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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 | import OAI.MathematicalPhysics.Boltzmann.Compactness.LocalMomentIntegrability
namespace OAI
/-! # A fixed positive set of angular pairs reaching the target velocity ball -/
noncomputable section
open MeasureTheory Set Metric
open scoped ENNReal
namespace BoltzmannNonuniqueness.ReferenceGeometry
def accessAngles (κ ρ : ℝ) : Set (Sphere × Sphere) :=
twoCollisionMap beamAxis ⁻¹' ball (κ • firstAxis) ρ ∩ firstAngleSet beamAxis
theorem accessAngles_measurable (κ ρ : ℝ) : MeasurableSet (accessAngles κ ρ) :=
((twoCollisionMap_measurable beamAxis) measurableSet_ball).inter
(firstAngleSet_measurable beamAxis)
theorem accessAngles_positive : ∃ δ : ℝ, 0 < δ ∧ δ ≤ 1/4 ∧
∀ κ ρ : ℝ, 0 < κ → 0 < ρ → κ+ρ ≤ δ →
0 < (sphereProbability.prod sphereProbability) (accessAngles κ ρ) := by
obtain ⟨δ,B,hδ,hδsmall,_hB0,_hBt,hvolume⟩ := localTwoCollisionLaw_volume_domination
refine ⟨δ,hδ,hδsmall,?_⟩
intro κ ρ hκ hρ hκρ
have hsub : ball (κ • firstAxis) ρ ⊆ ball (0 : V) δ := by
intro v hv
have ht := dist_triangle v (κ • firstAxis) 0
simp only [dist_zero_right,norm_smul,Real.norm_eq_abs,abs_of_pos hκ,axis_norms.1,mul_one] at ht
change dist v 0 < δ
rw [dist_zero_right]
have hv' : dist v (κ • firstAxis) < ρ := hv
linarith
have hv := hvolume beamAxis (mem_ball_self hδ) (ball (κ • firstAxis) ρ)
rw [Measure.restrict_apply measurableSet_ball,inter_eq_left.mpr hsub,
Measure.smul_apply,smul_eq_mul] at hv
have hp : 0 < volume (ball (κ • firstAxis) ρ) := measure_ball_pos volume _ hρ
have hl : 0 < localTwoCollisionLaw beamAxis (ball (κ • firstAxis) ρ) := by
by_contra hz
have hz' : localTwoCollisionLaw beamAxis (ball (κ • firstAxis) ρ) = 0 :=
bot_unique (not_lt.mp hz)
rw [hz',mul_zero] at hv
exact (not_le_of_gt hp) hv
simpa only [localTwoCollisionLaw,Measure.map_apply (twoCollisionMap_measurable beamAxis)
measurableSet_ball,Measure.restrict_apply
((twoCollisionMap_measurable beamAxis) measurableSet_ball),accessAngles] using hl
end BoltzmannNonuniqueness.ReferenceGeometry
end
end OAI
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