openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Compactness /LocalizedGainSmoothingAE.lean
| import OAI.MathematicalPhysics.Boltzmann.Branching.SmoothedPrePostIdentity | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.TransportCutoff | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.WeightedNormalizedGainTranslation | |
| import OAI.MathematicalPhysics.Boltzmann.Collision.CollisionDissipation | |
| import OAI.MathematicalPhysics.Boltzmann.Collision.CollisionMeasure | |
| import OAI.MathematicalPhysics.Boltzmann.Collision.VelocityCutoffEntropyError | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.NormalizedCollisionRates | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.PairBoundedTruncation | |
| namespace OAI | |
| section | |
| /-! # Incoming-velocity localization of the actual physical gain -/ | |
| noncomputable section | |
| open MeasureTheory | |
| namespace BoltzmannNonuniqueness.CollisionCompactness | |
| open Compactness | |
| variable {X : Type*} [MeasurableSpace X] {μ : Measure X} [SFinite μ] | |
| def localizedPostRate (η : C(Velocity,ℝ)) (f : X × Velocity → ℝ) | |
| (z : (X × (Velocity × Velocity)) × Sphere) : ℝ := | |
| (η (scatterLeft z.1.2.1 z.1.2.2 z.2)*η (scatterRight z.1.2.1 z.1.2.2 z.2))* | |
| massNormalizedPostRate f z | |
| def localizedPhysicalGain (η : C(Velocity,ℝ)) (f : X × Velocity → ℝ) (p : X × Velocity) : ℝ := | |
| ∫ q : Velocity × Sphere, localizedPostRate η f (collisionGrouping.symm (p,q)) | |
| ∂(volume : Measure Velocity).prod sphereArea | |
| theorem localizedPostRate_measurable (η : C(Velocity,ℝ)) | |
| {f : X × Velocity → ℝ} (hfm : Measurable f) : Measurable (localizedPostRate η f) := by | |
| have hm : Measurable (fun z : (X × (Velocity × Velocity)) × Sphere => | |
| η (scatterLeft z.1.2.1 z.1.2.2 z.2)*η (scatterRight z.1.2.1 z.1.2.2 z.2)) := by | |
| unfold scatterLeft scatterRight normalComponent | |
| fun_prop | |
| exact hm.mul (massNormalizedPostRate_measurable hfm) | |
| theorem localizedPostRate_integrable (η : C(Velocity,ℝ)) (hη : HasCompactSupport η) | |
| {f : X × Velocity → ℝ} (hfm : Measurable f) | |
| (hi : Integrable (weightedNormalizedPair f) (μ.prod (volume : Measure (Velocity × Velocity)))) : | |
| Integrable (localizedPostRate η f) ((μ.prod (volume : Measure (Velocity × Velocity))).prod sphereArea) := by | |
| obtain ⟨B,hB⟩ := η.continuous.bounded_above_of_compact_support hη | |
| have hB0 : 0 ≤ B := (abs_nonneg (η 0)).trans (hB 0) | |
| have hm : Measurable (fun z : (X × (Velocity × Velocity)) × Sphere => | |
| η (scatterLeft z.1.2.1 z.1.2.2 z.2)*η (scatterRight z.1.2.1 z.1.2.2 z.2)) := by | |
| unfold scatterLeft scatterRight normalComponent | |
| fun_prop | |
| apply (massNormalizedPostRate_integrable hfm hi).bdd_mul hm.aestronglyMeasurable | |
| filter_upwards [] with z | |
| rw [Real.norm_eq_abs,abs_mul] | |
| exact mul_le_mul (hB _) (hB _) (abs_nonneg _) hB0 | |
| theorem localizedPhysicalGain_integrable (η : C(Velocity,ℝ)) (hη : HasCompactSupport η) | |
| {f : X × Velocity → ℝ} (hfm : Measurable f) | |
| (hi : Integrable (weightedNormalizedPair f) (μ.prod (volume : Measure (Velocity × Velocity)))) : | |
| Integrable (localizedPhysicalGain η f) (μ.prod volume) := by | |
| have h := collision_test_marginal (localizedPostRate η f) (fun _ => (1:ℝ)) | |
| (by simpa only [one_mul] using localizedPostRate_integrable η hη hfm hi) | |
| change Integrable (fun p => ∫ q : Velocity × Sphere, | |
| localizedPostRate η f (collisionGrouping.symm (p,q)) ∂(volume : Measure Velocity).prod sphereArea) | |
| (μ.prod volume) | |
| simpa only [one_mul] using h.1 | |
| end BoltzmannNonuniqueness.CollisionCompactness | |
| end | |
| end | |
| section | |
| /-! # Identifying the localized physical gain with the linear pair operator -/ | |
| noncomputable section | |
| open MeasureTheory | |
| namespace BoltzmannNonuniqueness.CollisionCompactness.DoubleGain | |
| open Compactness | |
| variable {X : Type*} | |
| def localizedPair (η : C(Velocity,ℝ)) (f : X × Velocity → ℝ) | |
| (z : X × (Velocity × Velocity)) : ℝ := | |
| (η z.2.1*η z.2.2)*normalizedPair volume f z | |
| theorem localizedPair_preRate (η : C(Velocity,ℝ)) (f : X × Velocity → ℝ) | |
| (z : (X × (Velocity × Velocity)) × Sphere) : | |
| pairPreRate (localizedPair η f) z = | |
| (η z.1.2.1*η z.1.2.2)*massNormalizedPreRate f z := by | |
| rw [massNormalizedPreRate_eq_pair] | |
| unfold pairPreRate localizedPair | |
| ring | |
| theorem localizedPair_postRate (η : C(Velocity,ℝ)) (f : X × Velocity → ℝ) | |
| (z : (X × (Velocity × Velocity)) × Sphere) : | |
| pairPostRate (localizedPair η f) z = localizedPostRate η f z := by | |
| rw [pairPostRate,localizedPair_preRate] | |
| rfl | |
| theorem localizedPhysicalGain_eq_pairGain [MeasurableSpace X] (η : C(Velocity,ℝ)) (f : X × Velocity → ℝ) : | |
| localizedPhysicalGain η f = pairGain (localizedPair η f) := by | |
| funext p | |
| unfold localizedPhysicalGain pairGain | |
| apply integral_congr_ae | |
| exact ae_of_all _ (fun q => (localizedPair_postRate η f _).symm) | |
| end BoltzmannNonuniqueness.CollisionCompactness.DoubleGain | |
| end | |
| end | |
| section | |
| /-! # The entropy velocity tail removes incoming localization of the gain -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| open scoped ENNReal | |
| namespace BoltzmannNonuniqueness.CollisionCompactness.DoubleGain | |
| open Compactness PeriodicApproximation | |
| variable {X : Type*} [MeasurableSpace X] {μ : Measure X} [SFinite μ] | |
| theorem exists_velocity_unit_cutoff {R : ℝ} (hR : 0 < R) : | |
| ∃ η : C(Velocity,ℝ), HasCompactSupport η ∧ | |
| (∀ v, 0 ≤ η v) ∧ (∀ v, η v ≤ 1) ∧ (∀ v, ‖v‖ ≤ R → η v = 1) := by | |
| let b : ContDiffBump (0 : Velocity) := ⟨R,R+1,hR,by linarith⟩ | |
| let η : C(Velocity,ℝ) := ⟨b,b.continuous⟩ | |
| refine ⟨η,b.hasCompactSupport,fun v => b.nonneg,fun v => b.le_one,?_⟩ | |
| intro v hv | |
| apply b.one_of_mem_closedBall | |
| simpa only [Metric.mem_closedBall,dist_zero_right] using hv | |
| omit [MeasurableSpace X] in | |
| theorem localizedPair_weight (η : C(Velocity,ℝ)) (f : X × Velocity → ℝ) : | |
| momentPair (localizedPair η f) = | |
| fun z => (η z.2.1*η z.2.2)*weightedNormalizedPair f z := by | |
| funext z | |
| unfold momentPair localizedPair weightedNormalizedPair pairVelocityWeight | |
| ring | |
| theorem localizedGain_tail_le {R C : ℝ} (hR : 0 < R) | |
| (η : C(Velocity,ℝ)) (hη0 : ∀ v, 0 ≤ η v) (hη1 : ∀ v, η v ≤ 1) | |
| (hone : ∀ v, ‖v‖ ≤ R → η v = 1) | |
| {f : X × Velocity → ℝ} (hfm : Measurable f) (hf : ∀ p, 0 ≤ f p) | |
| (hi : Integrable (entropyGauge f) (μ.prod volume)) | |
| (hb : (∫ p, entropyGauge f p ∂μ.prod volume) ≤ C) : | |
| eLpNorm (fun p => pairGain (normalizedPair volume f) p-localizedPhysicalGain η f p) | |
| 1 (μ.prod volume) ≤ sphereArea univ*ENNReal.ofReal (8*C/R) := by | |
| let w : X × (Velocity × Velocity) → ℝ := fun z => η z.2.1*η z.2.2 | |
| have hwm : Measurable w := | |
| (η.continuous.measurable.comp measurable_snd.fst).mul | |
| (η.continuous.measurable.comp measurable_snd.snd) | |
| have hw0 z : 0 ≤ w z := mul_nonneg (hη0 _) (hη0 _) | |
| have hw1 z : w z ≤ 1 := by | |
| dsimp only [w] | |
| nlinarith [hη0 z.2.1,hη0 z.2.2,hη1 z.2.1,hη1 z.2.2] | |
| have hpm : Measurable (normalizedPair volume f) := | |
| normalizedTensor_measurable (fiberMass_measurable hfm) hfm hfm | |
| have hqm : Measurable (localizedPair η f) := hwm.mul hpm | |
| have hwi : Integrable (weightedNormalizedPair f) | |
| (μ.prod (volume : Measure (Velocity × Velocity))) := by | |
| simpa only [Measure.volume_eq_prod] using weightedNormalizedPair_integrable hfm hf hi | |
| have hqi : Integrable (momentPair (localizedPair η f)) | |
| (μ.prod (volume : Measure (Velocity × Velocity))) := by | |
| rw [localizedPair_weight] | |
| apply hwi.bdd_mul hwm.aestronglyMeasurable (c := 1) | |
| exact ae_of_all _ (fun z => by rw [Real.norm_of_nonneg (hw0 z)]; exact hw1 z) | |
| have hpn z : 0 ≤ weightedNormalizedPair f z := by | |
| have hd : 0 ≤ 1+fiberMass volume f z.1 := by | |
| have hm : 0 ≤ fiberMass volume f z.1 := integral_nonneg (fun v => hf (z.1,v)) | |
| linarith | |
| exact mul_nonneg (by dsimp [pairVelocityWeight]; positivity) | |
| (div_nonneg (mul_nonneg (hf _) (hf _)) hd) | |
| have he : eLpNorm (momentPair (fun z => normalizedPair volume f z-localizedPair η f z)) | |
| 1 (μ.prod (volume : Measure (Velocity × Velocity))) ≤ | |
| eLpNorm (fun z => weightedNormalizedPair f z-pairVelocityCut R f z) | |
| 1 (μ.prod (volume : Measure (Velocity × Velocity))) := by | |
| apply eLpNorm_mono_ae (momentPair_measurable (hpm.sub hqm)).aestronglyMeasurable | |
| filter_upwards [] with z | |
| change ‖momentPair (fun y => normalizedPair volume f y-localizedPair η f y) z‖ ≤ _ | |
| have heq : momentPair (fun y => normalizedPair volume f y-localizedPair η f y) z = | |
| (1-w z)*weightedNormalizedPair f z := by | |
| dsimp only [momentPair,localizedPair,weightedNormalizedPair,pairVelocityWeight,w] | |
| ring | |
| rw [heq] | |
| by_cases hv : ‖z.2.1‖ ≤ R ∧ ‖z.2.2‖ ≤ R | |
| · have hw : w z = 1 := by simp only [w,hone _ hv.1,hone _ hv.2,one_mul] | |
| simp only [hw,sub_self,zero_mul,norm_zero] | |
| exact norm_nonneg _ | |
| · rw [pairVelocityCut,ite_eq_right hv,sub_zero, | |
| Real.norm_of_nonneg (mul_nonneg (sub_nonneg.mpr (hw1 z)) (hpn z)), | |
| Real.norm_of_nonneg (hpn z)] | |
| nlinarith [mul_nonneg (hw0 z) (hpn z)] | |
| rw [localizedPhysicalGain_eq_pairGain] | |
| exact (pairGain_moment_difference_le hpm hqm hwi hqi).trans | |
| (mul_le_mul' le_rfl (he.trans (pairVelocityCut_L1_error hR hfm hf hi hb))) | |
| end BoltzmannNonuniqueness.CollisionCompactness.DoubleGain | |
| end | |
| end | |
| section | |
| /-! # Causal localized transport sources and their velocity support -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| open BoltzmannNonuniqueness.Published | |
| namespace BoltzmannNonuniqueness.TransportCutoff | |
| def localizedSource (U : ℝ) (χ f Q : KineticPhase → ℝ) (z : KineticPhase) : ℝ := | |
| if z.1.1 ∈ Icc 0 U then χ z*Q z+(transportDerivative χ z+χ z)*f z else 0 | |
| theorem localizedSource_causal (U : ℝ) (χ f Q : KineticPhase → ℝ) | |
| (z : KineticPhase) (ht : z.1.1 < 0) : localizedSource U χ f Q z = 0 := by | |
| apply ite_eq_right | |
| exact fun h => not_le_of_gt ht h.1 | |
| theorem localizedSource_on {U : ℝ} (χ f Q : KineticPhase → ℝ) | |
| (z : KineticPhase) (ht : z.1.1 ∈ Icc 0 U) : | |
| localizedSource U χ f Q z = χ z*Q z+(transportDerivative χ z+χ z)*f z := ite_eq_left ht | |
| theorem localizedSource_velocity_support {U R : ℝ} {χ : KineticPhase → ℝ} | |
| (hχ : tsupport χ ⊆ {z | ‖z.2‖ ≤ R}) (f Q : KineticPhase → ℝ) | |
| (z : KineticPhase) (hv : R < ‖z.2‖) : localizedSource U χ f Q z = 0 := by | |
| have hz : z ∉ tsupport χ := fun h => not_le_of_gt hv (hχ h) | |
| have hc : χ z = 0 := image_eq_zero_of_notMem_tsupport hz | |
| have hd := transportDerivative_eq_zero hz | |
| unfold localizedSource | |
| split_ifs <;> simp only [hc,hd,zero_mul,add_zero] | |
| theorem localizedSource_measurable {U : ℝ} {χ f Q : KineticPhase → ℝ} | |
| (hχ : ContDiff ℝ (⊤ : ℕ∞) χ) (hf : Measurable f) (hQ : Measurable Q) : | |
| Measurable (localizedSource U χ f Q) := by | |
| apply Measurable.ite | |
| · exact measurableSet_Icc.preimage measurable_fst.fst | |
| · exact (hχ.continuous.measurable.mul hQ).add | |
| (((transportDerivative_continuous hχ).measurable.add hχ.continuous.measurable).mul hf) | |
| · exact measurable_const | |
| end BoltzmannNonuniqueness.TransportCutoff | |
| end | |
| end | |
| section | |
| /-! # Velocity translations of the actual localized physical gains -/ | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped Topology NNReal | |
| namespace BoltzmannNonuniqueness.CollisionCompactness.DoubleGain | |
| open Compactness PeriodicApproximation | |
| variable {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsFiniteMeasure μ] | |
| theorem localizedGain_translation | |
| {f : ℕ → X × Velocity → ℝ} {C : ℝ} | |
| (hfm : ∀ n, Measurable (f n)) (hf : ∀ n p, 0 ≤ f n p) | |
| (hi : ∀ n, Integrable (entropyGauge (f n)) (μ.prod volume)) | |
| (hb : ∀ n, (∫ p, entropyGauge (f n) p ∂μ.prod volume) ≤ C) | |
| (η : C(Velocity,ℝ)) (hη : HasCompactSupport η) | |
| {a : ℕ → Velocity} (ha : Tendsto a atTop (𝓝 0)) : | |
| Tendsto (fun n => ∫ p, | |
| |localizedPhysicalGain η (f n) (p.1,p.2+a n)-localizedPhysicalGain η (f n) p| | |
| ∂μ.prod volume) atTop (𝓝 0) := by | |
| obtain ⟨B,hB⟩ := η.continuous.bounded_above_of_compact_support hη | |
| have hB0 : 0 ≤ B := (abs_nonneg (η 0)).trans (hB 0) | |
| let K : ℝ≥0 := ⟨B^2,sq_nonneg B⟩ | |
| have hw : Measurable (fun z : X × (Velocity × Velocity) => η z.2.1*η z.2.2) := by fun_prop | |
| have hK (z : X × (Velocity × Velocity)) : |η z.2.1*η z.2.2| ≤ K := by | |
| change |η z.2.1*η z.2.2| ≤ B^2 | |
| rw [abs_mul,pow_two] | |
| exact mul_le_mul (hB _) (hB _) (abs_nonneg _) hB0 | |
| have ht := weighted_normalizedGain_translation (μ := μ) hfm hf hi hb hw K hK ha | |
| simp_rw [localizedPhysicalGain_eq_pairGain] | |
| change Tendsto (fun n => ∫ p, | |
| |pairGain (fun z => η z.2.1*η z.2.2*normalizedPair volume (f n) z) (p.1,p.2+a n)- | |
| pairGain (fun z => η z.2.1*η z.2.2*normalizedPair volume (f n) z) p| | |
| ∂μ.prod volume) atTop (𝓝 0) | |
| exact ht | |
| end BoltzmannNonuniqueness.CollisionCompactness.DoubleGain | |
| end | |
| end | |
| section | |
| /-! # Velocity smoothing of the genuine localized gain equals its incoming pair formula -/ | |
| noncomputable section | |
| open MeasureTheory | |
| namespace BoltzmannNonuniqueness.CollisionCompactness | |
| open Compactness | |
| variable {X : Type*} [MeasurableSpace X] | |
| theorem localizedPhysicalGain_smoothing (η χ : C(Velocity,ℝ)) | |
| (hη : HasCompactSupport η) (hχ : HasCompactSupport χ) | |
| (f : X × Velocity → ℝ) (x : X) (v : Velocity) | |
| (hpre : Integrable (fun z : (Velocity × Velocity) × Sphere => | |
| massNormalizedPreRate f ((x,z.1),z.2)) ((volume : Measure (Velocity × Velocity)).prod sphereArea)) | |
| (hpost : Integrable (fun z : (Velocity × Velocity) × Sphere => | |
| localizedPostRate η f ((x,z.1),z.2)) ((volume : Measure (Velocity × Velocity)).prod sphereArea)) : | |
| (∫ u : Velocity, χ (v-u)*localizedPhysicalGain η f (x,u)) = | |
| ∫ y : Velocity × Velocity, smoothedAngularKernel η χ (v,y)*normalizedPair volume f (x,y) := by | |
| obtain ⟨B,hB⟩ := η.continuous.bounded_above_of_compact_support hη | |
| obtain ⟨D,hD⟩ := χ.continuous.bounded_above_of_compact_support hχ | |
| have hB0 : 0 ≤ B := (abs_nonneg (η 0)).trans (hB 0) | |
| have hD0 : 0 ≤ D := (abs_nonneg (χ 0)).trans (hD 0) | |
| have hm : Measurable (fun z : (Velocity × Velocity) × Sphere => | |
| (η z.1.1*η z.1.2)*χ (v-scatterLeft z.1.1 z.1.2 z.2)) := by | |
| unfold scatterLeft normalComponent | |
| fun_prop | |
| have hi : Integrable (fun z : (Velocity × Velocity) × Sphere => | |
| (η z.1.1*η z.1.2)*χ (v-scatterLeft z.1.1 z.1.2 z.2)* | |
| massNormalizedPreRate f ((x,z.1),z.2)) | |
| ((volume : Measure (Velocity × Velocity)).prod sphereArea) := by | |
| apply hpre.bdd_mul hm.aestronglyMeasurable (c := B*B*D) | |
| filter_upwards [] with z | |
| simp only [Real.norm_eq_abs,abs_mul] | |
| exact mul_le_mul (mul_le_mul (hB _) (hB _) (abs_nonneg _) hB0) (hD _) | |
| (abs_nonneg _) (mul_nonneg hB0 hB0) | |
| have hposteq := collision_fiber_test_fubini hpost | |
| (χ.continuous.measurable.comp (measurable_const.sub measurable_id)) D (fun u => hD (v-u)) | |
| calc | |
| _ = ∫ ω : Sphere, (∫ y : Velocity × Velocity, | |
| (η (scatterLeft y.1 y.2 ω)*η (scatterRight y.1 y.2 ω))*χ (v-y.1)* | |
| massNormalizedPostRate f ((x,y),ω)) ∂sphereArea := by | |
| change (∫ u : Velocity, χ (v-u)*localizedPhysicalGain η f (x,u)) = _ at hposteq | |
| rw [hposteq] | |
| apply integral_congr_ae | |
| filter_upwards [] with ω | |
| apply integral_congr_ae | |
| filter_upwards [] with y | |
| dsimp only [localizedPostRate,Function.comp_apply,Pi.sub_apply,Function.const_apply,id_eq] | |
| ring | |
| _ = ∫ ω : Sphere, (∫ y : Velocity × Velocity, | |
| (η y.1*η y.2)*χ (v-scatterLeft y.1 y.2 ω)*massNormalizedPreRate f ((x,y),ω)) ∂sphereArea := | |
| (smoothed_pre_post_normals η χ f x v).symm | |
| _ = ∫ y : Velocity × Velocity, ∫ ω : Sphere, | |
| (η y.1*η y.2)*χ (v-scatterLeft y.1 y.2 ω)*massNormalizedPreRate f ((x,y),ω) ∂sphereArea := | |
| (integral_prod_symm _ hi).symm.trans (integral_prod _ hi) | |
| _ = _ := smoothed_preRate_pair_identity η χ f x v | |
| end BoltzmannNonuniqueness.CollisionCompactness | |
| end | |
| end | |
| section | |
| /-! # The physical gain convolution identity on a common full-measure base set -/ | |
| noncomputable section | |
| open MeasureTheory | |
| namespace BoltzmannNonuniqueness.CollisionCompactness | |
| open Compactness | |
| variable {X : Type*} [MeasurableSpace X] {μ : Measure X} [SFinite μ] | |
| local instance collisionRateFiberSFinite : SFinite ((volume : Measure (Velocity × Velocity)).prod sphereArea) := | |
| Measure.prod.instSFinite | |
| theorem collision_rate_fibers {r : (X × (Velocity × Velocity)) × Sphere → ℝ} | |
| (hi : Integrable r ((μ.prod (volume : Measure (Velocity × Velocity))).prod sphereArea)) : | |
| ∀ᵐ x ∂μ, Integrable (fun z : (Velocity × Velocity) × Sphere => r ((x,z.1),z.2)) | |
| ((volume : Measure (Velocity × Velocity)).prod sphereArea) := by | |
| have hmp := measurePreserving_prodAssoc μ (volume : Measure (Velocity × Velocity)) sphereArea | |
| have hh := hmp.symm.integrable_comp_of_integrable hi | |
| exact hh.prod_right_ae | |
| theorem localizedPhysicalGain_smoothing_ae (η χ : C(Velocity,ℝ)) | |
| (hη : HasCompactSupport η) (hχ : HasCompactSupport χ) | |
| {f : X × Velocity → ℝ} (hfm : Measurable f) | |
| (hi : Integrable (weightedNormalizedPair f) (μ.prod (volume : Measure (Velocity × Velocity)))) : | |
| ∀ᵐ x ∂μ, ∀ v : Velocity, | |
| (∫ u : Velocity, χ (v-u)*localizedPhysicalGain η f (x,u)) = | |
| ∫ y : Velocity × Velocity, smoothedAngularKernel η χ (v,y)*normalizedPair volume f (x,y) := by | |
| filter_upwards [collision_rate_fibers (massNormalizedPreRate_integrable hfm hi), | |
| collision_rate_fibers (localizedPostRate_integrable η hη hfm hi)] with x hx hy | |
| intro v | |
| exact localizedPhysicalGain_smoothing η χ hη hχ f x v hx hy | |
| end BoltzmannNonuniqueness.CollisionCompactness | |
| end | |
| end | |
| end OAI | |