File size: 6,517 Bytes
a458723 | 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 | import Mathlib
import OAI.Analysis.Conductivity.Geometry.PhysicalCollarBand
import OAI.Analysis.Conductivity.Fourier.PoissonTraceRate
import OAI.Analysis.Conductivity.Fourier.AttachedEndPoisson
namespace OAI
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Filter Topology UnitAddTorus
open scoped ENNReal NNReal
lemma sourceAngular_time_displacement (t b : ℝ) (θ : UnitAddTorus (Fin 2)) :
‖sourceAngularCollar t θ-sourceAngularCollar b θ‖≤8*|t-b| := by
rw [show sourceAngularCollar t θ=sourceAngularCollar b θ+(t-b) • sourceAngularVelocity θ by
simpa only [add_sub_cancel] using sourceAngular_affine b (t-b) θ,add_sub_cancel_left,norm_smul,
Real.norm_eq_abs]
simpa only [mul_comm] using mul_le_mul_of_nonneg_left (sourceAngularVelocity_norm θ) (abs_nonneg (t-b))
lemma smooth_sourceCollar_lipschitz {q : (Fin 3 → ℝ) → ℝ}
(hq : ContDiff ℝ (↑(⊤:ℕ∞)) q) :
∃ K : ℝ≥0,LipschitzOnWith K q (sourceClosedCollarBand (-(1:ℝ)/100) (1/100)) := by
obtain ⟨R,hR⟩ := (isCompact_sourceClosedCollarBand (le_refl _) (le_refl _)).isBounded.subset_closedBall 0
obtain ⟨K,hK⟩ := hq.contDiffOn.exists_lipschitzOnWith (s:=Metric.closedBall 0 R)
(by simp) (convex_closedBall (0:Fin 3 → ℝ) R) (isCompact_closedBall _ _)
exact ⟨K,hK.mono hR⟩
lemma smooth_sourceCollar_value_displacement {q : (Fin 3 → ℝ) → ℝ} {K : ℝ≥0}
(hK : LipschitzOnWith K q (sourceClosedCollarBand (-(1:ℝ)/100) (1/100)))
{t b : ℝ} (ht : t∈Icc (-(1:ℝ)/100) (1/100)) (hb : b∈Icc (-(1:ℝ)/100) (1/100))
(θ : UnitAddTorus (Fin 2)) :
|q (sourceAngularCollar t θ)-q (sourceAngularCollar b θ)|≤8*(K:ℝ)*|t-b| := by
have ht' : sourceAngularCollar t θ∈sourceClosedCollarBand (-(1:ℝ)/100) (1/100) := by
change sourceCollarTime (sourceAngularCollar t θ)∈Icc (-(1:ℝ)/100) (1/100)
rw [sourceAngular_time ht]; exact ht
have hb' : sourceAngularCollar b θ∈sourceClosedCollarBand (-(1:ℝ)/100) (1/100) := by
change sourceCollarTime (sourceAngularCollar b θ)∈Icc (-(1:ℝ)/100) (1/100)
rw [sourceAngular_time hb]; exact hb
have H := (hK.dist_le_mul _ ht' _ hb').trans
(mul_le_mul_of_nonneg_left (sourceAngular_time_displacement t b θ) K.coe_nonneg)
simpa only [Real.dist_eq,dist_eq_norm,Real.norm_eq_abs,mul_assoc,mul_left_comm] using H
local instance smoothCollarRateMeasureSpace : MeasureSpace UnitAddCircle := ⟨AddCircle.haarAddCircle⟩
local instance smoothCollarRateProbabilityMeasure : IsProbabilityMeasure (volume : Measure UnitAddCircle) :=
inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle)
lemma endPoissonField_eq_traceFlow {s : Fin 3 → ℝ}
(hs : ∀ u v : ℝ,(1/2)*(u^2+v^2) ≤ s 0*u^2+2*s 1*u*v+s 2*v^2)
(f : spectralTraceGraph (torusRate s)) (q : C(UnitAddTorus (Fin 2),ℂ))
(hq : ∀ h,f.val 0 h=mFourierCoeff q h) {t : ℝ} (ht : 0<t) (θ : UnitAddTorus (Fin 2)) :
endPoissonField s f 0 (t,θ)=torusPoissonContinuous s t (q.toLp 2 volume ℂ) θ := by
rw [torusPoissonContinuous_apply hs ht]
unfold endPoissonField
apply tsum_congr
intro h
simp only [endPoissonModeField,endModeCoefficient,Matrix.cons_val_zero,hq,mFourierCoeff_toLp]
lemma smooth_attachedEnd_fiber_rate {s : Fin 3 → ℝ}
(hs : ∀ u v : ℝ,(1/2)*(u^2+v^2) ≤ s 0*u^2+2*s 1*u*v+s 2*v^2)
(f : spectralTraceGraph (torusRate s)) {q : (Fin 3 → ℝ) → ℝ}
(hq : ContDiff ℝ (↑(⊤:ℕ∞)) q) {K : ℝ≥0}
(hK : LipschitzOnWith K q (sourceClosedCollarBand (-(1:ℝ)/100) (1/100)))
{a b t : ℝ} (hb : b∈Icc (-(1:ℝ)/100) (1/100)) (ht : t∈Icc (-(1:ℝ)/100) (1/100))
(hpos : 0<a*(t-b)) (htrace : ∀ h,f.val 0 h=mFourierCoeff
(fun θ => (q (sourceAngularCollar b θ):ℂ)) h)
(g : SpectralL2 TorusModes) (hg : ∀ h,g h=(torusRate s h:ℂ)*f.val 0 h) :
(∫ θ,((attachedEndPoissonField s f a b 0 (sourceAngularCollar t θ)).re-
q (sourceAngularCollar t θ))^2)≤(2*a^2*‖g‖^2+2*(8*(K:ℝ))^2)*(t-b)^2 := by
let Q : C(UnitAddTorus (Fin 2),ℂ) := ⟨fun θ => (q (sourceAngularCollar b θ):ℂ),
Complex.continuous_ofReal.comp (hq.continuous.comp (continuous_sourceAngularCollar b))⟩
let P := torusPoissonContinuous s (a*(t-b)) (Q.toLp 2 volume ℂ)
have he (θ : UnitAddTorus (Fin 2)) :
attachedEndPoissonField s f a b 0 (sourceAngularCollar t θ)=P θ :=
(attachedEndPoissonField_angular s f a b 0 ht θ).trans
(endPoissonField_eq_traceFlow hs f Q htrace hpos θ)
have hP : Continuous (fun θ => (P θ).re-q (sourceAngularCollar t θ)) :=
(Complex.continuous_re.comp P.continuous).sub (hq.continuous.comp (continuous_sourceAngularCollar t))
simp_rw [he]
have hbound (θ : UnitAddTorus (Fin 2)) :
((P θ).re-q (sourceAngularCollar t θ))^2≤
2*‖P θ-Q θ‖^2+2*(8*(K:ℝ)*|t-b|)^2 := by
have hp := Complex.abs_re_le_norm (P θ-Q θ)
have hq' := smooth_sourceCollar_value_displacement hK ht hb θ
have hq'' : |q (sourceAngularCollar b θ)-q (sourceAngularCollar t θ)|≤8*(K:ℝ)*|t-b| := by
rwa [abs_sub_comm] at hq'
have h1 := pow_le_pow_left₀ (abs_nonneg _) hp 2
have h2 := pow_le_pow_left₀ (abs_nonneg _) hq'' 2
simp only [sq_abs,Complex.sub_re,Complex.ofReal_re,Q,ContinuousMap.coe_mk] at h1 h2
dsimp only [Q,ContinuousMap.coe_mk]
nlinarith [sq_nonneg ((P θ).re-q (sourceAngularCollar b θ)-
(q (sourceAngularCollar b θ)-q (sourceAngularCollar t θ)))]
have hpint : Integrable (fun θ => ‖P θ-Q θ‖^2) :=
(((P.continuous.sub Q.continuous).norm).pow 2).integrable_of_hasCompactSupport (isCompact_univ.of_isClosed_subset (isClosed_tsupport _) (subset_univ _))
have hfin : Integrable (fun θ => ((P θ).re-q (sourceAngularCollar t θ))^2) :=
(hP.pow 2).integrable_of_hasCompactSupport (isCompact_univ.of_isClosed_subset (isClosed_tsupport _) (subset_univ _))
have H := integral_mono hfin ((hpint.const_mul 2).add (integrable_const _)) hbound
simp only [Pi.add_apply] at H
rw [integral_add (hpint.const_mul 2) (integrable_const _),integral_const_mul,integral_const] at H
have hrate : (∫ θ,‖P θ-Q θ‖^2)≤(a*(t-b))^2*‖g‖^2 :=
torusPoissonContinuous_trace_sq hs hpos Q g (fun h => by rw [hg,mFourierCoeff_toLp,htrace]; rfl)
simp only [probReal_univ,one_smul] at H
calc
_ ≤ 2*((a*(t-b))^2*‖g‖^2)+2*(8*(K:ℝ)*|t-b|)^2 := by linarith
_ = _ := by rw [mul_pow,mul_pow,sq_abs]; ring
end ScalarConductivity
end
end OAI
|