openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Approximation /CapMixedPositiveBound.lean
| import OAI.MathematicalPhysics.Boltzmann.Approximation.CapLimitTransport | |
| import OAI.MathematicalPhysics.Boltzmann.Approximation.CapMixedCollisionLimits | |
| import OAI.MathematicalPhysics.Boltzmann.Approximation.CapNonlinearHotLimit | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.DampedLateBoxPhysicalBlock | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.HotChartHomeomorph | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.MainPositiveRemainderBound | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.RescaledBoundedGain | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.RescaledColdMajorant | |
| import OAI.MathematicalPhysics.Boltzmann.Collision.CollisionSources | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.NormalizedPairCompactness | |
| namespace OAI | |
| section | |
| /-! # The actual cold density in the Euclidean dilation coordinates -/ | |
| noncomputable section | |
| open MeasureTheory | |
| open BoltzmannNonuniqueness.Published BoltzmannNonuniqueness.VelocityWeight | |
| namespace BoltzmannNonuniqueness.JetCopyDatum | |
| def CapApproximation.dilatedCold {s₀ η K α γ d₀ q T : ℝ} {N : ℕ} | |
| (F : CapApproximation s₀ η K α γ d₀ q T N) (r : ℝ) (z : KineticPhase) : ℝ := | |
| r*(H q z.2*F.coldTotal (r*z.1.1,(JetPeriodicDatum.spatialMap (r • z.1.2),z.2))) | |
| theorem CapApproximation.dilatedCold_measurable {s₀ η K α γ d₀ q T : ℝ} {N : ℕ} | |
| (F : CapApproximation s₀ η K α γ d₀ q T N) | |
| (hs₀ : 0 < s₀) (hsmax : s₀ ≤ 1/8) (hη : 0 < η) (hηmax : η ≤ Real.log 2) | |
| (hK : 0 ≤ K) (hd₀ : 0 ≤ d₀) (hq : q < 2) (hT : 0 ≤ T) (r : ℝ) : | |
| Measurable (F.dilatedCold r) := by | |
| have hc := (F.coldTotal_properties hs₀ hsmax hη hηmax hK hd₀ hq hT).1 | |
| have hx : Measurable (fun z : KineticPhase => | |
| (r*z.1.1,(JetPeriodicDatum.spatialMap (r • z.1.2),z.2))) := by | |
| apply Measurable.prodMk (by fun_prop) | |
| exact ((JetPeriodicDatum.spatialMap_measurePreserving.measurable.comp | |
| (by fun_prop)).prodMk measurable_snd) | |
| exact measurable_const.mul (((ChartKernel.measurable_H q).comp measurable_snd).mul (hc.comp hx)) | |
| theorem CapApproximation.dilatedCold_nonneg {s₀ η K α γ d₀ q T r : ℝ} {N : ℕ} | |
| (F : CapApproximation s₀ η K α γ d₀ q T N) (hr : 0 ≤ r) | |
| (hs₀ : 0 < s₀) (hsmax : s₀ ≤ 1/8) (hη : 0 < η) (hηmax : η ≤ Real.log 2) | |
| (hK : 0 ≤ K) (hd₀ : 0 ≤ d₀) (hq : q < 2) (hT : 0 ≤ T) (z : KineticPhase) : | |
| 0 ≤ F.dilatedCold r z := | |
| mul_nonneg hr (mul_nonneg (H_nonneg q z.2) | |
| ((F.coldTotal_properties hs₀ hsmax hη hηmax hK hd₀ hq hT).2 _).2.1) | |
| end BoltzmannNonuniqueness.JetCopyDatum | |
| end | |
| end | |
| section | |
| /-! # The positive Duhamel gain uses the actual hot and atomic cold particles -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| open BoltzmannNonuniqueness.EuclideanFlight | |
| namespace BoltzmannNonuniqueness.JetCopyDatum.AttainedCapSequence | |
| variable {s₀ η K α γ d₀ q : ℝ} {F : ShrinkingCapFamily s₀ η K α γ d₀ q} | |
| {S : AttainedCapSequence F} | |
| theorem HotLimitExtraction.damped_gain_hot_integral (E : HotLimitExtraction S) | |
| {a A R C s t : ℝ} (ha : 0 < a) (haA : a ≤ A) (hK : 0 ≤ K) (hα : 0 < α) (hγ : 0 < γ) | |
| (hq : q < 2) (hC : 0 ≤ C) (hs : a < s) (ht : t < A) | |
| {ψ β : WeakTime.Phase → ℝ} (hψ : Measurable ψ) (hβ : Measurable β) | |
| (hβn : ∀ p, 0 ≤ β p) {P B : ℝ} (hψB : ∀ p, |ψ p| ≤ P) (hβB : ∀ p, |β p| ≤ B) : | |
| (∫ p, ψ p ∂WeakTime.dampedSourceIncrement a A C s t | |
| (WeakTime.weightedPhaseSource (E.characteristicGainMeasure ha hK hα hγ A R) β)) = | |
| ∫ x, ∫ w, E.cappedHot (x,w)* | |
| (∫ v, ∫ ω, collisionKernel v w ω* | |
| (WeakTime.dampedPhaseTest C s t ψ β | |
| (hotCharacteristicEquiv.symm (x,scatterLeft v w ω))+ | |
| WeakTime.dampedPhaseTest C s t ψ β | |
| (hotCharacteristicEquiv.symm (x,scatterRight v w ω))) ∂sphereArea | |
| ∂E.background (max a x.1) x.2) ∂volume ∂coldCylinderVolume a A R := by | |
| rw [WeakTime.dampedSourceIncrement_weighted_integral _ hs ht hψ hβ hβn] | |
| exact E.characteristicGain_hot_integral ha haA hK hα hγ hq | |
| (WeakTime.dampedPhaseTest_measurable C s t hψ hβ) | |
| (WeakTime.dampedPhaseTest_bound hC s t hψB hβB) | |
| end BoltzmannNonuniqueness.JetCopyDatum.AttainedCapSequence | |
| end | |
| end | |
| section | |
| /-! # The actual hot collision equation in the coordinates of positive iteration -/ | |
| noncomputable section | |
| open MeasureTheory | |
| open BoltzmannNonuniqueness.Published BoltzmannNonuniqueness.VelocityWeight | |
| open BoltzmannNonuniqueness.CollisionCompactness | |
| namespace BoltzmannNonuniqueness.JetCopyDatum.AttainedCapSequence | |
| variable {s₀ η K α γ d₀ q : ℝ} {F : ShrinkingCapFamily s₀ η K α γ d₀ q} | |
| theorem euclidean_hot_collision (S : AttainedCapSequence F) | |
| (hs₀ : 0 < s₀) (hsmax : s₀ ≤ 1/8) (hη : 0 < η) (hηmax : η ≤ Real.log 2) | |
| (hK : 0 ≤ K) (hd₀ : 0 ≤ d₀) (hq : q < 2) | |
| (n : ℕ) (z : KineticPhase) (hz : z.2 ∈ HotRepresentative.good q) : | |
| S.euclideanHotSource n z = | |
| 2*mixedGainDensity (F.indices n+1) ((F.approximation n).dilatedCold (S.radius n)) | |
| (S.euclideanHot n) z- | |
| S.euclideanHot n z*KineticCollisionNullSets.frequency (F.indices n+1) | |
| ((F.approximation n).dilatedCold (S.radius n)) z+ | |
| (S.radius n)^2*collisionSource (F.indices n+1) (S.euclideanHot n) z+ | |
| H q z.2*S.rescaledColdSource n (z.1.1,EuclideanFlight.phaseEquiv (z.1.2,z.2)) := by | |
| let τ : SpaceTime → ℝ × JetPrototype.Space := fun x => (x.1,JetPeriodicDatum.spatialMap x.2) | |
| have hc : (fun w : KineticPhase => S.rescaledColdDensity n (τ w.1,w.2)) = | |
| (F.approximation n).dilatedCold (S.radius n) := by | |
| funext w | |
| unfold rescaledColdDensity CapApproximation.dilatedCold τ | |
| change S.radius n*H q w.2*(F.approximation n).coldTotal _ = _ | |
| rw [show S.radius n • JetPeriodicDatum.spatialMap w.1.2 = | |
| JetPeriodicDatum.spatialMap (S.radius n • w.1.2) from rfl] | |
| ring | |
| have hh : (fun w : KineticPhase => S.physicalHotDensity n (τ w.1,w.2)) = | |
| S.euclideanHot n := by | |
| funext w | |
| rw [euclideanHot_eq_rescaledHot,EuclideanFlight.phaseEquiv_apply] | |
| rfl | |
| have hi := S.rescaled_hot_collision hs₀ hsmax hη hηmax hK hd₀ hq n (τ z.1,z.2) hz | |
| rw [← mixedGain_base_comp (F.indices n+1) (S.rescaledColdDensity n) | |
| (S.physicalHotDensity n) τ z, | |
| ← cutoffFrequency_base_comp (F.indices n+1) (S.rescaledColdDensity n) τ z, | |
| ← collisionSource_base_comp (F.indices n+1) (S.physicalHotDensity n) τ z,hc,hh] at hi | |
| have hhpoint : S.physicalHotDensity n (τ z.1,z.2) = S.euclideanHot n z := congrFun hh z | |
| rw [hhpoint] at hi | |
| simpa only [euclideanHotSource,EuclideanFlight.phaseEquiv_apply,JetPeriodicDatum.phaseMap,τ] using hi | |
| end BoltzmannNonuniqueness.JetCopyDatum.AttainedCapSequence | |
| end | |
| end | |
| section | |
| /-! # An upper bound for the actual normalized source after discarding losses -/ | |
| noncomputable section | |
| open MeasureTheory | |
| open BoltzmannNonuniqueness.Published BoltzmannNonuniqueness.VelocityWeight | |
| open BoltzmannNonuniqueness.CollisionCompactness | |
| namespace BoltzmannNonuniqueness.JetCopyDatum.AttainedCapSequence | |
| variable {s₀ η K α γ d₀ q : ℝ} {F : ShrinkingCapFamily s₀ η K α γ d₀ q} | |
| theorem euclideanHotSource_le_gains (S : AttainedCapSequence F) | |
| (hs₀ : 0 < s₀) (hsmax : s₀ ≤ 1/8) (hη : 0 < η) (hηmax : η ≤ Real.log 2) | |
| (hK : 0 ≤ K) (hd₀ : 0 ≤ d₀) (hq : q < 2) | |
| (n : ℕ) (z : KineticPhase) (hz : z.2 ∈ HotRepresentative.good q) : | |
| S.euclideanHotSource n z ≤ | |
| 2*mixedGainDensity (F.indices n+1) ((F.approximation n).dilatedCold (S.radius n)) | |
| (S.euclideanHot n) z+ | |
| (S.radius n)^2*KineticCollisionNullSets.gain (F.indices n+1) (S.euclideanHot n) z+ | |
| H q z.2*S.rescaledColdSource n (z.1.1,EuclideanFlight.phaseEquiv (z.1.2,z.2)) := by | |
| rw [S.euclidean_hot_collision hs₀ hsmax hη hηmax hK hd₀ hq n z hz] | |
| exact hot_source_le_gain _ _ _ | |
| ((F.approximation n).dilatedCold_nonneg (S.radius_pos hs₀ n).le hs₀ hsmax hη hηmax | |
| hK hd₀ hq (shrinkingHorizon_pos F.horizonScale_pos n).le) | |
| (S.euclideanHot_nonneg hs₀ hsmax hη hηmax hK hd₀ hq n) z | |
| theorem normalizedHotSource_le_gains (S : AttainedCapSequence F) | |
| (hs₀ : 0 < s₀) (hsmax : s₀ ≤ 1/8) (hη : 0 < η) (hηmax : η ≤ Real.log 2) | |
| (hK : 0 ≤ K) (hd₀ : 0 ≤ d₀) (hq : q < 2) | |
| (n : ℕ) (z : KineticPhase) (hz : z.2 ∈ HotRepresentative.good q) : | |
| S.euclideanHotSource n z/H q z.2 ≤ | |
| (2*mixedGainDensity (F.indices n+1) ((F.approximation n).dilatedCold (S.radius n)) | |
| (S.euclideanHot n) z)/H q z.2+ | |
| (S.radius n)^2*(KineticCollisionNullSets.gain (F.indices n+1) (S.euclideanHot n) z/H q z.2)+ | |
| S.rescaledColdSource n (z.1.1,EuclideanFlight.phaseEquiv (z.1.2,z.2)) := by | |
| apply (div_le_div_of_nonneg_right | |
| (S.euclideanHotSource_le_gains hs₀ hsmax hη hηmax hK hd₀ hq n z hz) (H_nonneg q z.2)).trans_eq | |
| field_simp [(H_pos q z.2).ne'] | |
| end BoltzmannNonuniqueness.JetCopyDatum.AttainedCapSequence | |
| end | |
| end | |
| section | |
| /-! # The actual mixed cutoff gain is controlled by the main positive action -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| open scoped ENNReal | |
| open BoltzmannNonuniqueness.Published BoltzmannNonuniqueness.VelocityWeight | |
| namespace BoltzmannNonuniqueness.JetCopyDatum | |
| theorem CapApproximation.mixed_gain_le_selected_action (hcarleman : Published.GPV2009CarlemanInput) | |
| {s₀ η K α γ d₀ q T r : ℝ} {N : ℕ} | |
| (F : CapApproximation s₀ η K α γ d₀ q T N) | |
| (hr : 0 < r) (hs₀ : 0 < s₀) (hsmax : s₀ ≤ 1/8) | |
| (hη : 0 < η) (hηmax : η ≤ Real.log 2) | |
| (hK : 0 ≤ K) (hα : 0 ≤ α) (hαmax : α ≤ 1) (hγ : 0 ≤ γ) | |
| (hd₀ : 0 < d₀) (hdmax : d₀ ≤ 1) | |
| (hsmall : JetComparisonCoefficient.coefficient K*d₀*s₀^9 ≤ 1) | |
| (hq₁ : 1 < q) (hq₂ : q < 2) (hT : 0 ≤ T) (hTmax : T ≤ 1/32) | |
| (x : SpaceTime) (ht : r*x.1 ∈ Ioc 0 T) | |
| (b : ℝ) {f : KineticPhase → ℝ} (hf : Measurable f) (hfn : ∀ z, 0 ≤ f z) | |
| (hftwo : ∀ z, f z ≤ 2) : | |
| ∀ᵐ v : Velocity, ENNReal.ofReal (CollisionCompactness.mixedGainDensity b | |
| (F.dilatedCold r) (fun z => H q z.2*f z) (x,v)) ≤ | |
| ENNReal.ofReal (H q v)* | |
| (ChartKernel.action (selectedFreeKernel r s₀ η K α γ d₀ q N) | |
| (fun z => ENNReal.ofReal (f z)) (x,v)+ | |
| 2*(ENNReal.ofReal r*coldGainErrorConstant s₀ η K α d₀ q)) := by | |
| have hc := F.dilatedCold_measurable hs₀ hsmax hη hηmax hK hd₀.le hq₂ hT r | |
| have hcn := F.dilatedCold_nonneg hr.le hs₀ hsmax hη hηmax hK hd₀.le hq₂ hT | |
| have hg := F.rescaled_cold_gain_majorant_two hcarleman hr hs₀ hsmax hη hηmax hK hα hαmax | |
| hγ hd₀ hdmax hsmall hq₁ hq₂ hT hTmax x ht hf.ennreal_ofReal | |
| (fun z => (ENNReal.ofReal_le_ofReal (hftwo z)).trans_eq (by norm_num)) | |
| filter_upwards [hg,CollisionCompactness.normalized_mixed_cutoff_le_chart hcarleman b q | |
| hc hf hcn hfn x] with v hv hm | |
| exact hm.trans (mul_le_mul' le_rfl hv) | |
| end BoltzmannNonuniqueness.JetCopyDatum | |
| end | |
| end | |
| section | |
| /-! # The normalized mixed source is bounded by the positive main kernel -/ | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal Topology | |
| open BoltzmannNonuniqueness.Published BoltzmannNonuniqueness.VelocityWeight | |
| namespace BoltzmannNonuniqueness.JetCopyDatum | |
| theorem CapApproximation.normalized_mixed_gain_le_positive (hcarleman : Published.GPV2009CarlemanInput) | |
| {s₀ η K α γ d₀ q T r : ℝ} {N : ℕ} | |
| (F : CapApproximation s₀ η K α γ d₀ q T N) | |
| (hr : 0 < r) (hs₀ : 0 < s₀) (hsmax : s₀ ≤ 1/8) | |
| (hη : 0 < η) (hηmax : η ≤ Real.log 2) | |
| (hK : 0 ≤ K) (hα : 0 ≤ α) (hαmax : α ≤ 1) (hγ : 0 ≤ γ) | |
| (hd₀ : 0 < d₀) (hdmax : d₀ ≤ 1) | |
| (hsmall : JetComparisonCoefficient.coefficient K*d₀*s₀^9 ≤ 1) | |
| (hq₁ : 1 < q) (hq₂ : q < 2) (hT : 0 ≤ T) (hTmax : T ≤ 1/32) | |
| (x : SpaceTime) (ht : r*x.1 ∈ Ioc 0 T) | |
| (b : ℝ) {f : KineticPhase → ℝ} (hf : Measurable f) (hfn : ∀ z, 0 ≤ f z) | |
| (hftwo : ∀ z, f z ≤ 2) : | |
| ∀ᵐ v : Velocity, ENNReal.ofReal ((2*CollisionCompactness.mixedGainDensity b | |
| (F.dilatedCold r) (fun z => H q z.2*f z) (x,v))/H q v) ≤ | |
| ChartKernel.action (positiveMainKernel r s₀ η K α γ d₀ q N) | |
| (fun z => ENNReal.ofReal (f z)) (x,v)+ | |
| 4*(ENNReal.ofReal r*coldGainErrorConstant s₀ η K α d₀ q) := by | |
| filter_upwards [F.mixed_gain_le_selected_action hcarleman hr hs₀ hsmax hη hηmax hK hα | |
| hαmax hγ hd₀ hdmax hsmall hq₁ hq₂ hT hTmax x ht b hf hfn hftwo] with v hv | |
| rw [ENNReal.ofReal_div_of_pos (H_pos q v)] | |
| apply (ENNReal.div_le_iff' (ne_of_gt (ENNReal.ofReal_pos.mpr (H_pos q v))) | |
| ENNReal.ofReal_ne_top).mpr | |
| rw [ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 2)] | |
| apply (mul_le_mul' le_rfl hv).trans_eq | |
| rw [positiveMain_action] | |
| norm_num only [ENNReal.ofReal_ofNat] | |
| ring | |
| theorem cold_gain_error_tendsto_zero {r : ℕ → ℝ} (hr : Tendsto r atTop (𝓝 0)) | |
| (s₀ η K α d₀ q : ℝ) : | |
| Tendsto (fun n => 4*(ENNReal.ofReal (r n)*coldGainErrorConstant s₀ η K α d₀ q)) | |
| atTop (𝓝 0) := by | |
| have h := ENNReal.continuous_ofReal.continuousAt.tendsto.comp hr | |
| have hm := ENNReal.Tendsto.mul_const h (Or.inr (coldGainErrorConstant_ne_top s₀ η K α d₀ q)) | |
| have hh := ENNReal.Tendsto.const_mul hm (Or.inr (by norm_num : (4:ℝ≥0∞) ≠ ∞)) | |
| simpa only [Function.comp_def,ENNReal.ofReal_zero,zero_mul,mul_zero] using hh | |
| end BoltzmannNonuniqueness.JetCopyDatum | |
| end | |
| end | |
| end OAI | |