| import Mathlib |
| import OAI.Analysis.Conductivity.Fourier.TorusFlatMode |
| import OAI.Analysis.Conductivity.Sobolev.CollarLpPullback |
| import OAI.Analysis.Conductivity.Fourier.EndPoissonJets |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace ScalarConductivity |
|
|
| section |
| open Set MeasureTheory Filter Topology UnitAddTorus |
| open scoped ENNReal |
|
|
| local instance : MeasureSpace UnitAddCircle := ⟨AddCircle.haarAddCircle⟩ |
| local instance : IsProbabilityMeasure (volume : Measure UnitAddCircle) := |
| inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle) |
|
|
| lemma lp_hasSum_ae_unique {X E ι : Type*} [MeasurableSpace X] |
| [NormedAddCommGroup E] [Countable ι] [Infinite ι] |
| {μ : Measure X} {p : ℝ≥0∞} [Fact (1≤p)] |
| {u : Lp E p μ} {f : ι → Lp E p μ} {a : ι → X → E} {g : X → E} |
| (hf : HasSum f u) (ha : ∀ i,f i=ᵐ[μ] a i) |
| (hg : ∀ᵐ x∂μ,HasSum (fun i => a i x) (g x)) : u=ᵐ[μ] g := by |
| obtain ⟨e⟩ := nonempty_equiv_of_countable (α:=ℕ) (β:=ι) |
| have ht := (e.hasSum_iff.mpr hf).tendsto_sum_nat |
| obtain ⟨ns,hns,hn⟩ := (tendstoInMeasure_of_tendsto_Lp ht).exists_seq_tendsto_ae |
| have hs : ∀ᵐ x∂μ,∀ n : ℕ,(∑ i∈Finset.range n,f (e i)) x= |
| ∑ i∈Finset.range n,a (e i) x := by |
| have hA : ∀ᵐ x∂μ,∀ i : ℕ,f (e i) x=a (e i) x := ae_all_iff.mpr (fun i => ha (e i)) |
| have hF : ∀ᵐ x∂μ,∀ n : ℕ,(∑ i∈Finset.range n,f (e i)) x= |
| ∑ i∈Finset.range n,f (e i) x := |
| ae_all_iff.mpr (fun n => Lp.coeFn_fun_finsetSum (Finset.range n) (fun i => f (e i))) |
| filter_upwards [hA,hF] with x hx hy n |
| rw [hy] |
| exact Finset.sum_congr rfl (fun i _ => hx i) |
| filter_upwards [hg,hn,hs] with x hx hn hs |
| have hpoint := ((e.hasSum_iff.mpr hx).tendsto_sum_nat).comp hns.tendsto_atTop |
| have hco : (fun i => (∑ k∈Finset.range (ns i),f (e k)) x)= |
| (fun i => ∑ k∈Finset.range (ns i),a (e k) x) := funext (fun i => hs (ns i)) |
| change Tendsto (fun i => (∑ k∈Finset.range (ns i),f (e k)) x) atTop (𝓝 (u x)) at hn |
| rw [hco] at hn |
| exact tendsto_nhds_unique hn hpoint |
|
|
| def endPoissonModeField (s : Fin 3 → ℝ) (h : TorusModes) (a : ℂ) (j : Fin 4) |
| (z : ℝ × UnitAddTorus (Fin 2)) : ℂ := |
| (Real.exp (-torusRate s h*z.1):ℂ)*endModeCoefficient s h a j*mFourier h z.2 |
|
|
| def endPoissonField (s : Fin 3 → ℝ) (f : spectralTraceGraph (torusRate s)) (j : Fin 4) |
| (z : ℝ × UnitAddTorus (Fin 2)) : ℂ := |
| ∑' h,endPoissonModeField s h (f.val 0 h) j z |
| |
| lemma endPoissonModeJet_ae (s : Fin 3 → ℝ) (R : ℝ) (h : TorusModes) (a : ℂ) (j : Fin 4) : |
| endPoissonModeJet s R h a j=ᵐ[(FiniteAxisMeasure R).prod volume] |
| endPoissonModeField s h a j := by |
| have h1 := (Measure.quasiMeasurePreserving_fst |
| (μ := FiniteAxisMeasure R) (ν := (volume : Measure (UnitAddTorus (Fin 2))))).ae |
| (finiteDecayLp_ae (torusRate s h) R (endModeCoefficient s h a j)) |
| have h2 := (Measure.quasiMeasurePreserving_snd |
| (μ := FiniteAxisMeasure R) (ν := (volume : Measure (UnitAddTorus (Fin 2))))).ae (coeFn_mFourierLp 2 h) |
| filter_upwards [cylinderTensor_ae |
| (finiteDecayLp (torusRate s h) R (endModeCoefficient s h a j)) (mFourierLp 2 h),h1,h2] |
| with z hz h1 h2 |
| change cylinderTensor _ _ z=_ |
| rw [hz,h1,h2] |
| rfl |
| |
| lemma endModeCoefficient_norm_bound (s : Fin 3 → ℝ) (h : TorusModes) (a : ℂ) (j : Fin 4) : |
| ‖endModeCoefficient s h a j‖ ≤ (1+torusRate s h+torusSize h)*‖a‖ := by |
| have hm : 0≤torusRate s h := Real.sqrt_nonneg _ |
| have hsize := torusSize_nonneg h |
| fin_cases j |
| · change ‖a‖≤_ |
| nlinarith [mul_nonneg hm (norm_nonneg a),mul_nonneg hsize (norm_nonneg a)] |
| · change ‖-(torusRate s h:ℂ)*a‖≤_ |
| rw [norm_mul,norm_neg,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg hm] |
| exact mul_le_mul_of_nonneg_right (by linarith) (norm_nonneg a) |
| · change ‖Complex.I*(h 0:ℂ)*a‖≤_ |
| rw [norm_mul,norm_mul,Complex.norm_I,one_mul] |
| have he : ‖(h 0:ℂ)‖=|(h 0:ℝ)| := by simp |
| rw [he] |
| have hh : |(h 0:ℝ)|≤torusSize h := le_add_of_nonneg_right (abs_nonneg _) |
| exact mul_le_mul_of_nonneg_right (by linarith) (norm_nonneg a) |
| · change ‖Complex.I*(h 1:ℂ)*a‖≤_ |
| rw [norm_mul,norm_mul,Complex.norm_I,one_mul] |
| have he : ‖(h 1:ℂ)‖=|(h 1:ℝ)| := by simp |
| rw [he] |
| have hh : |(h 1:ℝ)|≤torusSize h := le_add_of_nonneg_left (abs_nonneg _) |
| exact mul_le_mul_of_nonneg_right (by linarith) (norm_nonneg a) |
| |
| lemma endPoissonField_hasSum (s : Fin 3 → ℝ) |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2) ≤ s 0*x^2+2*s 1*x*y+s 2*y^2) |
| (f : spectralTraceGraph (torusRate s)) (j : Fin 4) |
| {z : ℝ × UnitAddTorus (Fin 2)} (hz : 0<z.1) : |
| HasSum (fun h => endPoissonModeField s h (f.val 0 h) j z) (endPoissonField s f j z) := by |
| apply Summable.hasSum |
| apply Summable.of_norm |
| have hb : Summable (fun h => ‖f.val 0‖*((1+torusRate s h+torusSize h)* |
| Real.exp (-z.1*torusRate s h))) := by |
| have h0 := torusRate_exp_summable hs hz |
| have h1 := flatMode_derivative_summable hs 1 hz |
| simpa only [pow_one,add_mul,one_mul,add_assoc] using (h0.add h1).mul_left ‖f.val 0‖ |
| apply hb.of_nonneg_of_le (fun h => norm_nonneg _) (fun h => ?_) |
| have hc := endModeCoefficient_norm_bound s h (f.val 0 h) j |
| have ha := lp.norm_apply_le_norm (by norm_num : (2:ℝ≥0∞)≠0) (f.val 0) h |
| have hp : 0≤1+torusRate s h+torusSize h := by |
| have hm : 0≤torusRate s h := Real.sqrt_nonneg _ |
| linarith [torusSize_nonneg h] |
| have hc' := hc.trans (mul_le_mul_of_nonneg_left ha hp) |
| have hm : ‖mFourier h z.2‖≤1 := by simpa only [mFourier_norm] using (mFourier h).norm_coe_le_norm z.2 |
| unfold endPoissonModeField |
| rw [norm_mul,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)] |
| calc |
| _ ≤ (Real.exp (-torusRate s h*z.1)*((1+torusRate s h+torusSize h)*‖f.val 0‖))*1 := |
| mul_le_mul (mul_le_mul_of_nonneg_left hc' (Real.exp_nonneg _)) hm (norm_nonneg _) |
| (mul_nonneg (Real.exp_nonneg _) (mul_nonneg hp (norm_nonneg _))) |
| _ = _ := by rw [show -torusRate s h*z.1= -z.1*torusRate s h by ring]; ring |
| |
| theorem endPoissonJet_ae (s : Fin 3 → ℝ) |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2) ≤ s 0*x^2+2*s 1*x*y+s 2*y^2) |
| (R : ℝ) (hR : 0≤R) (f : spectralTraceGraph (torusRate s)) (j : Fin 4) : |
| endPoissonJet s hs R hR f j=ᵐ[(FiniteAxisMeasure R).prod volume] endPoissonField s f j := by |
| apply lp_hasSum_ae_unique (endPoissonJet_component_hasSum s hs R hR f j) |
| (fun h => endPoissonModeJet_ae s R h (f.val 0 h) j) |
| have ht : ∀ᵐ t∂FiniteAxisMeasure R,0<t := by |
| filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht using ht.1 |
| filter_upwards [Measure.quasiMeasurePreserving_fst.ae ht] with z hz |
| exact endPoissonField_hasSum s hs f j hz |
| |
| end |
| |
| open Set MeasureTheory Filter Topology UnitAddTorus |
| open scoped NNReal ENNReal |
| |
| lemma continuous_endPoissonModeField (s : Fin 3 → ℝ) (h : TorusModes) |
| (a : ℂ) (j : Fin 4) : Continuous (endPoissonModeField s h a j) := by |
| unfold endPoissonModeField |
| fun_prop |
| |
| lemma measurable_endPoissonField (s : Fin 3 → ℝ) |
| (f : spectralTraceGraph (torusRate s)) (j : Fin 4) : Measurable (endPoissonField s f j) := by |
| exact Measurable.tsum (fun h => (continuous_endPoissonModeField s h (f.val 0 h) j).measurable) |
| |
| def physicalEndPoissonField (s : Fin 3 → ℝ) |
| (f : spectralTraceGraph (torusRate s)) (j : Fin 4) : (Fin 3 → ℝ) → ℂ := |
| endPoissonField s f j ∘ sourcePhysicalCoordinates |
| |
| lemma measurable_physicalEndPoissonField (s : Fin 3 → ℝ) |
| (f : spectralTraceGraph (torusRate s)) (j : Fin 4) : Measurable (physicalEndPoissonField s f j) := |
| (measurable_endPoissonField s f j).comp measurable_sourcePhysicalCoordinates |
| |
| lemma physicalEndPoissonField_angular (s : Fin 3 → ℝ) |
| (f : spectralTraceGraph (torusRate s)) (j : Fin 4) {t : ℝ} |
| (ht : t∈Icc (-(1:ℝ)/100) (1/100)) (θ : UnitAddTorus (Fin 2)) : |
| physicalEndPoissonField s f j (sourceAngularCollar t θ)=endPoissonField s f j (t,θ) := by |
| simp only [physicalEndPoissonField,Function.comp_apply,sourcePhysicalCoordinates_angular ht] |
| |
| def physicalEndPoissonJet (s : Fin 3 → ℝ) |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2) ≤ s 0*x^2+2*s 1*x*y+s 2*y^2) |
| (R : ℝ) (hR : 0≤R) (hR' : R≤1/100) (f : spectralTraceGraph (torusRate s)) |
| (j : Fin 4) : Lp ℂ 2 (sourcePhysicalCollarMeasure 0 R) := |
| sourceCollarPullback hR (by norm_num) hR' (endPoissonJet s hs R hR f j) |
| |
| lemma physicalEndPoissonJet_ae (s : Fin 3 → ℝ) |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2) ≤ s 0*x^2+2*s 1*x*y+s 2*y^2) |
| (R : ℝ) (hR : 0≤R) (hR' : R≤1/100) (f : spectralTraceGraph (torusRate s)) |
| (j : Fin 4) : physicalEndPoissonJet s hs R hR hR' f j=ᵐ[sourcePhysicalCollarMeasure 0 R] |
| physicalEndPoissonField s f j := |
| sourceCollarPullback_ae_of_ae hR (by norm_num) hR' _ (endPoissonJet_ae s hs R hR f j) |
|
|
| lemma physicalEndPoissonField_memLp (s : Fin 3 → ℝ) |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2) ≤ s 0*x^2+2*s 1*x*y+s 2*y^2) |
| (R : ℝ) (hR : 0≤R) (hR' : R≤1/100) (f : spectralTraceGraph (torusRate s)) |
| (j : Fin 4) : MemLp (physicalEndPoissonField s f j) 2 (sourcePhysicalCollarMeasure 0 R) := |
| (memLp_congr_ae (physicalEndPoissonJet_ae s hs R hR hR' f j)).mp |
| (Lp.memLp (physicalEndPoissonJet s hs R hR hR' f j)) |
| |
| lemma physicalEndPoissonJet_hasSum (s : Fin 3 → ℝ) |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2) ≤ s 0*x^2+2*s 1*x*y+s 2*y^2) |
| (R : ℝ) (hR : 0≤R) (hR' : R≤1/100) (f : spectralTraceGraph (torusRate s)) |
| (j : Fin 4) : |
| HasSum (fun h => sourceCollarPullback hR (by norm_num) hR' |
| (endPoissonModeJet s R h (f.val 0 h) j)) |
| (physicalEndPoissonJet s hs R hR hR' f j) := |
| sourceCollarPullback_hasSum hR (by norm_num) hR' (endPoissonJet_component_hasSum s hs R hR f j) |
| |
| end ScalarConductivity |
| |
| end |
| |
| end OAI |
| |
