File size: 6,478 Bytes
59d1c80 | 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 | import OAI.Geometry.IsometricImmersion.Estimates.ActualQApplicability
import OAI.Geometry.IsometricImmersion.Comparison.QApproximationMargins
import OAI.Geometry.IsometricImmersion.Comparison.ComparisonEnergyScaling
namespace OAI
noncomputable section
open Set Filter Function MeasureTheory
open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise
namespace SmoothLocal.Perturbation
open SmoothLocal.Geometry SmoothLocal.Pulse SmoothLocal.HighEquation SmoothLocal.Flow
open SmoothLocal.ODE SmoothLocal.Weighted SmoothLocal.Hyperbolic SmoothLocal.Taylor
theorem exists_actual_class_comparison_source_bound_at_radius
{gStar : MetricField} {V : Set Coord}
(hgStar : SmoothPositiveOn gStar V) (hV : IsOpen V) (hSV : modelSquare ⊆ V)
{G d kappa q0 r : ℝ} (M : ℕ) (hG : 0 ≤ G) (hd : 0 < d)
(hgStarB : ∀ i j k, k ≤ 2 → ∀ p ∈ modelSquare,
‖iteratedFDeriv ℝ k (fun q => gStar q i j) p‖ ≤ G)
(hdStar : ∀ p ∈ modelSquare, d ≤ (gStar p).det)
(hkappa : 0 < kappa) (hM : 0 < M) (hq0 : |q0| ≤ 1/20)
(hr : 0 < r) (hrhalf : r < 1/2) (hLr : boundedClassWidth kappa M*r ≤ 1/20)
(hrsmall : heightQuotientJetBound G (M : ℝ) d (1/(M : ℝ))*
(r+107*(boundedClassWidth kappa M*r)/100) ≤ 9/(100*boundedClassWidth kappa M))
(N : ℕ) (hN : 2 < N) (Cres : ℝ) (hCres : 0 ≤ Cres) :
∃ Csource : ℝ, 0 < Csource ∧
∀ delta : ℝ, 0 < delta → delta ≤ 1/2 →
∀ᶠ tau : ℕ in atTop,
1 ≤ tau ∧
∀ (g0 : MetricField) (eta : metricPatchSet g0 kappa) (U W : Set Coord)
(z : Coord → ℝ) (Y : ℝ → ℝ → ℝ),
SmoothPositiveOn (perturbedMetric g0 eta.val) U → IsOpen U →
CapInductionHeight (perturbedMetric g0 eta.val) U (M : ℝ) (1/(M : ℝ)) (1/(M : ℝ)) z →
CapInductionFlow (perturbedMetric g0 eta.val) U G (M : ℝ) d (1/(M : ℝ)) (1/(M : ℝ)) kappa z Y W →
BoundedAdmissibleHeight (perturbedMetric g0 eta.val) M z →
(∀ i j k, k ≤ 4 → ∀ p ∈ modelSquare,
‖iteratedFDeriv ℝ k (fun q => perturbedMetric g0 eta.val q i j) p‖ ≤ G) →
(∀ p ∈ modelSquare, d ≤ |(perturbedMetric g0 eta.val p).det|) →
|hessianQuotient (perturbedMetric g0 eta.val) z 0-q0| ≤ 1/(100*boundedClassWidth kappa M) →
(∀ i j : Fin 2, ∀ k ≤ tau, ∀ p ∈ modelSquare,
‖iteratedFDeriv ℝ k (fun q => perturbedMetric g0 eta.val q i j-
testMetric gStar q0 (boundedClassWidth kappa M*r/16) N delta (tau : ℝ) q i j) p‖ ≤
metricApproximationAccuracy tau) →
∀ P : Coord → ℝ,
(∀ p ∈ pulseStrip (boundedClassWidth kappa M*r/2) delta (tau : ℝ),
|qResidual (metricInShearCoordinates gStar q0) P p| ≤ Cres/(tau : ℝ)^N) →
∀ p ∈ pulseStrip (boundedClassWidth kappa M*r/2) delta (tau : ℝ),
|actualComparisonSource gStar (perturbedMetric g0 eta.val) z q0 P p| ≤
Csource*(tau : ℝ)^2/(tau : ℝ)^N := by
let L := boundedClassWidth kappa M
let a := L*r/16
let c := (boundedClassSpeed kappa M)^2/(4*(M : ℝ))
let BQ := max (8*G) (boundedClassShearedStateBudget q0 M)
have ha : 0 < a := div_pos (mul_pos (boundedClassWidth_pos kappa M) hr) (by norm_num)
have hc : 0 < c := by
have hh := boundedClassShearHxxFloor_pos hkappa hM
have he : c=((boundedClassSpeed kappa M)^2/(2*(M : ℝ)))/2 := by dsimp only [c]; ring
rw [he]
exact half_pos hh
obtain ⟨_,A,D,dcompare,_,hA,hD,hdcompare,happlicable⟩ :=
exists_actual_Q_applicability_at_radius hgStar hV hSV M hG hd hgStarB hdStar
hkappa hM hq0 hr hrhalf hLr hrsmall N hN
obtain ⟨H,Cfirst,Ctest,Ccompare,_,hCf,_,hCc,hsource⟩ :=
exists_actual_comparison_source_pointwise_bound (8*G) (G+1) BQ D A Cres
hd hdcompare hc hD hA ha N (by omega)
let Csource := 2*A*pulseLeadingBound a ha+1
have hCs : 0 < Csource := by
have hpulse := pulseLeadingBound_nonneg ha
dsimp only [Csource]
positivity
refine ⟨Csource,hCs,?_⟩
intro delta hdelt hdhalf
obtain ⟨T,_,hsourceT⟩ := hsource delta hdelt
let Q := 8+scalarPulseFirstJetBound a ha delta
have hlarge : ∀ᶠ tau : ℕ in atTop,
T ≤ (tau : ℝ) ∧
comparisonLowerSourceCoefficient A Cfirst Ctest Ccompare Cres 1 Q a ha delta ≤ (tau : ℝ) := by
apply (tendsto_natCast_atTop_atTop : Tendsto (fun tau : ℕ => (tau : ℝ)) atTop atTop).eventually
(p := fun t : ℝ => T ≤ t ∧
comparisonLowerSourceCoefficient A Cfirst Ctest Ccompare Cres 1 Q a ha delta ≤ t)
filter_upwards [eventually_ge_atTop T,
eventually_ge_atTop (comparisonLowerSourceCoefficient A Cfirst Ctest Ccompare Cres 1 Q a ha delta)]
with t ht hscale
exact ⟨ht,hscale⟩
filter_upwards [happlicable delta hdelt hdhalf,
actual_Q_approximation_eventual_margins (G+1) BQ H hd hdcompare hc ha N (by omega) delta,hlarge]
with tau hactual hmarg hlargeTau
rcases hmarg with ⟨ht2,_,_,heps1,hepsSmall,hepsScale,_,hepsQ1,hepsQSmall,hxxSmall⟩
have ht1 : 1 ≤ tau := by omega
have htR : (1 : ℝ) ≤ tau := by exact_mod_cast ht1
refine ⟨ht1,?_⟩
intro g0 eta U W z Y hg hU hh hf hclass hgB hdet hcenter happ P hres p hp
let g := perturbedMetric g0 eta.val
have hSU : modelSquare ⊆ U := hf.squareSubset.trans hf.domainSubset
have hOU : modelOpenSquare ⊆ U := modelOpenSquare_subset.trans hSU
have hOV : modelOpenSquare ⊆ V := modelOpenSquare_subset.trans hSV
have hgO : SmoothPositiveOn g modelOpenSquare :=
⟨fun i j => (hg.1 i j).mono hOU,fun p hp => hg.2 p (hOU hp)⟩
have hgStarO : SmoothPositiveOn gStar modelOpenSquare :=
⟨fun i j => (hgStar.1 i j).mono hOV,fun p hp => hgStar.2 p (hOV hp)⟩
have hzO := hh.smooth.mono hOU
have hpoint := hactual.2.2.2 g0 eta U W z Y hg hU hh hf hclass hgB hdet hcenter happ
have hb := hsourceT (tau : ℝ) hlargeTau.1 gStar g q0 modelOpenSquare hgStarO hactual.2.2.1 hgO
modelOpenSquare_isOpen z P hzO (metricApproximationAccuracy tau)
(qPulseFirstJetBudget a ha N delta (tau : ℝ))
heps1 hepsSmall hepsQ1 hepsQSmall hxxSmall p (hpoint p hp).1 (hres p hp)
apply hb.trans
apply comparisonSourceBudget_le_leading (E := 1) (Q := Q) ha N hA hCf hCc hCres htR
· simpa only [one_mul] using hepsScale
· exact le_rfl
· exact hlargeTau.2
end SmoothLocal.Perturbation
end
end OAI
|