| 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 |
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| end |
|
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| end OAI |
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