openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Compactness /FiniteRatesWindowUI.lean
| import OAI.MathematicalPhysics.Boltzmann.Approximation.MassNormalizedCollisionLimits | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.BoundedCollisionBlocks | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.BoundedMildEntropyLimit | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.EntropyRegularizationLimit | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.KineticEntropyErrorLimit | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.UniformIntegrableCoordinates | |
| import OAI.MathematicalPhysics.Boltzmann.Jets.TorusVelocityWindowAE | |
| namespace OAI | |
| section | |
| /-! # Regularized entropy inequality for the actual periodic finite approximations -/ | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped Topology | |
| open BoltzmannNonuniqueness.VelocityWeight | |
| namespace BoltzmannNonuniqueness | |
| namespace WeightedLocalControl.PeriodicMildState | |
| def torusEvolution {q b L U R : ℝ} {f₀ : JetPrototype.Phase → ℝ} | |
| (F : PeriodicMildState q b L U R f₀) : Evolution := | |
| fun t p => JetCopyDatum.torusPhysical q F.value ((t,p.1),p.2) | |
| theorem regularized_entropy_inequality (hcarleman : Published.GPV2009CarlemanInput) | |
| {q b L T U R δ p : ℝ} {f₀ : JetPrototype.Phase → ℝ} | |
| (F : PeriodicMildState q b L U R f₀) | |
| (hq₁ : 1 < q) (hq₂ : q < 2) (hb : 0 ≤ b) (hR : 0 ≤ R) (hT : 0 < T) (hTU : T ≤ U) | |
| (hloss : R*CutoffLipschitz.frequencyConstant q b ≤ L) | |
| (hf₀ : Measurable f₀) (hf₀b : ∀ z, 0 ≤ f₀ z ∧ f₀ z ≤ R) | |
| (hδ : 0 < δ) (hδone : δ ≤ 1) (hp : 1 < p) | |
| (hpi : Integrable (fun v : Velocity => H q v^p)) : | |
| (∫ z, regularizedRelativeEntropyDensity δ (F.torusEvolution T) z)- | |
| (∫ z, regularizedRelativeEntropyDensity δ (TorusFlight.initialDensity q f₀) z) ≤ | |
| (δ*entropyErrorCoefficient q R b)*∫ z : (ℝ × Position) × Velocity, | |
| logarithmicMajorant q (R+tiltConstant) p δ z.2 | |
| ∂(AnnularEnvelope.timePositionVolume T).prod volume := by | |
| let E : Evolution := F.torusEvolution | |
| let a : Density := TorusFlight.initialDensity q f₀ | |
| let r := TorusFlight.along (cutoffPhysicalSource b E) | |
| have hEm : Measurable (fun z : ℝ × Phase => E z.1 z.2) := | |
| (JetCopyDatum.torusPhysical_measurable F.measurable_value q).comp MeasurableEquiv.prodAssoc.symm.measurable | |
| have hEn : ∀ t z, 0 ≤ E t z := fun t z => | |
| JetCopyDatum.torusPhysical_nonneg (fun u => (F.value_bounds u).1) q _ | |
| have hEb : ∀ t z, E t z ≤ R*H q z.2 := by | |
| intro t z | |
| have h := mul_le_mul_of_nonneg_left (F.value_bounds | |
| (t,(JetPeriodicDatum.spatialMap (TorusChart.representative z.1),z.2))).2 (H_nonneg q z.2) | |
| exact h.trans_eq (mul_comm _ _) | |
| have ham : Measurable a := TorusFlight.initialDensity_measurable hf₀ q | |
| have han : ∀ z, 0 ≤ a z := fun z => mul_nonneg (H_nonneg q z.2) (hf₀b _).1 | |
| have hab : ∀ z, a z ≤ R*H q z.2 := by | |
| intro z | |
| exact (mul_le_mul_of_nonneg_left (hf₀b _).2 (H_nonneg q z.2)).trans_eq (mul_comm _ _) | |
| have hcT : (fun z => E T (TorusFlight.flight T z)) =ᵐ[volume] entropyPath a r T := | |
| TorusFlight.characteristic hcarleman F hq₁ hq₂ hb hR hT.le hTU hloss hf₀ | |
| have hc : (fun z : ℝ × Phase => E z.1 (TorusFlight.flight z.1 z.2)) =ᵐ[ | |
| (volume.restrict (Icc 0 T)).prod volume] (fun z => entropyPath a r z.1 z.2) := by | |
| have h := TorusFlight.spacetime_mild hcarleman F hq₁ hq₂ hb hR hT.le hTU hloss hf₀ | |
| have ht : ∀ᵐ z : ℝ × Phase ∂(volume.restrict (Icc 0 T)).prod volume, z.1 ∈ Icc 0 T := | |
| Measure.quasiMeasurePreserving_fst.ae (ae_restrict_mem measurableSet_Icc) | |
| filter_upwards [h,ht] with z hz hzt | |
| rw [Published.timePrimitive_eq_interval hzt] at hz | |
| exact hz | |
| have hbal := bounded_mild_entropy_balance hcarleman hT hq₁ hq₂ hR hδ hδone hb | |
| hEm hEn hEb ham han hab hc hcT | |
| have hprod := physical_regularized_entropy_production_bound (T := T) hcarleman hq₁ hq₂ hR hδ hδone hb | |
| hp hpi hEm hEn hEb | |
| exact hbal.trans_le hprod | |
| /-- Every finite periodic mild approximation has nonincreasing relative entropy. -/ | |
| theorem entropy_nonincrease (hcarleman : Published.GPV2009CarlemanInput) | |
| {q b L T U R : ℝ} {f₀ : JetPrototype.Phase → ℝ} | |
| (F : PeriodicMildState q b L U R f₀) | |
| (hq₁ : 1 < q) (hq₂ : q < 2) (hb : 0 ≤ b) (hR : 0 ≤ R) (hT : 0 < T) (hTU : T ≤ U) | |
| (hloss : R*CutoffLipschitz.frequencyConstant q b ≤ L) | |
| (hf₀ : Measurable f₀) (hf₀b : ∀ z, 0 ≤ f₀ z ∧ f₀ z ≤ R) : | |
| relativeEntropy (F.torusEvolution T) ≤ relativeEntropy (TorusFlight.initialDensity q f₀) := by | |
| obtain ⟨p,hp,hpi⟩ := exists_integrable_rpow_H hq₁ hq₂ | |
| have hEm : Measurable (fun z : ℝ × Phase => F.torusEvolution z.1 z.2) := | |
| (JetCopyDatum.torusPhysical_measurable F.measurable_value q).comp MeasurableEquiv.prodAssoc.symm.measurable | |
| have hEn : ∀ z, 0 ≤ F.torusEvolution T z := fun z => | |
| JetCopyDatum.torusPhysical_nonneg (fun u => (F.value_bounds u).1) q _ | |
| have hEb : ∀ z, F.torusEvolution T z ≤ R*H q z.2 := by | |
| intro z | |
| exact (mul_le_mul_of_nonneg_left (F.value_bounds | |
| (T,(JetPeriodicDatum.spatialMap (TorusChart.representative z.1),z.2))).2 | |
| (H_nonneg q z.2)).trans_eq (mul_comm _ _) | |
| have ham : Measurable (TorusFlight.initialDensity q f₀) := TorusFlight.initialDensity_measurable hf₀ q | |
| have han : ∀ z, 0 ≤ TorusFlight.initialDensity q f₀ z := | |
| fun z => mul_nonneg (H_nonneg q z.2) (hf₀b _).1 | |
| have hab : ∀ z, TorusFlight.initialDensity q f₀ z ≤ R*H q z.2 := by | |
| intro z | |
| exact (mul_le_mul_of_nonneg_left (hf₀b _).2 (H_nonneg q z.2)).trans_eq (mul_comm _ _) | |
| have hFt := regularizedRelativeEntropyDensity_tendsto hq₂ hR hp hpi | |
| (hEm.comp (measurable_const.prodMk measurable_id)) hEn hEb | |
| have hAt := regularizedRelativeEntropyDensity_tendsto hq₂ hR hp hpi ham han hab | |
| have herror := (kinetic_entropy_error_tendsto (AnnularEnvelope.timePositionVolume T) | |
| (B := R+tiltConstant) (entropyErrorCoefficient q R b) hq₂ hpi).mono_left | |
| (show 𝓝[>] (0:ℝ) ≤ 𝓝 0 from nhdsWithin_le_nhds) | |
| have hle := le_of_tendsto_of_tendsto (hFt.sub hAt) herror (show | |
| ∀ᶠ δ : ℝ in 𝓝[>] 0, (∫ z, regularizedRelativeEntropyDensity δ (F.torusEvolution T) z)- | |
| (∫ z, regularizedRelativeEntropyDensity δ (TorusFlight.initialDensity q f₀) z) ≤ | |
| (δ*entropyErrorCoefficient q R b)*∫ z : (ℝ × Position) × Velocity, | |
| logarithmicMajorant q (R+tiltConstant) p δ z.2 | |
| ∂(AnnularEnvelope.timePositionVolume T).prod volume from by | |
| filter_upwards [Ioo_mem_nhdsGT (show (0:ℝ) < 1 by norm_num)] with δ hδ | |
| exact F.regularized_entropy_inequality hcarleman hq₁ hq₂ hb hR hT hTU hloss hf₀ hf₀b | |
| hδ.1 hδ.2.le hp hpi) | |
| have he (f : Density) : (∫ z, scalarRelativeEntropy (maxwellian z.2) (f z)) = relativeEntropy f := by | |
| unfold relativeEntropy | |
| apply integral_congr_ae | |
| filter_upwards [] with z | |
| exact scalarRelativeEntropy_eq (maxwellian_pos z.2) (f z) | |
| rw [he,he] at hle | |
| change relativeEntropy (F.torusEvolution T)-relativeEntropy (TorusFlight.initialDensity q f₀) ≤ 0 at hle | |
| linarith only [hle] | |
| end WeightedLocalControl.PeriodicMildState | |
| end BoltzmannNonuniqueness | |
| end | |
| end | |
| section | |
| /-! # The actual finite periodic approximations satisfy entropy with dissipation -/ | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal Topology | |
| open BoltzmannNonuniqueness.VelocityWeight | |
| namespace BoltzmannNonuniqueness.WeightedLocalControl.PeriodicMildState | |
| theorem entropy_dissipation_inequality (hcarleman : Published.GPV2009CarlemanInput) | |
| {q b L T U R : ℝ} {f₀ : JetPrototype.Phase → ℝ} | |
| (F : PeriodicMildState q b L U R f₀) | |
| (hq₁ : 1 < q) (hq₂ : q < 2) (hb : 0 ≤ b) (hR : 0 ≤ R) (hT : 0 < T) (hTU : T ≤ U) | |
| (hloss : R*CutoffLipschitz.frequencyConstant q b ≤ L) | |
| (hf₀ : Measurable f₀) (hf₀b : ∀ z, 0 ≤ f₀ z ∧ f₀ z ≤ R) : | |
| ENNReal.ofReal (relativeEntropy (F.torusEvolution T))+ | |
| CollisionCompactness.cutoffExtendedDissipation (AnnularEnvelope.timePositionVolume T) b | |
| (kineticDensity F.torusEvolution) ≤ | |
| ENNReal.ofReal (relativeEntropy (TorusFlight.initialDensity q f₀)) := by | |
| let E : Evolution := F.torusEvolution | |
| let a : Density := TorusFlight.initialDensity q f₀ | |
| let r := TorusFlight.along (cutoffPhysicalSource b E) | |
| have hEm : Measurable (fun z : ℝ × Phase => E z.1 z.2) := | |
| (JetCopyDatum.torusPhysical_measurable F.measurable_value q).comp MeasurableEquiv.prodAssoc.symm.measurable | |
| have hEn : ∀ t z, 0 ≤ E t z := fun t z => | |
| JetCopyDatum.torusPhysical_nonneg (fun u => (F.value_bounds u).1) q _ | |
| have hEb : ∀ t z, E t z ≤ R*H q z.2 := by | |
| intro t z | |
| have h := mul_le_mul_of_nonneg_left (F.value_bounds | |
| (t,(JetPeriodicDatum.spatialMap (TorusChart.representative z.1),z.2))).2 (H_nonneg q z.2) | |
| exact h.trans_eq (mul_comm _ _) | |
| have ham : Measurable a := TorusFlight.initialDensity_measurable hf₀ q | |
| have han : ∀ z, 0 ≤ a z := fun z => mul_nonneg (H_nonneg q z.2) (hf₀b _).1 | |
| have hab : ∀ z, a z ≤ R*H q z.2 := by | |
| intro z | |
| exact (mul_le_mul_of_nonneg_left (hf₀b _).2 (H_nonneg q z.2)).trans_eq (mul_comm _ _) | |
| have hcT : (fun z => E T (TorusFlight.flight T z)) =ᵐ[volume] entropyPath a r T := | |
| TorusFlight.characteristic hcarleman F hq₁ hq₂ hb hR hT.le hTU hloss hf₀ | |
| have hc : (fun z : ℝ × Phase => E z.1 (TorusFlight.flight z.1 z.2)) =ᵐ[ | |
| (volume.restrict (Icc 0 T)).prod volume] (fun z => entropyPath a r z.1 z.2) := by | |
| have h := TorusFlight.spacetime_mild hcarleman F hq₁ hq₂ hb hR hT.le hTU hloss hf₀ | |
| have ht : ∀ᵐ z : ℝ × Phase ∂(volume.restrict (Icc 0 T)).prod volume, z.1 ∈ Icc 0 T := | |
| Measure.quasiMeasurePreserving_fst.ae (ae_restrict_mem measurableSet_Icc) | |
| filter_upwards [h,ht] with z hz hzt | |
| rw [Published.timePrimitive_eq_interval hzt] at hz | |
| exact hz | |
| exact bounded_mild_entropy_dissipation_limit hcarleman hq₁ hq₂ hb hR hT | |
| hEm hEn hEb ham han hab hc hcT | |
| end BoltzmannNonuniqueness.WeightedLocalControl.PeriodicMildState | |
| end | |
| end | |
| section | |
| /-! # Uniform integrability of actual finite rates on elastic windows -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| open scoped ENNReal | |
| namespace BoltzmannNonuniqueness.PeriodicApproximation | |
| open Compactness CollisionCompactness | |
| variable {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsFiniteMeasure μ] | |
| theorem scatterTriple_window_measurePreserving (R : ℝ) : | |
| MeasurePreserving (scatterTriple (X := X)) | |
| (((μ.prod (volume : Measure (Velocity × Velocity))).prod sphereArea).restrict (collisionWindow R)) | |
| (((μ.prod (volume : Measure (Velocity × Velocity))).prod sphereArea).restrict (collisionWindow R)) := by | |
| have he : scatterTriple (X := X) ⁻¹' collisionWindow R = collisionWindow R := by | |
| ext z | |
| change (‖scatterLeft z.1.2.1 z.1.2.2 z.2‖^2+ | |
| ‖scatterRight z.1.2.1 z.1.2.2 z.2‖^2 ≤ R^2) ↔ _ | |
| rw [scatter_energy _ _ _ (mem_sphere_zero_iff_norm.mp z.2.property)] | |
| rfl | |
| have h := (scatterTriple_measurePreserving μ).restrict_preimage (collisionWindow_measurable R) | |
| rwa [he] at h | |
| theorem massRegularizedPreRate_window_uniformIntegrable | |
| {f : ℕ → X × Velocity → ℝ} {N : ℕ → 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) | |
| (hN : ∀ n, 0 < N n) {R : ℝ} (hR : 0 ≤ R) : | |
| UniformIntegrable (fun n => massRegularizedPreRate (N n) (f n)) 1 | |
| (((μ.prod (volume : Measure (Velocity × Velocity))).prod sphereArea).restrict (collisionWindow R)) := by | |
| have hu := uniformIntegrable_restrict | |
| (massNormalizedPreRate_local_uniformIntegrable R hfm hf hi hb) (collisionWindow R) | |
| rw [Measure.restrict_restrict_of_subset (collisionWindow_subset_angularPairBox hR)] at hu | |
| have hm n := (massRegularizedPreRate_measurable (N n) (hfm n)).aestronglyMeasurable | |
| (μ := (((μ.prod (volume : Measure (Velocity × Velocity))).prod sphereArea).restrict (collisionWindow R))) | |
| have he n : ∀ᵐ z ∂(((μ.prod (volume : Measure (Velocity × Velocity))).prod sphereArea).restrict | |
| (collisionWindow R)), ‖massRegularizedPreRate (N n) (f n) z‖ₑ ≤ ‖massNormalizedPreRate (f n) z‖ₑ := by | |
| filter_upwards [] with z | |
| apply enorm_le_iff_norm_le.mpr | |
| rw [Real.norm_of_nonneg (massRegularizedPreRate_bounds (hN n) (hf n) z).1,Real.norm_eq_abs] | |
| exact (massRegularizedPreRate_bounds (hN n) (hf n) z).2.trans (le_abs_self _) | |
| obtain ⟨hu,B,hB⟩ := hu | |
| exact ⟨hu.ae_mono hm he,B,fun n => (eLpNorm_mono_enorm_ae (hm n) (he n)).trans (hB n)⟩ | |
| theorem massRegularizedPostRate_window_uniformIntegrable | |
| {f : ℕ → X × Velocity → ℝ} {N : ℕ → 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) | |
| (hN : ∀ n, 0 < N n) {R : ℝ} (hR : 0 ≤ R) : | |
| UniformIntegrable (fun n => massRegularizedPostRate (N n) (f n)) 1 | |
| (((μ.prod (volume : Measure (Velocity × Velocity))).prod sphereArea).restrict (collisionWindow R)) := by | |
| let ν := ((μ.prod (volume : Measure (Velocity × Velocity))).prod sphereArea).restrict (collisionWindow R) | |
| have he : MeasurePreserving (scatterTripleEquiv (X := X)) ν ν := | |
| scatterTriple_window_measurePreserving R | |
| have hp : UniformIntegrable (fun n => massRegularizedPreRate (N n) (f n)) 1 ν := | |
| massRegularizedPreRate_window_uniformIntegrable hfm hf hi hb hN hR | |
| have h := Published.uniformIntegrable_comp_equiv (μ := ν) (ν := ν) | |
| scatterTripleEquiv he (fun n => massRegularizedPreRate (N n) (f n)) hp | |
| exact h | |
| end BoltzmannNonuniqueness.PeriodicApproximation | |
| end | |
| end | |
| end OAI | |