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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 | import OAI.MathematicalPhysics.Boltzmann.Branching.ProductKineticCutoff
import OAI.MathematicalPhysics.Boltzmann.Branching.WeightedJetStability
import OAI.MathematicalPhysics.Boltzmann.Compactness.LocalMomentIntegrability
import OAI.MathematicalPhysics.Boltzmann.Compactness.PairPatchIdentity
namespace OAI
section
/-! # Identification of the actual cap-family weak pair limit -/
noncomputable section
open MeasureTheory Set
open BoltzmannNonuniqueness.TransportCutoff
namespace BoltzmannNonuniqueness.JetCopyDatum
/-- The velocity-averaging step identifies the genuine nonlinear pair product. -/
theorem TightCapFamily.exists_nonlinear_weak_limit
(hcarleman : Published.GPV2009CarlemanInput) (hGLPS : Published.GLPS1988Proposition3)
(hDP : Published.DunfordPettis1940Theorem321)
{s₀ η K α γ d₀ q : ℝ} (F : TightCapFamily s₀ η K α γ d₀ q)
(hq₁ : 1 < q) (hq₂ : q < 2) :
∃ (φ : ℕ → ℕ) (g : (ℝ × Position) × Velocity → ℝ), StrictMono φ ∧
Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume)
(fun n => torusPhysical q (F.approximation (φ n)).solution.value) g ∧
Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume)
(fun n => CollisionCompactness.pairProduct
(torusPhysical q (F.approximation (φ n)).solution.value))
(CollisionCompactness.pairProduct g) ∧
(∀ᵐ z ∂(AnnularEnvelope.timePositionVolume F.time).prod volume, 0 ≤ g z) := by
obtain ⟨ψ,hψ,hψc,hdet⟩ := Compactness.exists_countable_smooth_determining_tests
choose β hβ hβc _hβn hβtest using fun i => Compactness.exists_velocity_cutoff (hψc i)
obtain ⟨χ,hχ,hχc,hχs,hcover⟩ := Compactness.exists_spacetime_cutoff_cover
TorusKineticChart.interiorRegion_open TorusKineticChart.interiorRegion_nonempty
let cuts : ℕ → Published.KineticPhase → ℝ := fun n =>
Compactness.productCutoff (χ (Nat.unpair n).1) (β (Nat.unpair n).2)
let tests : ℕ → Velocity → ℝ := fun n => ψ (Nat.unpair n).2
have hcuts (n) : ContDiff ℝ (⊤ : ℕ∞) (cuts n) :=
Compactness.productCutoff_smooth (hχ _) (hβ _)
have hcutsc (n) : HasCompactSupport (cuts n) :=
Compactness.productCutoff_compact (hχc _) (hβc _)
have hcutss (n) : tsupport (cuts n) ⊆ {z | z.1.2 ∈ TorusChart.cube} :=
Compactness.productCutoff_in_cell _ (hχs _)
have hcuts0 (n) (p : Velocity × Velocity) : cuts n (TransportCutoff.characteristic p 0) = 0 :=
Compactness.productCutoff_initial_zero _ (hχs _) p
obtain ⟨φ,g,W,hφ,hg,hW,hmom,hgn,_hWn⟩ := F.exists_factorized_pair_moments
hcarleman hGLPS hDP hq₁ hq₂ cuts hcuts hcutsc hcutss hcuts0 tests
(fun n => hψ _) (fun n => hψc _)
have hw (s i : ℕ) : averageWeight (cuts (Nat.pair s i)) (tests (Nat.pair s i)) =
fun z => χ s z.1*ψ i z.2 := by
dsimp only [cuts,tests]
rw [Nat.unpair_pair]
exact Compactness.productCutoff_preserves_test (χ s) (hβtest i)
have hgw := TorusKineticChart.weak_density_transfer
(fun n => (F.approximation (φ n)).solution.measurable_value) hg
have hWw := TorusKineticChart.weak_pair_transfer
(fun n => (F.approximation (φ n)).solution.measurable_value) hW
have hscaled (s i j : ℕ) : ∀ᵐ x ∂TorusKineticChart.baseVolume F.time,
(∫ y : Velocity × Velocity, ((χ s x*ψ i y.1)*(χ s x*ψ j y.2))*TorusKineticChart.liftPair W (x,y)) =
(∫ v, (χ s x*ψ i v)*TorusKineticChart.liftDensity g (x,v))*
(∫ w, (χ s x*ψ j w)*TorusKineticChart.liftDensity g (x,w)) := by
apply Compactness.ae_of_all_compact_restrict
intro S hS
have ht := hmom (Nat.pair s i) (Nat.pair s j) S hS
simp only [hw s i,hw s j] at ht
filter_upwards [ht] with x hx
exact hx
have hpatch (s) : ∀ᵐ x ∂TorusKineticChart.baseVolume F.time, χ s x ≠ 0 →
(fun y => TorusKineticChart.liftPair W (x,y)) =ᵐ[volume]
(fun y : Velocity × Velocity => TorusKineticChart.liftDensity g (x,y.1)*
TorusKineticChart.liftDensity g (x,y.2)) := by
apply Compactness.pair_identity_on_patch hψ hψc hdet hWw.1 hgw.1 hWw.2.1 hgw.2.1 (χ s)
exact fun i j => (hscaled s i j).mono (fun _ h _ => h)
have hfib : ∀ᵐ x ∂TorusKineticChart.baseVolume F.time,
(fun y => TorusKineticChart.liftPair W (x,y)) =ᵐ[volume]
(fun y : Velocity × Velocity => TorusKineticChart.liftDensity g (x,y.1)*
TorusKineticChart.liftDensity g (x,y.2)) := by
filter_upwards [ae_all_iff.mpr hpatch,TorusKineticChart.interiorRegion_ae F.time] with x hx hxin
obtain ⟨s,hs⟩ := hcover x hxin
exact hx s hs
have he : TorusKineticChart.liftPair W =ᵐ[TorusKineticChart.pairCellVolume F.time]
CollisionCompactness.pairProduct (TorusKineticChart.liftDensity g) := by
apply (Measure.ae_prod_iff_ae_ae ?_).mpr hfib
exact measurableSet_eq_fun hWw.1 (CollisionCompactness.pairProduct_measurable hgw.1)
have hback := (TorusKineticChart.pair_measurePreserving F.time).quasiMeasurePreserving.ae he
have hEq : W =ᵐ[(AnnularEnvelope.timePositionVolume F.time).prod volume]
CollisionCompactness.pairProduct g := by
filter_upwards [hback] with z hz
simpa only [TorusKineticChart.liftPair,TorusKineticChart.pairChart,
CollisionCompactness.pairProduct,TorusKineticChart.liftDensity,
TorusChart.projection_representative] using hz
refine ⟨φ,g,hφ,hg,?_,hgn⟩
refine ⟨CollisionCompactness.pairProduct_measurable hg.1,hW.2.1.congr hEq,?_⟩
intro θ hθ hθB
have ht := hW.2.2 θ hθ hθB
have hi : (∫ z, θ z*W z ∂(AnnularEnvelope.timePositionVolume F.time).prod volume) =
∫ z, θ z*CollisionCompactness.pairProduct g z
∂(AnnularEnvelope.timePositionVolume F.time).prod volume :=
integral_congr_ae (hEq.mono (fun z hz => congrArg (fun y => θ z*y) hz))
rw [hi] at ht
exact ht
end BoltzmannNonuniqueness.JetCopyDatum
end
end
section
/-! # The physical pair weight also converges to the genuine nonlinear product -/
noncomputable section
open MeasureTheory
namespace BoltzmannNonuniqueness.JetCopyDatum
theorem TightCapFamily.exists_weighted_nonlinear_weak_limit
(hcarleman : Published.GPV2009CarlemanInput) (hGLPS : Published.GLPS1988Proposition3)
(hDP : Published.DunfordPettis1940Theorem321)
{s₀ η K α γ d₀ q : ℝ} (F : TightCapFamily s₀ η K α γ d₀ q)
(hq₁ : 1 < q) (hq₂ : q < 2) :
∃ (φ : ℕ → ℕ) (g : (ℝ × Position) × Velocity → ℝ), StrictMono φ ∧
Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume)
(fun n => torusPhysical q (F.approximation (φ n)).solution.value) g ∧
Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume)
(fun n => CollisionCompactness.pairDensity
(torusPhysical q (F.approximation (φ n)).solution.value))
(CollisionCompactness.pairDensity g) ∧
(∀ᵐ z ∂(AnnularEnvelope.timePositionVolume F.time).prod volume, 0 ≤ g z) := by
obtain ⟨a,g,ha,hg,hraw,hgn⟩ := F.exists_nonlinear_weak_limit hcarleman hGLPS hDP hq₁ hq₂
have hm (n) := CollisionCompactness.pairDensity_measurable
(torusPhysical_measurable (F.approximation (a n)).solution.measurable_value q)
obtain ⟨b,G,hb,hG⟩ := (hDP F.time F.time_pos.le).2
(fun n => CollisionCompactness.pairDensity (torusPhysical q (F.approximation (a n)).solution.value))
hm (Compactness.uniformIntegrable_comp F.pairs_ui a) (F.pairs_tight.comp a)
let w : (ℝ × Position) × (Velocity × Velocity) → ℝ := fun z => 1+‖z.2.1‖+‖z.2.2‖
have hw : Measurable w := by dsimp [w]; fun_prop
have hw1 : ∀ z, 1 ≤ w z := by
intro z
dsimp [w]
linarith [norm_nonneg z.2.1,norm_nonneg z.2.2]
have hfun (u : (ℝ × Position) × Velocity → ℝ) :
(fun z => w z*CollisionCompactness.pairProduct u z) = CollisionCompactness.pairDensity u := by
funext z
simp only [w,CollisionCompactness.pairProduct,CollisionCompactness.pairDensity,mul_assoc]
have hraw' := hraw.comp hb.tendsto_atTop
have hG' : Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume)
(fun n z => w z*CollisionCompactness.pairProduct
(torusPhysical q (F.approximation (a (b n))).solution.value) z) G := by
have hseq : (fun n z => w z*CollisionCompactness.pairProduct
(torusPhysical q (F.approximation (a (b n))).solution.value) z) =
(fun n => CollisionCompactness.pairDensity (torusPhysical q
(F.approximation (a (b n))).solution.value)) := funext (fun n => hfun _)
rw [hseq]
exact hG
have he := Published.identify_weighted_weak_limit hraw' hG' hw hw1
have he' : G =ᵐ[(AnnularEnvelope.timePositionVolume F.time).prod volume]
CollisionCompactness.pairDensity g := by
rw [hfun] at he
exact he
exact ⟨a ∘ b,g,ha.comp hb,hg.comp hb.tendsto_atTop,
hG.congr_limit (CollisionCompactness.pairDensity_measurable hg.1) he',hgn⟩
end BoltzmannNonuniqueness.JetCopyDatum
end
end
end OAI
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