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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 | import OAI.MathematicalPhysics.Boltzmann.Compactness.LocalizedGainSmoothingAE
import OAI.MathematicalPhysics.Boltzmann.Compactness.StrongCompactPairTests
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Tactic
namespace OAI
section
/-! # Scalar majorants for the gain discrepancy of two positive profiles -/
namespace BoltzmannNonuniqueness.Compactness
theorem normalized_difference_abs_le {a b u v : ℝ} (hu : 0 ≤ u) (hv : 0 ≤ v) :
|(a-b)/(1+u+v)| ≤ |a-b| := by
rw [abs_div,abs_of_pos (by positivity : 0 < 1+u+v)]
exact div_le_self (abs_nonneg _) (by linarith)
theorem normalized_difference_source_bound {a b u v : ℝ}
(ha : 0 ≤ a) (hb : 0 ≤ b) (hu : 0 ≤ u) (hv : 0 ≤ v) :
|(a-b)/(1+u+v)| ≤ a/(1+u)+b/(1+v) := by
have hd : 0 < 1+u+v := by positivity
calc
_ = |a-b|/(1+u+v) := by rw [abs_div,abs_of_pos hd]
_ ≤ (a+b)/(1+u+v) := by
apply div_le_div_of_nonneg_right _ hd.le
simpa only [abs_of_nonneg ha,abs_of_nonneg hb] using abs_sub a b
_ = a/(1+u+v)+b/(1+u+v) := add_div _ _ _
_ ≤ _ := add_le_add
(div_le_div_of_nonneg_left ha (by positivity) (by linarith))
(div_le_div_of_nonneg_left hb (by positivity) (by linarith))
end BoltzmannNonuniqueness.Compactness
end
section
/-! # The linear normalized-pair gain is the physical gain divided by spatial mass -/
noncomputable section
open MeasureTheory
namespace BoltzmannNonuniqueness.CollisionCompactness.DoubleGain
open Compactness
variable {X : Type*} [MeasurableSpace X]
omit [MeasurableSpace X] in
theorem normalizedPair_postRate (f : X × Velocity → ℝ)
(z : (X × (Velocity × Velocity)) × Sphere) :
pairPostRate (normalizedPair volume f) z = massNormalizedPostRate f z := by
change pairPreRate (normalizedPair volume f) (scatterTriple z) =
massNormalizedPreRate f (scatterTriple z)
exact (massNormalizedPreRate_eq_pair f (scatterTriple z)).symm
theorem normalizedPair_gain (f : X × Velocity → ℝ) (p : X × Velocity) :
pairGain (normalizedPair volume f) p = gainDensity f p/(1+fiberMass volume f p.1) := by
unfold pairGain
simp_rw [normalizedPair_postRate]
change (∫ q : Velocity × Sphere,
postRate f (collisionGrouping.symm (p,q))/(1+fiberMass volume f p.1)
∂(volume : Measure Velocity).prod sphereArea) = _
simp_rw [div_eq_mul_inv]
rw [integral_mul_const]
rfl
end BoltzmannNonuniqueness.CollisionCompactness.DoubleGain
end
end
section
/-! # Strong convergence of full normalized gains from their localized limits -/
noncomputable section
open MeasureTheory Set Filter
open scoped ENNReal Topology
namespace BoltzmannNonuniqueness.CollisionCompactness.DoubleGain
open Compactness PeriodicApproximation
variable {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsFiniteMeasure μ]
theorem normalizedGain_strong_of_localized
{f : ℕ → X × Velocity → ℝ} {g : X × Velocity → ℝ} {C : ℝ}
(hfm : ∀ n, Measurable (f n)) (hf : ∀ n p, 0 ≤ f n p)
(hi : ∀ n, Integrable (entropyGauge (f n)) (μ.prod volume))
(hb : ∀ n, (∫ p, entropyGauge (f n) p ∂μ.prod volume) ≤ C)
(hgm : Measurable g) (hg : ∀ p, 0 ≤ g p)
(hgi : Integrable (entropyGauge g) (μ.prod volume))
(hgb : (∫ p, entropyGauge g p ∂μ.prod volume) ≤ C)
{ν : Measure (X × Velocity)} (hν : ν ≤ μ.prod volume)
(hc : ∀ η : C(Velocity,ℝ), HasCompactSupport η →
Tendsto (fun n => eLpNorm
(localizedPhysicalGain η (f n)-localizedPhysicalGain η g) 1 ν) atTop (𝓝 0)) :
Tendsto (fun n => eLpNorm
(pairGain (normalizedPair volume (f n))-pairGain (normalizedPair volume g)) 1 ν)
atTop (𝓝 0) := by
choose η hηc hη0 hη1 hηeq using
(fun j : ℕ => exists_velocity_unit_cutoff (show 0 < (j:ℝ)+1 by positivity))
have hδ : Tendsto (fun j : ℕ => sphereArea univ*ENNReal.ofReal (8*C/((j:ℝ)+1)))
atTop (𝓝 0) := by
have he : Tendsto (fun j : ℕ => 8*C/((j:ℝ)+1)) atTop (𝓝 0) := by
exact tendsto_const_nhds.div_atTop
(tendsto_natCast_atTop_atTop.atTop_add tendsto_const_nhds)
have ht := ENNReal.Tendsto.const_mul (a := sphereArea univ) (ENNReal.tendsto_ofReal he)
(Or.inr (measure_ne_top sphereArea univ))
simpa only [ENNReal.ofReal_zero,mul_zero] using ht
apply strongL1_of_two_sided_approximations hδ
(u := fun j n => localizedPhysicalGain (η j) (f n))
(v := fun j => localizedPhysicalGain (η j) g)
· intro j n
exact (eLpNorm_mono_measure _ hν).trans
(localizedGain_tail_le (by positivity) (η j) (hη0 j) (hη1 j) (hηeq j)
(hfm n) (hf n) (hi n) (hb n))
· intro j
rw [eLpNorm_sub_comm]
exact (eLpNorm_mono_measure _ hν).trans
(localizedGain_tail_le (by positivity) (η j) (hη0 j) (hη1 j) (hηeq j) hgm hg hgi hgb)
· exact fun j => hc (η j) (hηc j)
end BoltzmannNonuniqueness.CollisionCompactness.DoubleGain
end
end
end OAI
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