| import OAI.MathematicalPhysics.Boltzmann.Jets.ColdMildRepresentative |
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| namespace OAI |
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| /-! # Initial traces of the a.e. characteristic equations -/ |
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| noncomputable section |
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| open MeasureTheory Set |
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| namespace BoltzmannNonuniqueness |
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| namespace PositiveIntegral |
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| theorem value_zero {T : ℝ} (_hT : 0 ≤ T) (L : ℝ) |
| (f₀ : JetPrototype.Phase → ℝ) (g : ℝ × JetPrototype.Phase → ℝ) (p : JetPrototype.Phase) : |
| value T L JetBarrier.flight f₀ g (0,p) = f₀ p := by |
| simp [value,time,JetBarrier.flight] |
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| end PositiveIntegral |
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| namespace WeightedLocal |
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| theorem MildAE.initial_ae {q b L T : ℝ} (hT : 0 ≤ T) |
| {F : ℝ × JetPrototype.Phase → ℝ} {F₀ : JetPrototype.Phase → ℝ} |
| (he : MildAE q b L T F₀ F) : |
| (fun p => F (0,p)) =ᵐ[FlightMeasure.phaseVolume] F₀ := by |
| simpa only [PositiveIntegral.value_zero hT] using he 0 ⟨le_rfl,hT⟩ |
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| theorem ColdMildAE.initial_ae {q b L T : ℝ} (hT : 0 ≤ T) |
| {F c : ℝ × JetPrototype.Phase → ℝ} {c₀ : JetPrototype.Phase → ℝ} |
| (he : ColdMildAE q b L T F c₀ c) : |
| (fun p => c (0,p)) =ᵐ[FlightMeasure.phaseVolume] c₀ := by |
| simpa only [PositiveIntegral.value_zero hT] using he 0 ⟨le_rfl,hT⟩ |
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| end WeightedLocal |
| end BoltzmannNonuniqueness |
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| end |
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| end OAI |
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|
