openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Approximation /CappedDensitySequence.lean
| import OAI.MathematicalPhysics.Boltzmann.Approximation.CapMomentBound | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.CommonMomentEnvelope | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.DensityCompactness | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.TimeSliceIntegration | |
| import OAI.MathematicalPhysics.Boltzmann.Jets.AnnularCollisionIntegrability | |
| import OAI.MathematicalPhysics.Boltzmann.Jets.JetFiniteCap | |
| import OAI.MathematicalPhysics.Boltzmann.Jets.JetFamilyBounds | |
| import OAI.MathematicalPhysics.Boltzmann.Jets.TorusTailRegions | |
| namespace OAI | |
| section | |
| /-! # A common velocity moment for the actual capped torus densities -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| open scoped ENNReal | |
| open BoltzmannNonuniqueness.VelocityWeight BoltzmannNonuniqueness.VelocityMoment | |
| namespace BoltzmannNonuniqueness.JetCopyDatum | |
| def momentBase (s₀ η K α d₀ q : ℝ) : ℝ := | |
| (2+escapedConstant s₀ η K α d₀)*hotConstant q | |
| theorem momentBase_nonneg {s₀ K d₀ : ℝ} (hs₀ : 0 < s₀) (hK : 0 ≤ K) (hd₀ : 0 < d₀) | |
| (η α q : ℝ) : 0 ≤ momentBase s₀ η K α d₀ q := | |
| mul_nonneg (add_nonneg (by norm_num) (JetEscapedEnvelope.constant_nonneg hs₀ hK hd₀ η α _)) | |
| (hotConstant_nonneg q) | |
| theorem capped_torus_moment {s₀ η K α γ d₀ q b L T R δ : 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) (hd₀max : d₀ ≤ 1) | |
| (hsmall : JetComparisonCoefficient.coefficient K*d₀*s₀^9 ≤ 1) | |
| (hq : q < 2) (hb : 0 ≤ b) (hL : 0 ≤ L) (hT : 0 ≤ T) | |
| (hR : 0 ≤ R) (hTmax : T ≤ 1/32) (hδ : 0 < δ) (hδmax : δ ≤ 1) | |
| (hloss : R*CutoffLipschitz.frequencyConstant q b ≤ L) | |
| (N : ℕ) {h₀ : JetPrototype.Phase → ℝ} (hh₀m : Measurable h₀) (hh₀ : ∀ p, 0 ≤ h₀ p) | |
| (hF₀ : ∀ p, initial s₀ η K α γ d₀ q N h₀ p ≤ R) | |
| (F : WeightedLocalControl.PeriodicMildState q b L T R (initial s₀ η K α γ d₀ q N h₀)) | |
| (hcap : F.hotPeak δ (color s₀ η K α γ d₀ q N) ≤ 1) | |
| {t : ℝ} (ht : t ∈ Ioc 0 T) : | |
| ∀ᵐ x : Position, | |
| moment (fun v => H q v*F.value | |
| (t,(JetPeriodicDatum.spatialMap (TorusChart.representative x),v))) ≤ | |
| ENNReal.ofReal (AnnularEnvelope.commonMoment s₀ η | |
| (momentBase s₀ η K α d₀ q) (JetEnvelopeGain.momentConstant K α γ) | |
| (AnnularEnvelope.torusChart (t,x))) := by | |
| have hm := capped_cell_moment hs₀ hsmax hη hηmax hK hα hαmax hγ hd₀ hd₀max hsmall | |
| hq hb hL hT hR hTmax hδ hδmax hloss N hh₀m hh₀ hF₀ F hcap ht | |
| have hp := TorusChart.representative_measurePreserving.quasiMeasurePreserving.ae | |
| (hm.filter_mono MeasureTheory.ae_restrict_le) | |
| filter_upwards [hp] with x hx | |
| apply (hx (TorusChart.representative_mem x)).trans (ENNReal.ofReal_le_ofReal ?_) | |
| exact AnnularEnvelope.affine_le_commonMoment hs₀ hη.le ht.1.le | |
| (JetEnvelopeGain.momentConstant_nonneg hK α γ) _ _ _ | |
| end BoltzmannNonuniqueness.JetCopyDatum | |
| end | |
| end | |
| section | |
| /-! # Spacetime velocity moments of the actual capped periodic solutions -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| open scoped ENNReal | |
| open BoltzmannNonuniqueness.VelocityWeight BoltzmannNonuniqueness.VelocityMoment | |
| namespace BoltzmannNonuniqueness.JetCopyDatum | |
| def torusPhysical (q : ℝ) (F : ℝ × JetPrototype.Phase → ℝ) | |
| (z : (ℝ × Position) × Velocity) : ℝ := | |
| H q z.2*F (z.1.1,(JetPeriodicDatum.spatialMap (TorusChart.representative z.1.2),z.2)) | |
| theorem torusPhysical_measurable {F : ℝ × JetPrototype.Phase → ℝ} (hFm : Measurable F) (q : ℝ) : | |
| Measurable (torusPhysical q F) := by | |
| have hrep : Measurable (fun z : (ℝ × Position) × Velocity => TorusChart.representative z.1.2) := | |
| TorusChart.representative_measurePreserving.measurable.comp measurable_fst.snd | |
| have hspace := JetPeriodicDatum.spatialMap_measurePreserving.measurable.comp hrep | |
| have harg : Measurable (fun z : (ℝ × Position) × Velocity => | |
| (z.1.1,(JetPeriodicDatum.spatialMap (TorusChart.representative z.1.2),z.2))) := | |
| measurable_fst.fst.prodMk (hspace.prodMk measurable_snd) | |
| have hw : Measurable (fun z : (ℝ × Position) × Velocity => H q z.2) := by | |
| unfold H singularity | |
| fun_prop | |
| exact hw.mul (hFm.comp harg) | |
| theorem torusPhysical_nonneg {F : ℝ × JetPrototype.Phase → ℝ} (hF : ∀ u, 0 ≤ F u) | |
| (q : ℝ) (z : (ℝ × Position) × Velocity) : 0 ≤ torusPhysical q F z := | |
| mul_nonneg (H_nonneg q z.2) (hF _) | |
| theorem capped_spacetime_moment {s₀ η K α γ d₀ q b L T R δ : 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) (hd₀max : d₀ ≤ 1) | |
| (hsmall : JetComparisonCoefficient.coefficient K*d₀*s₀^9 ≤ 1) | |
| (hq : q < 2) (hb : 0 ≤ b) (hL : 0 ≤ L) (hT : 0 ≤ T) | |
| (hR : 0 ≤ R) (hTmax : T ≤ 1/32) (hδ : 0 < δ) (hδmax : δ ≤ 1) | |
| (hloss : R*CutoffLipschitz.frequencyConstant q b ≤ L) | |
| (N : ℕ) {h₀ : JetPrototype.Phase → ℝ} (hh₀m : Measurable h₀) (hh₀ : ∀ p, 0 ≤ h₀ p) | |
| (hF₀ : ∀ p, initial s₀ η K α γ d₀ q N h₀ p ≤ R) | |
| (F : WeightedLocalControl.PeriodicMildState q b L T R (initial s₀ η K α γ d₀ q N h₀)) | |
| (hcap : F.hotPeak δ (color s₀ η K α γ d₀ q N) ≤ 1) | |
| : | |
| ∀ᵐ u ∂AnnularEnvelope.timePositionVolume T, | |
| moment (fun v => torusPhysical q F.value (u,v)) ≤ | |
| ENNReal.ofReal (AnnularEnvelope.commonMoment s₀ η | |
| (momentBase s₀ η K α d₀ q) (JetEnvelopeGain.momentConstant K α γ) | |
| (AnnularEnvelope.torusChart u)) := by | |
| apply Compactness.ae_interval_of_slices | |
| · exact measurableSet_le (measurable_moment (torusPhysical_measurable F.measurable_value q)) | |
| (ENNReal.measurable_ofReal.comp ((AnnularEnvelope.commonMoment_measurable s₀ η _ _).comp | |
| AnnularEnvelope.torusChart_measurable)) | |
| · intro t ht | |
| exact capped_torus_moment hs₀ hsmax hη hηmax hK hα hαmax hγ hd₀ hd₀max hsmall | |
| hq hb hL hT hR hTmax hδ hδmax hloss N hh₀m hh₀ hF₀ F hcap ht | |
| end BoltzmannNonuniqueness.JetCopyDatum | |
| end | |
| end | |
| section | |
| /-! # Fixed outer-annulus velocity majorants for the actual capped torus densities -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| open scoped ENNReal | |
| open BoltzmannNonuniqueness.WeightedLocal BoltzmannNonuniqueness.VelocityWeight | |
| open BoltzmannNonuniqueness.WeightedLocalControl | |
| namespace BoltzmannNonuniqueness.JetCopyDatum | |
| theorem capped_torus_outer_bound {s₀ η K α γ d₀ q b L T R δ : 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) (hd₀max : d₀ ≤ 1) | |
| (hsmall : JetComparisonCoefficient.coefficient K*d₀*s₀^9 ≤ 1) | |
| (hq : q < 2) (hb : 0 ≤ b) (hL : 0 ≤ L) (hT : 0 ≤ T) | |
| (hR : 0 ≤ R) (hTmax : T ≤ 1/32) (hδ : 0 < δ) (hδmax : δ ≤ 1) | |
| (hloss : R*CutoffLipschitz.frequencyConstant q b ≤ L) | |
| (N J : ℕ) {h₀ : JetPrototype.Phase → ℝ} (hh₀m : Measurable h₀) (hh₀ : ∀ p, 0 ≤ h₀ p) | |
| (hF₀ : ∀ p, initial s₀ η K α γ d₀ q N h₀ p ≤ R) | |
| (F : WeightedLocalControl.PeriodicMildState q b L T R (initial s₀ η K α γ d₀ q N h₀)) | |
| (hcap : F.hotPeak δ (color s₀ η K α γ d₀ q N) ≤ 1) | |
| {t : ℝ} (ht : t ∈ Ioc 0 T) : | |
| ∀ᵐ x : Position, ∀ᵐ v : Velocity, | |
| (t,x) ∉ AnnularEnvelope.torusTail s₀ η J → torusPhysical q F.value ((t,x),v) ≤ | |
| (2+escapedConstant s₀ η K α d₀+finiteAmplitude s₀ η K α d₀ J)*H q v := by | |
| have hi := initial_color_bounds hs₀ hsmax hη hηmax hK hd₀.le α γ q N hh₀ | |
| have hc₀ (i) (p) : 0 ≤ color s₀ η K α γ d₀ q N i p ∧ | |
| color s₀ η K α γ d₀ q N i p ≤ R := ⟨(hi.1 i p).1,(hi.1 i p).2.trans (hF₀ p)⟩ | |
| have hm : Measurable (initial s₀ η K α γ d₀ q N h₀) := | |
| (measurable_seed s₀ η K α γ d₀ q N).add hh₀m | |
| have hz := F.hot_size_le_cap hq hb hL hT hR hδ hloss hm hF₀ | |
| (measurable_color s₀ η K α γ d₀ q N) hc₀ hi.2 hcap t ⟨ht.1.le,ht.2⟩ | |
| have hz2 : PhaseEssSup.size (fun p => F.value (t,p)-∑' i, | |
| rebuilt s₀ η K α γ d₀ q b L T N F.value i (t,p)) ≤ 2 := by | |
| exact hz.trans (by linarith [ht.2]) | |
| have ho := rebuilt_cell_outer_bound hs₀ hsmax hη hηmax hK hα hαmax hγ hd₀ hd₀max hsmall | |
| hq hb hL hT hR hTmax hloss N J hh₀ F.measurable_value F.value_bounds hF₀ | |
| (F.all_mild hT) ht hz2 | |
| have hp := TorusChart.representative_measurePreserving.quasiMeasurePreserving.ae | |
| (ho.filter_mono MeasureTheory.ae_restrict_le) | |
| filter_upwards [hp] with x hx | |
| by_cases hxt : (t,x) ∈ AnnularEnvelope.torusTail s₀ η J | |
| · exact ae_of_all _ (fun _ hn => False.elim (hn hxt)) | |
| · have houter (j : ℕ) (hj : J ≤ j) : | |
| transverse (TorusChart.representative x) ∉ AnnularEnvelope.shell s₀ η j := by | |
| intro hmem | |
| exact hxt ⟨j,hj,hmem⟩ | |
| have hr := hx (TorusChart.representative_mem x) houter | |
| filter_upwards [hr] with v hv | |
| intro _ | |
| exact (mul_le_mul_of_nonneg_left hv (H_nonneg q v)).trans_eq (mul_comm _ _) | |
| end BoltzmannNonuniqueness.JetCopyDatum | |
| end | |
| end | |
| section | |
| /-! # Spacetime outer-annulus majorants for actual capped periodic solutions -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| open scoped ENNReal | |
| open BoltzmannNonuniqueness.WeightedLocal BoltzmannNonuniqueness.VelocityWeight | |
| open BoltzmannNonuniqueness.WeightedLocalControl | |
| namespace BoltzmannNonuniqueness.JetCopyDatum | |
| theorem capped_spacetime_outer_bound {s₀ η K α γ d₀ q b L T R δ : 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) (hd₀max : d₀ ≤ 1) | |
| (hsmall : JetComparisonCoefficient.coefficient K*d₀*s₀^9 ≤ 1) | |
| (hq : q < 2) (hb : 0 ≤ b) (hL : 0 ≤ L) (hT : 0 ≤ T) | |
| (hR : 0 ≤ R) (hTmax : T ≤ 1/32) (hδ : 0 < δ) (hδmax : δ ≤ 1) | |
| (hloss : R*CutoffLipschitz.frequencyConstant q b ≤ L) | |
| (N J : ℕ) {h₀ : JetPrototype.Phase → ℝ} (hh₀m : Measurable h₀) (hh₀ : ∀ p, 0 ≤ h₀ p) | |
| (hF₀ : ∀ p, initial s₀ η K α γ d₀ q N h₀ p ≤ R) | |
| (F : WeightedLocalControl.PeriodicMildState q b L T R (initial s₀ η K α γ d₀ q N h₀)) | |
| (hcap : F.hotPeak δ (color s₀ η K α γ d₀ q N) ≤ 1) | |
| : | |
| ∀ᵐ u ∂AnnularEnvelope.timePositionVolume T, ∀ᵐ v : Velocity, | |
| u ∉ AnnularEnvelope.torusTail s₀ η J → torusPhysical q F.value (u,v) ≤ | |
| (2+escapedConstant s₀ η K α d₀+finiteAmplitude s₀ η K α d₀ J)*H q v := by | |
| apply Compactness.ae_interval_nested_of_slices (T := T) | |
| (ν := (volume : Measure Position)) (ρ := (volume : Measure Velocity)) | |
| (P := fun z : (ℝ × Position) × Velocity => | |
| z.1 ∉ AnnularEnvelope.torusTail s₀ η J → torusPhysical q F.value z ≤ | |
| (2+escapedConstant s₀ η K α d₀+finiteAmplitude s₀ η K α d₀ J)*H q z.2) | |
| · have hs := (AnnularEnvelope.torusTail_measurable s₀ η J).preimage | |
| (measurable_fst : Measurable (Prod.fst : (ℝ × Position) × Velocity → ℝ × Position)) | |
| have hm : Measurable (fun z : (ℝ × Position) × Velocity => | |
| (2+escapedConstant s₀ η K α d₀+finiteAmplitude s₀ η K α d₀ J)*H q z.2) := by | |
| unfold H singularity | |
| fun_prop | |
| convert hs.union (measurableSet_le (torusPhysical_measurable F.measurable_value q) hm) using 1 | |
| ext z | |
| simp only [mem_union,mem_preimage,mem_ofPred_eq,imp_iff_not_or,not_not] | |
| · intro t ht | |
| exact capped_torus_outer_bound hs₀ hsmax hη hηmax hK hα hαmax hγ hd₀ hd₀max hsmall | |
| hq hb hL hT hR hTmax hδ hδmax hloss N J hh₀m hh₀ hF₀ F hcap ht | |
| end BoltzmannNonuniqueness.JetCopyDatum | |
| end | |
| end | |
| section | |
| /-! # Uniform collision-pair integrability for the actual capped approximation family -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| open scoped ENNReal | |
| open BoltzmannNonuniqueness.VelocityWeight BoltzmannNonuniqueness.WeightedLocalControl | |
| namespace BoltzmannNonuniqueness.JetCopyDatum | |
| theorem capped_family_pairs_uniformIntegrable {s₀ η K α γ d₀ q T : 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) (hd₀max : d₀ ≤ 1) | |
| (hsmall : JetComparisonCoefficient.coefficient K*d₀*s₀^9 ≤ 1) | |
| (hq : q < 2) (hT : 0 ≤ T) (hTmax : T ≤ 1/32) | |
| (b L R δ : ℕ → ℝ) (h₀ : ℕ → JetPrototype.Phase → ℝ) | |
| (hb : ∀ n, 0 ≤ b n) (hL : ∀ n, 0 ≤ L n) (hR : ∀ n, 0 ≤ R n) | |
| (hδ : ∀ n, 0 < δ n) (hδmax : ∀ n, δ n ≤ 1) | |
| (hloss : ∀ n, R n*CutoffLipschitz.frequencyConstant q (b n) ≤ L n) | |
| (hh₀m : ∀ n, Measurable (h₀ n)) (hh₀ : ∀ n p, 0 ≤ h₀ n p) | |
| (hF₀ : ∀ n p, initial s₀ η K α γ d₀ q n (h₀ n) p ≤ R n) | |
| (F : ∀ n, PeriodicMildState q (b n) (L n) T (R n) (initial s₀ η K α γ d₀ q n (h₀ n))) | |
| (hcap : ∀ n, (F n).hotPeak (δ n) (color s₀ η K α γ d₀ q n) ≤ 1) : | |
| UniformIntegrable (fun n => CollisionCompactness.pairDensity (torusPhysical q (F n).value)) 1 | |
| ((AnnularEnvelope.timePositionVolume T).prod (volume : Measure (Velocity × Velocity))) := by | |
| apply CollisionCompactness.annular_pairs_uniformIntegrable hs₀ hη hηmax hT hq | |
| (momentBase_nonneg hs₀ hK hd₀ η α q) (JetEnvelopeGain.momentConstant_nonneg hK α γ) | |
| (fun n => torusPhysical_measurable (F n).measurable_value q) | |
| (fun n => torusPhysical_nonneg (fun u => ((F n).value_bounds u).1) q) | |
| · intro n | |
| exact capped_spacetime_moment hs₀ hsmax hη hηmax hK hα hαmax hγ hd₀ hd₀max hsmall | |
| hq (hb n) (hL n) hT (hR n) hTmax (hδ n) (hδmax n) (hloss n) n | |
| (hh₀m n) (hh₀ n) (hF₀ n) (F n) (hcap n) | |
| · intro J | |
| refine ⟨2+escapedConstant s₀ η K α d₀+finiteAmplitude s₀ η K α d₀ J,?_,?_⟩ | |
| · exact add_nonneg (add_nonneg (by norm_num) | |
| (JetEscapedEnvelope.constant_nonneg hs₀ hK hd₀ η α _)) | |
| (finiteAmplitude_nonneg hs₀ hK hd₀ η α J) | |
| · intro n | |
| exact capped_spacetime_outer_bound hs₀ hsmax hη hηmax hK hα hαmax hγ hd₀ hd₀max hsmall | |
| hq (hb n) (hL n) hT (hR n) hTmax (hδ n) (hδmax n) (hloss n) n J | |
| (hh₀m n) (hh₀ n) (hF₀ n) (F n) (hcap n) | |
| end BoltzmannNonuniqueness.JetCopyDatum | |
| end | |
| end | |
| section | |
| /-! # The constructed approximation sequence with uniformly integrable collision pairs -/ | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal Topology | |
| open BoltzmannNonuniqueness.WeightedLocalControl | |
| namespace BoltzmannNonuniqueness.JetCopyDatum | |
| structure CapApproximation (s₀ η K α γ d₀ q T : ℝ) (N : ℕ) where | |
| amplitude : ℝ | |
| coefficient : ℝ | |
| amplitude_pos : 0 < amplitude | |
| coefficient_mem : coefficient ∈ Icc 0 2 | |
| initial_bound : ∀ p, initial s₀ η K α γ d₀ q N | |
| (fun p => coefficient*HotSeedProfile.normalized q (DatumBounds.scale s₀ η (N+1)) p) p ≤ 2*amplitude | |
| solution : PeriodicMildState q (N+1) (damping q (N+1) amplitude) T (2*amplitude) | |
| (initial s₀ η K α γ d₀ q N | |
| (fun p => coefficient*HotSeedProfile.normalized q (DatumBounds.scale s₀ η (N+1)) p)) | |
| cap : solution.hotPeak (DatumBounds.scale s₀ η (N+1)) (color s₀ η K α γ d₀ q N) = 1 | |
| theorem cap_family_pairs_uniformIntegrable {s₀ η K α γ d₀ q T : 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) (hd₀max : d₀ ≤ 1) | |
| (hsmall : JetComparisonCoefficient.coefficient K*d₀*s₀^9 ≤ 1) | |
| (hq : q < 2) (hT : 0 ≤ T) (hTmax : T ≤ 1/32) | |
| (F : ∀ N, CapApproximation s₀ η K α γ d₀ q T N) : | |
| UniformIntegrable (fun N => CollisionCompactness.pairDensity (torusPhysical q (F N).solution.value)) 1 | |
| ((AnnularEnvelope.timePositionVolume T).prod (volume : Measure (Velocity × Velocity))) := by | |
| have hδ (n) : 0 < DatumBounds.scale s₀ η (n+1) := DatumBounds.scale_pos hs₀ _ | |
| have hδmax (n) : DatumBounds.scale s₀ η (n+1) ≤ 1 := by | |
| have h := JetGeometry.scale_antitone hs₀.le hη.le (Nat.zero_le (n+1)) | |
| rw [show DatumBounds.scale s₀ η 0 = s₀ by simp [DatumBounds.scale]] at h | |
| linarith | |
| have hhn (n) (p) : 0 ≤ (F n).coefficient* | |
| HotSeedProfile.normalized q (DatumBounds.scale s₀ η (n+1)) p := by | |
| obtain ⟨_,_,hb⟩ := HotSeedProfile.normalized_bounds (hδ n) (hδmax n) q | |
| exact mul_nonneg (F n).coefficient_mem.1 (hb p).1 | |
| exact capped_family_pairs_uniformIntegrable hs₀ hsmax hη hηmax hK hα hαmax hγ hd₀ hd₀max | |
| hsmall hq hT hTmax (fun n => n+1) (fun n => damping q (n+1) (F n).amplitude) | |
| (fun n => 2*(F n).amplitude) (fun n => DatumBounds.scale s₀ η (n+1)) | |
| (fun n p => (F n).coefficient*HotSeedProfile.normalized q (DatumBounds.scale s₀ η (n+1)) p) | |
| (fun _ => by positivity) (fun n => (bounds (by positivity) (F n).amplitude_pos).1) | |
| (fun n => mul_nonneg (by norm_num) (F n).amplitude_pos.le) hδ hδmax (fun _ => le_rfl) | |
| (fun n => (HotSeedProfile.normalized_measurable q _).const_mul _) hhn | |
| (fun n => (F n).initial_bound) (fun n => (F n).solution) (fun n => (F n).cap.le) | |
| theorem exists_integrable_cap_family (hcarleman : Published.GPV2009CarlemanInput) | |
| {s₀ η K α γ d₀ q : ℝ} | |
| (hs₀ : 0 < s₀) (hsmax : s₀ ≤ 1/8) (hη : 0 < η) (hηmax : η ≤ Real.log 2) | |
| (hK : 0 ≤ K) (hα : 0 < α) (hαmax : α ≤ 1/4) (hγ : 0 < γ) | |
| (hd₀ : 0 < d₀) (hd₀max : d₀ ≤ 1) (hwidth : d₀*(1/8:ℝ)^10 ≤ 1/64) | |
| (hsmall : JetComparisonCoefficient.coefficient K*d₀*s₀^9 ≤ 1) | |
| (hq₁ : 1 < q) (hq₂ : q < 2) : | |
| ∃ T : ℝ, 0 < T ∧ T ≤ 1/32 ∧ ∃ F : ∀ N, CapApproximation s₀ η K α γ d₀ q T N, | |
| UniformIntegrable (fun N => CollisionCompactness.pairDensity (torusPhysical q (F N).solution.value)) 1 | |
| ((AnnularEnvelope.timePositionVolume T).prod (volume : Measure (Velocity × Velocity))) ∧ | |
| Tendsto (fun N => ∫ p, ‖periodicInitial s₀ η K α γ d₀ q N (F N).coefficient p- | |
| JetPeriodicDatum.datum s₀ η K α γ d₀ p‖) atTop (𝓝 0) := by | |
| obtain ⟨T,hT,hTmax,hcap⟩ := exists_uniform_cap_solution hcarleman hs₀ hsmax hη hηmax | |
| hK hα hαmax hγ hd₀ hd₀max hwidth hsmall hq₁ hq₂ | |
| have hn (N : ℕ) : Nonempty (CapApproximation s₀ η K α γ d₀ q T N) := by | |
| obtain ⟨A,ell,hA,hell,hinit,F,hF⟩ := hcap N (N+1) T (by positivity) hT.le le_rfl | |
| exact ⟨⟨A,ell,hA,hell,hinit,F,hF⟩⟩ | |
| let F := fun N => Classical.choice (hn N) | |
| refine ⟨T,hT,hTmax,F,?_,?_⟩ | |
| · exact cap_family_pairs_uniformIntegrable hs₀ hsmax hη hηmax hK hα | |
| (hαmax.trans (by norm_num)) hγ hd₀ hd₀max hsmall hq₂ hT.le hTmax F | |
| · exact periodicInitial_L1_tendsto hs₀ hsmax hη hηmax hK hγ hd₀ | |
| (by linarith) hq₂ α (fun N => (F N).coefficient_mem) | |
| end BoltzmannNonuniqueness.JetCopyDatum | |
| end | |
| end | |
| section | |
| /-! # Uniform integrability of the actual capped one-particle densities -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| open scoped ENNReal | |
| open BoltzmannNonuniqueness.VelocityWeight BoltzmannNonuniqueness.WeightedLocalControl | |
| namespace BoltzmannNonuniqueness.JetCopyDatum | |
| theorem capped_family_density_uniformIntegrable {s₀ η K α γ d₀ q T : 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) (hd₀max : d₀ ≤ 1) | |
| (hsmall : JetComparisonCoefficient.coefficient K*d₀*s₀^9 ≤ 1) | |
| (hq : q < 2) (hT : 0 ≤ T) (hTmax : T ≤ 1/32) | |
| (b L R δ : ℕ → ℝ) (h₀ : ℕ → JetPrototype.Phase → ℝ) | |
| (hb : ∀ n, 0 ≤ b n) (hL : ∀ n, 0 ≤ L n) (hR : ∀ n, 0 ≤ R n) | |
| (hδ : ∀ n, 0 < δ n) (hδmax : ∀ n, δ n ≤ 1) | |
| (hloss : ∀ n, R n*CutoffLipschitz.frequencyConstant q (b n) ≤ L n) | |
| (hh₀m : ∀ n, Measurable (h₀ n)) (hh₀ : ∀ n p, 0 ≤ h₀ n p) | |
| (hF₀ : ∀ n p, initial s₀ η K α γ d₀ q n (h₀ n) p ≤ R n) | |
| (F : ∀ n, PeriodicMildState q (b n) (L n) T (R n) (initial s₀ η K α γ d₀ q n (h₀ n))) | |
| (hcap : ∀ n, (F n).hotPeak (δ n) (color s₀ η K α γ d₀ q n) ≤ 1) : | |
| UniformIntegrable (fun n => torusPhysical q (F n).value) 1 | |
| ((AnnularEnvelope.timePositionVolume T).prod (volume : Measure Velocity)) := by | |
| apply CollisionCompactness.annular_density_uniformIntegrable hs₀ hη hηmax hT hq | |
| (momentBase_nonneg hs₀ hK hd₀ η α q) (JetEnvelopeGain.momentConstant_nonneg hK α γ) | |
| (fun n => torusPhysical_measurable (F n).measurable_value q) | |
| (fun n => torusPhysical_nonneg (fun u => ((F n).value_bounds u).1) q) | |
| · intro n | |
| exact capped_spacetime_moment hs₀ hsmax hη hηmax hK hα hαmax hγ hd₀ hd₀max hsmall | |
| hq (hb n) (hL n) hT (hR n) hTmax (hδ n) (hδmax n) (hloss n) n | |
| (hh₀m n) (hh₀ n) (hF₀ n) (F n) (hcap n) | |
| · intro J | |
| refine ⟨2+escapedConstant s₀ η K α d₀+finiteAmplitude s₀ η K α d₀ J,?_,?_⟩ | |
| · exact add_nonneg (add_nonneg (by norm_num) | |
| (JetEscapedEnvelope.constant_nonneg hs₀ hK hd₀ η α _)) | |
| (finiteAmplitude_nonneg hs₀ hK hd₀ η α J) | |
| · intro n | |
| exact capped_spacetime_outer_bound hs₀ hsmax hη hηmax hK hα hαmax hγ hd₀ hd₀max hsmall | |
| hq (hb n) (hL n) hT (hR n) hTmax (hδ n) (hδmax n) (hloss n) n J | |
| (hh₀m n) (hh₀ n) (hF₀ n) (F n) (hcap n) | |
| end BoltzmannNonuniqueness.JetCopyDatum | |
| end | |
| end | |
| section | |
| /-! # One-particle uniform integrability for the constructed cap sequence -/ | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal Topology | |
| open BoltzmannNonuniqueness.WeightedLocalControl | |
| namespace BoltzmannNonuniqueness.JetCopyDatum | |
| theorem cap_family_density_uniformIntegrable {s₀ η K α γ d₀ q T : 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) (hd₀max : d₀ ≤ 1) | |
| (hsmall : JetComparisonCoefficient.coefficient K*d₀*s₀^9 ≤ 1) | |
| (hq : q < 2) (hT : 0 ≤ T) (hTmax : T ≤ 1/32) | |
| (F : ∀ N, CapApproximation s₀ η K α γ d₀ q T N) : | |
| UniformIntegrable (fun N => torusPhysical q (F N).solution.value) 1 | |
| ((AnnularEnvelope.timePositionVolume T).prod (volume : Measure Velocity)) := by | |
| have hδ (n) : 0 < DatumBounds.scale s₀ η (n+1) := DatumBounds.scale_pos hs₀ _ | |
| have hδmax (n) : DatumBounds.scale s₀ η (n+1) ≤ 1 := by | |
| have h := JetGeometry.scale_antitone hs₀.le hη.le (Nat.zero_le (n+1)) | |
| rw [show DatumBounds.scale s₀ η 0 = s₀ by simp [DatumBounds.scale]] at h | |
| linarith | |
| have hhn (n) (p) : 0 ≤ (F n).coefficient* | |
| HotSeedProfile.normalized q (DatumBounds.scale s₀ η (n+1)) p := by | |
| obtain ⟨_,_,hb⟩ := HotSeedProfile.normalized_bounds (hδ n) (hδmax n) q | |
| exact mul_nonneg (F n).coefficient_mem.1 (hb p).1 | |
| exact capped_family_density_uniformIntegrable hs₀ hsmax hη hηmax hK hα hαmax hγ hd₀ hd₀max | |
| hsmall hq hT hTmax (fun n => n+1) (fun n => damping q (n+1) (F n).amplitude) | |
| (fun n => 2*(F n).amplitude) (fun n => DatumBounds.scale s₀ η (n+1)) | |
| (fun n p => (F n).coefficient*HotSeedProfile.normalized q (DatumBounds.scale s₀ η (n+1)) p) | |
| (fun _ => by positivity) (fun n => (bounds (by positivity) (F n).amplitude_pos).1) | |
| (fun n => mul_nonneg (by norm_num) (F n).amplitude_pos.le) hδ hδmax (fun _ => le_rfl) | |
| (fun n => (HotSeedProfile.normalized_measurable q _).const_mul _) hhn | |
| (fun n => (F n).initial_bound) (fun n => (F n).solution) (fun n => (F n).cap.le) | |
| end BoltzmannNonuniqueness.JetCopyDatum | |
| end | |
| end | |
| end OAI | |