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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 | import OAI.MathematicalPhysics.Boltzmann.Collision.VelocitySmoothingError
import OAI.MathematicalPhysics.Boltzmann.Compactness.StrongPairMoment
namespace OAI
section
/-! # Four vanishing error terms have a vanishing nonnegative L1 sum -/
noncomputable section
open MeasureTheory Filter
open scoped ENNReal Topology
namespace BoltzmannNonuniqueness.Compactness
variable {X : Type*} [MeasurableSpace X] {μ : Measure X}
theorem real_eLpNorm_abs {f : X → ℝ} (hf : Measurable f) :
eLpNorm (fun x => |f x|) 1 μ = eLpNorm f 1 μ := by
simpa only [Real.norm_eq_abs] using eLpNorm_norm f hf.aestronglyMeasurable
theorem strongL1_sub_of_common_limit {f g : ℕ → X → ℝ} {h : X → ℝ}
(hf : Tendsto (fun n => eLpNorm (fun x => f n x-h x) 1 μ) atTop (𝓝 0))
(hg : Tendsto (fun n => eLpNorm (fun x => g n x-h x) 1 μ) atTop (𝓝 0)) :
Tendsto (fun n => eLpNorm (fun x => f n x-g n x) 1 μ) atTop (𝓝 0) := by
have ht := hf.add hg
simp only [zero_add] at ht
apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds ht (fun _ => zero_le)
intro n
have he : (fun x => f n x-g n x) = (fun x => f n x-h x)+(fun x => h x-g n x) := by
funext x
simp only [Pi.add_apply]
ring
dsimp only
rw [he]
exact (eLpNorm_add_le (by norm_num)).trans_eq
(congrArg (fun t => eLpNorm (fun x => f n x-h x) 1 μ+t) (eLpNorm_sub_comm h (g n) 1 μ))
theorem strongL1_abs_sum_four {a b c d : ℕ → X → ℝ}
(ham : ∀ n, Measurable (a n)) (hbm : ∀ n, Measurable (b n))
(hcm : ∀ n, Measurable (c n)) (hdm : ∀ n, Measurable (d n))
(ha : Tendsto (fun n => eLpNorm (a n) 1 μ) atTop (𝓝 0))
(hb : Tendsto (fun n => eLpNorm (b n) 1 μ) atTop (𝓝 0))
(hc : Tendsto (fun n => eLpNorm (c n) 1 μ) atTop (𝓝 0))
(hd : Tendsto (fun n => eLpNorm (d n) 1 μ) atTop (𝓝 0)) :
Tendsto (fun n => eLpNorm (fun x => |a n x|+|b n x|+|c n x|+|d n x|) 1 μ) atTop (𝓝 0) := by
have ht := ((ha.add hb).add hc).add hd
simp only [zero_add] at ht
apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds ht (fun _ => zero_le)
intro n
have he : (fun x => |a n x|+|b n x|+|c n x|+|d n x|) =
((fun x => |a n x|)+(fun x => |b n x|))+(fun x => |c n x|)+(fun x => |d n x|) := rfl
dsimp only
rw [he]
calc
_ ≤ (eLpNorm ((fun x => |a n x|)+(fun x => |b n x|)) 1 μ+
eLpNorm (fun x => |c n x|) 1 μ)+eLpNorm (fun x => |d n x|) 1 μ :=
(eLpNorm_add_le (by norm_num)).trans
(add_le_add (eLpNorm_add_le (by norm_num)) le_rfl)
_ ≤ ((eLpNorm (fun x => |a n x|) 1 μ+eLpNorm (fun x => |b n x|) 1 μ)+
eLpNorm (fun x => |c n x|) 1 μ)+eLpNorm (fun x => |d n x|) 1 μ :=
add_le_add (add_le_add (eLpNorm_add_le (by norm_num)) le_rfl) le_rfl
_ = _ := by rw [real_eLpNorm_abs (ham n),real_eLpNorm_abs (hbm n),
real_eLpNorm_abs (hcm n),real_eLpNorm_abs (hdm n)]
end BoltzmannNonuniqueness.Compactness
end
end
section
/-! # Passing velocity translation continuity through uniform L1 approximation -/
noncomputable section
open MeasureTheory Filter
open scoped Topology
namespace BoltzmannNonuniqueness.CollisionCompactness.DoubleGain
variable {X : Type*} [MeasurableSpace X] {μ : Measure X} [SFinite μ]
theorem velocityTranslation_compare {F H : X × Velocity → ℝ}
(hF : Integrable F (μ.prod volume)) (hH : Integrable H (μ.prod volume)) (a : Velocity) :
(∫ p, |F (p.1,p.2+a)-F p| ∂μ.prod volume) ≤
2*(∫ p, |F p-H p| ∂μ.prod volume)+
(∫ p, |H (p.1,p.2+a)-H p| ∂μ.prod volume) := by
have hFs := (velocityShift_preserving μ a).integrable_comp_of_integrable hF
have hHs := (velocityShift_preserving μ a).integrable_comp_of_integrable hH
have hA : Integrable (fun p => |F (p.1,p.2+a)-H (p.1,p.2+a)|) (μ.prod volume) :=
(hFs.sub hHs).abs
have hB : Integrable (fun p => |H (p.1,p.2+a)-H p|) (μ.prod volume) := (hHs.sub hH).abs
have hC : Integrable (fun p => |F p-H p|) (μ.prod volume) := (hF.sub hH).abs
have hs := (velocityShift_preserving μ a).integral_comp (velocityShift a).measurableEmbedding
(fun p => |F p-H p|)
calc
_ ≤ ∫ p, (|F (p.1,p.2+a)-H (p.1,p.2+a)|+|H (p.1,p.2+a)-H p|)+|F p-H p|
∂μ.prod volume := by
apply integral_mono (hFs.sub hF).abs ((hA.add hB).add hC)
intro p
change |F (p.1,p.2+a)-F p| ≤
(|F (p.1,p.2+a)-H (p.1,p.2+a)|+|H (p.1,p.2+a)-H p|)+|F p-H p|
have h1 := abs_sub_le (F (p.1,p.2+a)) (H (p.1,p.2+a)) (H p)
have h2 := abs_sub_le (F (p.1,p.2+a)) (H p) (F p)
rw [abs_sub_comm (H p) (F p)] at h2
linarith
_ = (∫ p, |F (p.1,p.2+a)-H (p.1,p.2+a)|+|H (p.1,p.2+a)-H p| ∂μ.prod volume)+
(∫ p, |F p-H p| ∂μ.prod volume) := integral_add (hA.add hB) hC
_ = ((∫ p, |F (p.1,p.2+a)-H (p.1,p.2+a)| ∂μ.prod volume)+
(∫ p, |H (p.1,p.2+a)-H p| ∂μ.prod volume))+
(∫ p, |F p-H p| ∂μ.prod volume) :=
congrArg (fun r => r+(∫ p, |F p-H p| ∂μ.prod volume)) (integral_add hA hB)
_ = _ := by
change (∫ p, |F (velocityShift a p)-H (velocityShift a p)| ∂μ.prod volume)+
(∫ p, |H (p.1,p.2+a)-H p| ∂μ.prod volume)+(∫ p, |F p-H p| ∂μ.prod volume) = _
rw [hs]
ring
theorem velocityTranslation_of_uniform_L1_approximation
{F : ℕ → X × Velocity → ℝ} (hF : ∀ n, Integrable (F n) (μ.prod volume))
{a : ℕ → Velocity}
(happrox : ∀ ε : ℝ, 0 < ε → ∃ H : ℕ → X × Velocity → ℝ,
(∀ n, Integrable (H n) (μ.prod volume)) ∧
(∀ n, (∫ p, |F n p-H n p| ∂μ.prod volume) ≤ ε) ∧
Tendsto (fun n => ∫ p, |H n (p.1,p.2+a n)-H n p| ∂μ.prod volume) atTop (𝓝 0)) :
Tendsto (fun n => ∫ p, |F n (p.1,p.2+a n)-F n p| ∂μ.prod volume) atTop (𝓝 0) := by
apply Metric.tendsto_atTop.mpr
intro ε hε
obtain ⟨H,hHi,hHe,hHt⟩ := happrox (ε/4) (by positivity)
obtain ⟨N,hN⟩ := Metric.tendsto_atTop.mp hHt (ε/2) (by positivity)
refine ⟨N,fun n hn => ?_⟩
have hnn := hN n hn
rw [Real.dist_eq,sub_zero,abs_of_nonneg (integral_nonneg (fun p => abs_nonneg _))] at hnn ⊢
have hc := velocityTranslation_compare (hF n) (hHi n) (a n)
linarith [hHe n]
end BoltzmannNonuniqueness.CollisionCompactness.DoubleGain
end
end
end OAI
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