‖ ≤ ε) ∧
ContinuousOn ζ {L : Radius | L₀ ≤ L.1} ∧
Continuous (fun p : Radius × FourierL2 => π p.1 p.2)
theorem continuous_torusLinearStep_of_frames (T : ℝ≥0) (A : Radius → FourierL2 →L[ℝ] FourierL2)
(ζ : Radius → F →L[ℝ] FourierL2) (π : Radius → FourierL2 →L[ℝ] F)
(D R : F →L[ℝ] F) (hinv : ∀ v, D (R v) = v) (hR : ‖R‖ ≤ 1)
(L₀ C Cπ CA : ℝ) (hC : 0 ≤ C) (hCπ : 0 ≤ Cπ) (hCA : 0 ≤ CA)
(hframe : ∀ L : Radius, L₀ ≤ L.1 →
(∀ v, π L (ζ L v) = v) ∧ ‖ζ L‖ ≤ C ∧ ‖π L‖ ≤ Cπ ∧
‖stableFrameProjection (ζ L) (π L)‖ ≤ C)
(hstable : ∀ L : Radius, L₀ ≤ L.1 →
‖stableProjectedBlock (ζ L) (ζ (torusEndpointRadius T L))
(π L) (π (torusEndpointRadius T L)) (A L)‖ ≤ 1 / 8)
(hA : ∀ L : Radius, ‖A L‖ ≤ CA)
(hdefect : ∀ ε : ℝ, 0 < ε → ∃ Lε : ℝ, ∀ L : Radius, Lε ≤ L.1 →
‖(π (torusEndpointRadius T L)).comp (A L) - D.comp (π L)‖ ≤ ε)
(hζc : ContinuousOn ζ {L : Radius | L₀ ≤ L.1})
(hπc : Continuous (fun p : Radius × FourierL2 => π p.1 p.2)) :
HasContinuousTorusLinearStep (F := F) T A := by
obtain ⟨K, hK, hscale⟩ := exists_uniform_frame_scale (E := FourierL2) (F := F) C CA C hC hCA hC
let ζ' := fun L => stableNormalizedFrame K (ζ L)
let π' := fun L => stableNormalizedCoordinate K (π L)
let B := 1 + C + K * Cπ
have hB : 0 < B := by dsimp [B]; positivity
refine ⟨B, hB, L₀, ζ', π', D, R, hinv, hR, ?_, ?_, ?_, ?_, ?_⟩
· intro L hL v
exact stableNormalized_rightInverse K hK.ne' (ζ L) (π L) (hframe L hL).1 v
· intro L hL
have hLR := hL.trans (torusEndpointRadius_ge T L)
obtain ⟨_, hζ, hπ, hP⟩ := hframe L hL
obtain ⟨_, _, _, hPR⟩ := hframe (torusEndpointRadius T L) hLR
obtain ⟨hζ', hmixed⟩ := hscale (ζ L) (ζ (torusEndpointRadius T L))
(π (torusEndpointRadius T L)) (A L) hPR (hA L) hζ
refine ⟨hζ', ?_, ?_, ?_, hmixed⟩
· exact (stableNormalizedCoordinate_norm_le K Cπ hK.le (π L) hπ).trans (by dsimp [B]; linarith)
· rw [stableNormalized_projection K hK.ne']
exact hP.trans (by dsimp [B]; nlinarith [mul_nonneg hK.le hCπ])
· rw [stableNormalized_projectedBlock K hK.ne']
exact hstable L hL
· intro ε hε
obtain ⟨Lε, hεL⟩ := hdefect (ε / K) (div_pos hε hK)
refine ⟨max L₀ Lε, le_max_left _ _, ?_⟩
intro L hL
apply (stableNormalized_defect_norm_le K (ε / K) hK.le (π L)
(π (torusEndpointRadius T L)) (A L) D
(hεL L ((le_max_right _ _).trans hL))).trans_eq
exact mul_div_cancel₀ ε hK.ne'
· change ContinuousOn (fun L => (K⁻¹ : ℝ) • ζ L) {L : Radius | L₀ ≤ L.1}
exact (continuousOn_const : ContinuousOn (fun _ : Radius => (K⁻¹ : ℝ))
{L : Radius | L₀ ≤ L.1}).smul hζc
· change Continuous (fun p : Radius × FourierL2 => (K : ℝ) • π p.1 p.2)
exact continuous_const.smul hπc
end DefocusingNLS
end OAI
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