| import OAI.MathematicalPhysics.Boltzmann.Branching.PerturbedFiniteBlock |
| import OAI.MathematicalPhysics.Boltzmann.Jets.BeamNormalOutgoing |
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| namespace OAI |
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|
| /-! # Uniform branching growth from coefficient bounds only on the visited cylinder -/ |
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| noncomputable section |
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| open MeasureTheory Set |
| open scoped ENNReal |
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| namespace BoltzmannNonuniqueness.ReferenceGeometry |
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| theorem uniform_local_beam_block {d κ : ℝ} |
| (hd : 0 < d) (hd1 : d ≤ 1) (hκ : 0 < κ) (hκ1 : κ < 1) |
| (hg : (referenceClock : ℝ)*κ < referenceGrowth) : |
| ∃ A₀ : ℝ, 0 < A₀ ∧ ∀ T : ℝ, A₀ ≤ T → ∃ N : ℕ, ∃ ε₀ : ℝ, 0 < ε₀ ∧ |
| ∀ ε ∈ Icc 0 ε₀, ∀ A ∈ Icc A₀ T, ∀ z : FlightState, |
| ∀ (b : BeamCoefficients) (endTime : ℝ), |
| (∀ y : ℝ × V, y.1 ∈ Icc (endTime-A) endTime → ‖y.2-z.1‖ ≤ 6*A → BeamNear b ε y) → |
| (6 : ℝ≥0∞) < mediumBlockMass b endTime T (A,z) κ d N := by |
| obtain ⟨A₀,hA₀,hgrowth⟩ := uniform_perturbed_block_growth hd hd1 hκ hκ1 hg |
| refine ⟨A₀,hA₀,?_⟩ |
| intro T hT |
| obtain ⟨N,ε₁,hε₁,hblock⟩ := hgrowth T hT |
| let L : ℝ := (N:ℝ)+1 |
| have hL : 0 < L := by dsimp [L]; positivity |
| refine ⟨N,min ε₁ (1/L),lt_min hε₁ (one_div_pos.mpr hL),?_⟩ |
| intro ε hε A hA z b endTime hnear |
| apply hblock ε ⟨hε.1,hε.2.trans (min_le_left _ _)⟩ A hA z b endTime |
| intro n hn a ha |
| have hnL : (n:ℝ) ≤ L := (Nat.cast_le.mpr hn.le).trans (le_add_of_nonneg_right (by norm_num)) |
| have hbudget : (n:ℝ)*ε ≤ 1 := |
| (mul_le_mul_of_nonneg_right hnL hε.1).trans (by |
| have hh := (le_div_iff₀ hL).mp (hε.2.trans (min_le_right _ _)) |
| simpa only [mul_comm] using hh) |
| exact wordNearby_from_start b endTime A z hε.1 (hA₀.le.trans hA.1) hnear n hbudget a ha |
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| theorem uniform_physical_beam_block {d κ : ℝ} |
| (hd : 0 < d) (hd1 : d ≤ 1) (hκ : 0 < κ) (hκ1 : κ < 1) |
| (hg : (referenceClock : ℝ)*κ < referenceGrowth) : |
| ∃ A₀ : ℝ, 0 < A₀ ∧ ∀ T : ℝ, A₀ ≤ T → ∃ N : ℕ, ∃ ε₀ : ℝ, 0 < ε₀ ∧ |
| ∀ ε ∈ Icc 0 ε₀, ∀ A ∈ Icc A₀ T, ∀ v : RetainedState, |
| ∀ (b : BeamCoefficients) (ρ t₀ : ℝ) (x₀ : V), 0 < ρ → |
| (∀ p : ℝ × V, p.1 ∈ Icc t₀ (t₀+A/ρ) → ‖p.2-x₀‖ ≤ 6*A/ρ → BeamNear b ε p) → |
| (6 : ℝ≥0∞) < mediumBlockMass (b.normalized ρ t₀ x₀) A T (A,(0,v)) κ d N := by |
| obtain ⟨A₀,hA₀,hgrowth⟩ := uniform_local_beam_block hd hd1 hκ hκ1 hg |
| refine ⟨A₀,hA₀,?_⟩ |
| intro T hT |
| obtain ⟨N,ε₀,hε₀,hblock⟩ := hgrowth T hT |
| refine ⟨N,ε₀,hε₀,?_⟩ |
| intro ε hε A hA v b ρ t₀ x₀ hρ hnear |
| apply hblock ε hε A hA (0,v) (b.normalized ρ t₀ x₀) A |
| intro p hp hx |
| exact normalized_beam_near b hρ x₀ hnear p (by simpa using hp) (by simpa using hx) |
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| end BoltzmannNonuniqueness.ReferenceGeometry |
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| end |
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| end OAI |
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