openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Compactness /LocalMomentIntegrability.lean
| import OAI.MathematicalPhysics.Boltzmann.Branching.Moments | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.PublishedInputs | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.ReferenceMinorization | |
| import OAI.MathematicalPhysics.Boltzmann.Jets.TorusTransport | |
| import Mathlib.Topology.MetricSpace.ProperSpace | |
| namespace OAI | |
| section | |
| /-! # Passing from all compact restrictions to an almost-everywhere assertion -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| namespace BoltzmannNonuniqueness.Compactness | |
| theorem ae_of_all_compact_restrict {μ : Measure Published.SpaceTime} {p : Published.SpaceTime → Prop} | |
| (h : ∀ K : Set Published.SpaceTime, IsCompact K → ∀ᵐ x ∂μ.restrict K, p x) : | |
| ∀ᵐ x ∂μ, p x := by | |
| have he : (⋃ n : ℕ, Metric.closedBall (0 : Published.SpaceTime) n) = univ := | |
| Metric.iUnion_closedBall_nat 0 | |
| have ht := (ae_restrict_iUnion_iff (fun n : ℕ => Metric.closedBall (0 : Published.SpaceTime) n) p).mpr | |
| (fun n => h _ (isCompact_closedBall _ _)) | |
| simpa only [he,Measure.restrict_univ] using ht | |
| end BoltzmannNonuniqueness.Compactness | |
| end | |
| end | |
| section | |
| /-! # Volume domination by the selected two-collision angular window -/ | |
| noncomputable section | |
| open MeasureTheory Set Metric ProbabilityTheory | |
| open scoped ENNReal NNReal ProbabilityTheory | |
| namespace BoltzmannNonuniqueness.ReferenceGeometry | |
| theorem localTwoCollisionLaw_volume_domination : | |
| ∃ (δ : ℝ) (B : ℝ≥0∞), 0 < δ ∧ δ ≤ 1/4 ∧ B ≠ 0 ∧ B ≠ ⊤ ∧ | |
| ∀ v ∈ ball beamAxis δ, | |
| volume.restrict (ball (0 : V) δ) ≤ B • | |
| localTwoCollisionLaw v := by | |
| obtain ⟨d,C,hd,hC0,hCt,hout,hin⟩ := paired_parameterSphere_dominated | |
| obtain ⟨ε,hε,hεsmall,hgood⟩ := collision_window_good | |
| obtain ⟨a,δ,K,_ha,hab,hδ,hδa,hK,hvolume⟩ := | |
| uniform_family_window_volume (lt_min hd hε) | |
| have had := hab.trans (min_le_left _ _) | |
| have haε := hab.trans (min_le_right _ _) | |
| have hδsmall := hδa.trans (haε.trans hεsmall) | |
| let I := ENNReal.ofReal (2*δ) | |
| let B := I⁻¹ * (K : ℝ≥0∞)^3 * (C^2) | |
| have hI0 : I ≠ 0 := by dsimp [I]; positivity | |
| have hIt : I ≠ ⊤ := ENNReal.ofReal_ne_top | |
| have hKn : (K : ℝ≥0∞) ≠ 0 := by exact_mod_cast hK.ne' | |
| have hB0 : B ≠ 0 := mul_ne_zero (mul_ne_zero (ENNReal.inv_ne_zero.mpr hIt) | |
| (pow_ne_zero 3 hKn)) (pow_ne_zero 2 hC0) | |
| have hBt : B ≠ ⊤ := ENNReal.mul_ne_top | |
| (ENNReal.mul_ne_top (ENNReal.inv_ne_top.mpr hI0) (ENNReal.pow_ne_top ENNReal.coe_ne_top)) | |
| (ENNReal.pow_ne_top hCt) | |
| refine ⟨δ,B,hδ,hδsmall,hB0,hBt,?_⟩ | |
| intro v hv | |
| have hupper := familyWindowMap_local_le haε hδa had hCt hgood hout hin hv | |
| apply Measure.le_iff.mpr | |
| intro A hA | |
| rw [Measure.restrict_apply hA,Measure.smul_apply,smul_eq_mul] | |
| have hlower := hvolume v hv (A ∩ ball (0 : V) δ) (hA.inter measurableSet_ball) inter_subset_right | |
| have hAupper : ((familyWindowMeasure a δ).map (familyWindowMap v)) (A ∩ ball (0 : V) δ) ≤ | |
| (C^2) * localTwoCollisionLaw v A := by | |
| exact (hupper _).trans (by | |
| rw [Measure.smul_apply,smul_eq_mul] | |
| exact mul_le_mul' le_rfl (measure_mono inter_subset_left)) | |
| have htotal : I * volume (A ∩ ball (0 : V) δ) ≤ (K : ℝ≥0∞)^3 * | |
| ((C^2) * localTwoCollisionLaw v A) := by | |
| exact hlower.trans (by gcongr) | |
| calc | |
| volume (A ∩ ball (0 : V) δ) = I⁻¹ * (I * volume (A ∩ ball (0 : V) δ)) := by | |
| rw [← mul_assoc,ENNReal.inv_mul_cancel hI0 hIt,one_mul] | |
| _ ≤ I⁻¹ * ((K : ℝ≥0∞)^3 * | |
| ((C^2) * localTwoCollisionLaw v A)) := by gcongr | |
| _ = B * localTwoCollisionLaw v A := by simp only [B,mul_assoc] | |
| end BoltzmannNonuniqueness.ReferenceGeometry | |
| end | |
| end | |
| section | |
| /-! # Uniform finite-entropy moments give spacetime transport integrability -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| namespace BoltzmannNonuniqueness | |
| theorem entropyMoment_spacetime_integrable {T : ℝ} {F : Evolution} | |
| (hFm : Measurable (fun z : ℝ × Phase => F z.1 z.2)) | |
| (hFn : ∀ t p, 0 ≤ F t p) | |
| (hfinite : ∀ t ∈ Icc 0 T, FiniteEntropyDatum (F t)) | |
| (hbound : ∃ C : ℝ, ∀ t ∈ Icc 0 T, (∫ p, entropyMoment (F t) p) ≤ C) : | |
| Integrable (fun z : ℝ × Phase => entropyMoment (F z.1) z.2) | |
| ((volume.restrict (Icc 0 T)).prod volume) := by | |
| let E : ℝ × Phase → ℝ := fun z => entropyMoment (F z.1) z.2 | |
| have hEm : Measurable E := by | |
| unfold E entropyMoment | |
| exact hFm.mul ((measurable_const.add (measurable_snd.snd.norm.pow_const 2)).add hFm.log.abs) | |
| have hEn (t : ℝ) (p : Phase) : 0 ≤ E (t,p) := entropyMoment_nonneg (hFn t p) | |
| obtain ⟨C,hC⟩ := hbound | |
| apply (integrable_prod_iff hEm.aestronglyMeasurable).mpr | |
| constructor | |
| · filter_upwards [ae_restrict_mem measurableSet_Icc] with t ht | |
| exact (hfinite t ht).2.2 | |
| · apply Integrable.of_bound (hEm.norm.stronglyMeasurable.integral_prod_right').aestronglyMeasurable C | |
| filter_upwards [ae_restrict_mem measurableSet_Icc] with t ht | |
| have he : (∫ p, ‖E (t,p)‖) = ∫ p, E (t,p) := by | |
| apply integral_congr_ae | |
| exact Filter.Eventually.of_forall (fun p => Real.norm_of_nonneg (hEn t p)) | |
| rw [he,Real.norm_of_nonneg (integral_nonneg (hEn t))] | |
| exact hC t ht | |
| theorem mass_transport_integrable {T : ℝ} {F : Evolution} | |
| (hFm : Measurable (fun z : ℝ × Phase => F z.1 z.2)) | |
| (hFn : ∀ t p, 0 ≤ F t p) | |
| (hEi : Integrable (fun z : ℝ × Phase => entropyMoment (F z.1) z.2) | |
| ((volume.restrict (Icc 0 T)).prod volume)) (φ : MassTest) : | |
| Integrable (fun z : ℝ × Phase => F z.1 z.2*φ.transportDerivative z.1 z.2) | |
| ((volume.restrict (Icc 0 T)).prod volume) := by | |
| obtain ⟨C,hC,hb⟩ := φ.bounded_on_time T | |
| apply (hEi.const_mul (2*C)).mono' (hFm.mul φ.measurable_transportDerivative).aestronglyMeasurable | |
| 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 [ht] with z hz | |
| change ‖F z.1 z.2*φ.transportDerivative z.1 z.2‖ ≤ 2*C*entropyMoment (F z.1) z.2 | |
| rw [Real.norm_eq_abs,abs_mul,abs_of_nonneg (hFn z.1 z.2)] | |
| have hφ := (hb z.1 hz z.2).2 | |
| have hf := hFn z.1 z.2 | |
| have hE := entropyMoment_nonneg hf | |
| have hv : (1+‖z.2.2‖)*F z.1 z.2 ≤ 2*entropyMoment (F z.1) z.2 := by | |
| unfold entropyMoment | |
| have hlog := abs_nonneg (Real.log (F z.1 z.2)) | |
| nlinarith [sq_nonneg (‖z.2.2‖-1),mul_nonneg hf hlog, | |
| mul_nonneg hf (sq_nonneg (‖z.2.2‖-1))] | |
| exact (mul_le_mul_of_nonneg_left hφ hf).trans (by nlinarith) | |
| end BoltzmannNonuniqueness | |
| end | |
| end | |
| end OAI | |