| import OAI.Analysis.HenonEmden.SymmetricVirial |
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| namespace OAI |
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| section LocalizationIntegral |
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| open MeasureTheory Set Metric |
| open scoped ENNReal |
| noncomputable section |
| namespace HenonLaneEmden |
| open Cutoff |
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| @[fun_prop] lemma measurable_localError (n : ℕ) : Measurable (localError : Space n × Space n → ℝ) := by |
| unfold localError |
| have hη := (continuous_eta n (E := Space n)).measurable |
| have hw := (continuous_w n (E := Space n)).measurable |
| fun_prop |
| lemma localError_nonneg {n : ℕ} (z : Space n × Space n) : 0 ≤ localError z := add_nonneg (abs_nonneg _) (abs_nonneg _) |
| lemma closeConstant_nonneg (n : ℕ) : 0 ≤ closeConstant n := by |
| obtain ⟨hD,hS,hW⟩ := constants_nonneg n |
| unfold closeConstant; positivity |
| lemma farLocalConstant_nonneg (n : ℕ) : 0 ≤ farLocalConstant n := by |
| have := (constants_nonneg n).1 |
| unfold farLocalConstant; positivity |
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| def localFarSum {n : ℕ} (δ : ℝ) (z : Space n × Space n) : ℝ := |
| farWeight δ (N n-1:ℕ) z+δ⁻¹*farWeight δ ((N n:ℝ)-(m n:ℝ)) z+ |
| farWeight δ (N n-1:ℕ) z.swap+δ⁻¹*farWeight δ ((N n:ℝ)-(m n:ℝ)) z.swap |
| lemma localFarSum_nonneg {n : ℕ} {δ : ℝ} (hδ : 0 ≤ δ) (z : Space n × Space n) : 0 ≤ localFarSum δ z := by |
| unfold localFarSum |
| exact add_nonneg (add_nonneg (add_nonneg (farWeight_nonneg _ _ _) (mul_nonneg (inv_nonneg.mpr hδ) |
| (farWeight_nonneg _ _ _))) (farWeight_nonneg _ _ _)) (mul_nonneg (inv_nonneg.mpr hδ) (farWeight_nonneg _ _ _)) |
| lemma localError_global {n : ℕ} (hn : 1 ≤ n) {δ : ℝ} (hδ : 0 < δ) (hδ1 : δ ≤ 1/4) |
| {z : Space n × Space n} (hne : z.1 ≠ z.2) : |
| localError z ≤ closeConstant n*δ*(eta n z.1+eta n z.2)+farLocalConstant n*localFarSum δ z := by |
| by_cases hc : 0 < pairGap z ∧ ‖z.1-z.2‖ ≤ δ*pairGap z^(m n) |
| · exact (localError_close hn hδ.le hδ1 hc.1 hc.2 hne).trans |
| (le_add_of_nonneg_right (mul_nonneg (farLocalConstant_nonneg n) (localFarSum_nonneg hδ.le z))) |
| · exact (localError_far hδ hne hc).trans |
| (le_add_of_nonneg_left (mul_nonneg (mul_nonneg (closeConstant_nonneg n) hδ.le) |
| (add_nonneg (eta_nonneg n _) (eta_nonneg n _)))) |
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| def localizationConstant (n : ℕ) (p q A B δ : ℝ) : ℝ := |
| farLocalConstant n*(1+δ⁻¹)*(farConstant n p q A B δ+farConstant n q p B A δ) |
| lemma localizationConstant_nonneg {n : ℕ} (p q A B : ℝ) {δ : ℝ} (hδ : 0 ≤ δ) : |
| 0 ≤ localizationConstant n p q A B δ := by |
| unfold localizationConstant |
| exact mul_nonneg (mul_nonneg (farLocalConstant_nonneg n) (by positivity)) |
| (add_nonneg (farConstant_nonneg _ _ _ _ hδ) (farConstant_nonneg _ _ _ _ hδ)) |
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| lemma IsSolution.localFarSum_bound {n : ℕ} (hn : 2 ≤ n) {p q A B δ : ℝ} |
| {u v : Space n → ℝ} (h : IsSolution n p q A B u v) (hp : 0 < p) (hq : 0 < q) |
| (hδ : 0 < δ) (hδ1 : δ ≤ 1) : |
| Integrable (localFarSum δ) (interaction (source u q B) (source v p A)) ∧ |
| (∫ z, localFarSum δ z ∂interaction (source u q B) (source v p A)) ≤ |
| (1+δ⁻¹)*(farConstant n p q A B δ+farConstant n q p B A δ) := by |
| have h1 := h.farWeight_bound hn hp hq hδ hδ1 (cutoff_far_exponents n).1 |
| have h2 := h.farWeight_bound hn hp hq hδ hδ1 (cutoff_far_exponents n).2.1 |
| have h3 := h.farWeight_swap_bound hn hp hq hδ hδ1 (cutoff_far_exponents n).1 |
| have h4 := h.farWeight_swap_bound hn hp hq hδ hδ1 (cutoff_far_exponents n).2.1 |
| have h12 : Integrable (fun z => farWeight δ (N n-1:ℕ) z+δ⁻¹*farWeight δ ((N n:ℝ)-(m n:ℝ)) z) _ := h1.1.add (h2.1.const_mul _) |
| have h123 : Integrable (fun z => farWeight δ (N n-1:ℕ) z+δ⁻¹*farWeight δ ((N n:ℝ)-(m n:ℝ)) z+farWeight δ (N n-1:ℕ) z.swap) _ := h12.add h3.1 |
| refine ⟨h123.add (h4.1.const_mul _),?_⟩ |
| unfold localFarSum |
| rw [integral_add h123 (h4.1.const_mul _), |
| integral_add h12 h3.1,integral_add h1.1 (h2.1.const_mul _), |
| integral_const_mul,integral_const_mul] |
| have hh := add_le_add (add_le_add (add_le_add h1.2 (mul_le_mul_of_nonneg_left h2.2 (inv_nonneg.mpr hδ.le))) h3.2) |
| (mul_le_mul_of_nonneg_left h4.2 (inv_nonneg.mpr hδ.le)) |
| exact hh.trans_eq (by ring) |
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| lemma IsSolution.localError_bound {n : ℕ} (hn : 2 ≤ n) {p q A B δ : ℝ} |
| {u v : Space n → ℝ} (h : IsSolution n p q A B u v) (hp : 0 < p) (hq : 0 < q) |
| (hδ : 0 < δ) (hδ1 : δ ≤ 1/4) : |
| Integrable localError (interaction (source u q B) (source v p A)) ∧ |
| (∫ z, localError z ∂interaction (source u q B) (source v p A)) ≤ |
| closeConstant n*δ*(weightedEnergy (eta n) u q B+weightedEnergy (eta n) v p A)+localizationConstant n p q A B δ := by |
| have hηf := h.weight_left hn hq (continuous_eta n) (compactSupport_eta n) (eta_nonneg n) |
| have hηg := h.weight_right hn hp (continuous_eta n) (compactSupport_eta n) (eta_nonneg n) |
| have hfar := h.localFarSum_bound hn hp hq hδ (by linarith) |
| have hηi : Integrable (fun z : Space n × Space n => eta n z.1+eta n z.2) _ := hηf.1.add hηg.1 |
| have hi : Integrable (fun z : Space n × Space n => closeConstant n*δ*(eta n z.1+eta n z.2)+farLocalConstant n*localFarSum δ z) _ := |
| (hηi.const_mul _).add (hfar.1.const_mul _) |
| have hb : ∀ᵐ z ∂interaction (source u q B) (source v p A), localError z ≤ |
| closeConstant n*δ*(eta n z.1+eta n z.2)+farLocalConstant n*localFarSum δ z := |
| (interaction_ae_ne (by omega : 1 ≤ n) _ _).mono (fun _ hz => localError_global (by omega) hδ hδ1 hz) |
| have he : Integrable localError (interaction (source u q B) (source v p A)) := hi.mono' |
| (measurable_localError n).aestronglyMeasurable (hb.mono (fun z hz => by |
| simpa only [Real.norm_eq_abs,abs_of_nonneg (localError_nonneg z)] using hz)) |
| refine ⟨he,(integral_mono_ae he hi hb).trans ?_⟩ |
| rw [integral_add (hηi.const_mul _) (hfar.1.const_mul _),integral_const_mul, |
| integral_const_mul,integral_add hηf.1 hηg.1,hηf.2,hηg.2] |
| calc |
| _ ≤ closeConstant n*δ*(weightedEnergy (eta n) u q B+weightedEnergy (eta n) v p A)+ |
| farLocalConstant n*((1+δ⁻¹)*(farConstant n p q A B δ+farConstant n q p B A δ)) := |
| add_le_add le_rfl (mul_le_mul_of_nonneg_left hfar.2 (farLocalConstant_nonneg n)) |
| _ = _ := by unfold localizationConstant; ring |
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| end HenonLaneEmden |
| end |
| end LocalizationIntegral |
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| end OAI |
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