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f18e7de | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 | import OAI.MathematicalPhysics.Boltzmann.Approximation.MildCharacteristics
namespace OAI
section
/-! # Scalar characteristic representatives for a.e. mild solutions -/
noncomputable section
open MeasureTheory Set
namespace BoltzmannNonuniqueness.PositiveIntegral
theorem ae_path_bound_of_slices {T : ℝ} {F : ℝ × JetPrototype.Phase → ℝ}
{A : ℝ → ℝ} (hFm : Measurable F) (hAm : Measurable A)
(hbound : ∀ t ∈ Icc 0 T, ∀ᵐ p ∂FlightMeasure.phaseVolume, F (t,p) ≤ A t) :
∀ᵐ p ∂FlightMeasure.phaseVolume, ∀ᵐ t ∂volume.restrict (Icc (0:ℝ) T),
F (t,JetBarrier.flight t p) ≤ A t := by
have hm : Measurable (fun z : ℝ × JetPrototype.Phase =>
F (z.1,JetBarrier.flight z.1 z.2)) :=
hFm.comp (measurable_fst.prodMk GaussianLocal.measurable_flight)
have hall : ∀ᵐ z ∂(volume.restrict (Icc (0:ℝ) T)).prod FlightMeasure.phaseVolume,
F (z.1,JetBarrier.flight z.1 z.2) ≤ A z.1 := by
apply (Measure.ae_prod_iff_ae_ae (measurableSet_le hm (hAm.comp measurable_fst))).mpr
filter_upwards [ae_restrict_mem measurableSet_Icc] with t ht
exact (FlightMeasure.flight_measurePreserving t).quasiMeasurePreserving.ae (hbound t ht)
exact Measure.ae_ae_of_ae_prod
((Measure.measurePreserving_swap (μ := FlightMeasure.phaseVolume)
(ν := volume.restrict (Icc 0 T))).quasiMeasurePreserving.ae hall)
theorem ae_characteristic_path {T t L : ℝ} (ht : t ∈ Icc 0 T)
{f₀ : JetPrototype.Phase → ℝ} (hf₀ : Measurable f₀)
{F g : ℝ × JetPrototype.Phase → ℝ} (hFm : Measurable F) (hgm : Measurable g)
(hmild : ∀ s ∈ Icc 0 T, (fun p => F (s,p)) =ᵐ[FlightMeasure.phaseVolume]
(fun p => value T L JetBarrier.flight f₀ g (s,p))) :
∀ᵐ p ∂FlightMeasure.phaseVolume,
((fun r => F (r,JetBarrier.flight r p)) =ᵐ[volume.restrict (Icc (0:ℝ) t)]
(fun r => TransportPrimitive.damped L (f₀ p)
(fun s => g (s,JetBarrier.flight s p)) r)) ∧
F (t,JetBarrier.flight t p) = TransportPrimitive.damped L (f₀ p)
(fun s => g (s,JetBarrier.flight s p)) t := by
let A : ℝ × JetPrototype.Phase → ℝ := fun z => F (z.1,JetBarrier.flight z.1 z.2)
let B : ℝ × JetPrototype.Phase → ℝ := fun z =>
value T L JetBarrier.flight f₀ g (z.1,JetBarrier.flight z.1 z.2)
have hflow : Measurable (fun z : ℝ × JetPrototype.Phase =>
(z.1,JetBarrier.flight z.1 z.2)) :=
measurable_fst.prodMk GaussianLocal.measurable_flight
have hAm : Measurable A := hFm.comp hflow
have hBm : Measurable B := (measurable_value (ht.1.trans ht.2)
GaussianLocal.measurable_flight hf₀ hgm).comp hflow
have hslice (r : ℝ) (hr : r ∈ Icc 0 T) :
(fun p => A (r,p)) =ᵐ[FlightMeasure.phaseVolume] (fun p => B (r,p)) :=
(FlightMeasure.flight_measurePreserving r).quasiMeasurePreserving.ae (hmild r hr)
have hall : A =ᵐ[(volume.restrict (Icc 0 t)).prod FlightMeasure.phaseVolume] B := by
apply (Measure.ae_prod_iff_ae_ae (measurableSet_eq_fun hAm hBm)).mpr
filter_upwards [ae_restrict_mem measurableSet_Icc] with r hr
exact hslice r ⟨hr.1,hr.2.trans ht.2⟩
have hp := Measure.ae_ae_of_ae_prod
((Measure.measurePreserving_swap (μ := FlightMeasure.phaseVolume)
(ν := volume.restrict (Icc 0 t))).quasiMeasurePreserving.ae hall)
filter_upwards [hp,hslice t ht] with p hpe hend
constructor
· filter_upwards [hpe,ae_restrict_mem measurableSet_Icc] with r hr hrmem
exact hr.trans (value_along_flight ⟨hrmem.1,hrmem.2.trans ht.2⟩ L f₀ g p)
· exact hend.trans (value_along_flight ht L f₀ g p)
end BoltzmannNonuniqueness.PositiveIntegral
end
end
section
/-! # Positive upper bounds for integrable signed characteristic sources -/
noncomputable section
open MeasureTheory Set
open scoped ENNReal
namespace BoltzmannNonuniqueness.PositiveIntegral
variable {X : Type*} [MeasurableSpace X] {μ : Measure X}
theorem ofReal_integral_le {f : X → ℝ} (hf : Integrable f μ) :
ENNReal.ofReal (∫ x, f x ∂μ) ≤ ∫⁻ x, ENNReal.ofReal (f x) ∂μ := by
have hm : Integrable (fun x => max (f x) 0) μ := hf.sup (integrable_zero X ℝ μ)
calc
_ ≤ ENNReal.ofReal (∫ x, max (f x) 0 ∂μ) :=
ENNReal.ofReal_le_ofReal (integral_mono hf hm (fun x => le_max_left _ _))
_ = ∫⁻ x, ENNReal.ofReal (max (f x) 0) ∂μ :=
ofReal_integral_eq_lintegral_ofReal hm (ae_of_all _ (fun x => le_max_right _ _))
_ = _ := by
apply lintegral_congr
intro x
rw [ENNReal.ofReal_max,ENNReal.ofReal_zero]
exact max_eq_left bot_le
theorem ofReal_add_intervalIntegral_le {a b c : ℝ} (hab : a ≤ b) {g : ℝ → ℝ}
(hg : IntervalIntegrable g volume a b) :
ENNReal.ofReal (c+∫ s in a..b, g s) ≤
ENNReal.ofReal c+∫⁻ s in Ioc a b, ENNReal.ofReal (g s) := by
apply ENNReal.ofReal_add_le.trans
apply add_le_add le_rfl
rw [intervalIntegral.integral_of_le hab]
exact ofReal_integral_le hg.1
end BoltzmannNonuniqueness.PositiveIntegral
end
end
end OAI
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