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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 | import OAI.Geometry.IsometricImmersion.Darboux.QSpatialSlabData
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped ContDiff Topology Interval ENNReal NNReal
namespace SmoothLocal.HighEquation
open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE SmoothLocal.Hyperbolic
theorem exists_Q_spatial_induction_step
(G R Z s0 M speed xl xr T lengthFloor : ℝ) (H A : ℝ≥0)
(hG : 0 ≤ G) (hR : 0 ≤ R) (hs0 : 0 < s0) (hM : 0 ≤ M)
(hspeed : 0 ≤ speed) (hT : 0 ≤ T) (hlen : 0 < lengthFloor)
(hwidth : lengthFloor ≤ xr - xl - 2 * speed * T)
{d c : ℝ} (hd : 0 < d) (hc : 0 < c) (m : ℕ) (hm : 8 ≤ m + 3) :
∃ Hnew : ℝ≥0, H ≤ Hnew ∧
∀ (g : MetricField) (z : Coord → ℝ) (radius lo hi a b : ℝ),
SmoothPositiveOn g (coordinateRectangle radius lo hi) →
ContDiffOn ℝ ∞ z (coordinateRectangle radius lo hi) →
(∀ p ∈ coordinateRectangle radius lo hi,
(covHessian g z p).det = gaussianCurvature g p * heightEnergy g z p) →
(∀ p ∈ coordinateRectangle radius lo hi, covHessian g z p 0 0 ≠ 0) →
0 < radius → a ≤ b → b - a ≤ T →
-radius < xl → xr < radius → a ∈ Ioo lo hi → b ∈ Ioo lo hi →
QLowBounds g z (shrinkingSlab xl xr a b speed) s0 M speed →
(∀ p ∈ shrinkingSlab xl xr a b speed, |p 0| ≤ R ∧ |p 1| ≤ R) →
(∀ i j, CoordinateBound (fun p => g p i j) (shrinkingSlab xl xr a b speed) (m + 5) G) →
CoordinateBound z (shrinkingSlab xl xr a b speed) 8 Z →
(∀ p ∈ shrinkingSlab xl xr a b speed, d ≤ |(g p).det|) →
(∀ p ∈ shrinkingSlab xl xr a b speed, c ≤ |covHessian g z p 0 0|) →
(∀ ds : List (Fin 2), ds.length ≤ m + 4 → ∀ x ∈ Icc xl xr,
|iteratedCoordPartial ds z (coordinatePoint x a)| ≤ (A : ℝ)) →
(∀ t ∈ Icc a b, SpatialSliceL2Bound (spatialFirstJet z) t
(inwardLeft xl a speed t) (inwardRight xr a speed t) (m + 2) H) →
∀ t ∈ Icc a b, SpatialSliceL2Bound (spatialFirstJet z) t
(inwardLeft xl a speed t) (inwardRight xr a speed t) (m + 3) Hnew := by
obtain ⟨B, hSource⟩ := exists_same_slice_actual_Q_source_norm_bound
G R Z lengthFloor (xr - xl) H hG hR hlen hd hc m hm
let I : ℝ≥0 := initialCutEnergyBudget (xr - xl) M A
let C : ℝ := 8 * (M * (m + 4)) * (1 + 1 / s0)
have hC : 0 ≤ C := by dsimp [C]; positivity
let Ebound : ℝ≥0 := ⟨Real.exp (C * T) * ((I : ℝ) + T * (B : ℝ)), by positivity⟩
let Hnew : ℝ≥0 := max H (spatialEnergyNormBudget s0 Ebound)
refine ⟨Hnew, le_max_left _ _, ?_⟩
intro g z radius lo hi a b hg hz hD hxx hradius hab habT hxl hxr ha hb
hLow hcoords hgB hzB hdet hden hcut hprev t ht
let S := shrinkingSlab xl xr a b speed
have hU := coordinateRectangle_isOpen radius lo hi
have hlength : 2 * speed * (b - a) < xr - xl := by
have hh := mul_le_mul_of_nonneg_left habT (by positivity : 0 ≤ 2 * speed)
linarith
have hSU : S ⊆ coordinateRectangle radius lo hi :=
shrinkingSlab_subset_rectangle hab hspeed hxl hxr ha hb hlength
have htime (v : ℝ) (hv : v ∈ Icc a b) : v ∈ Ioo lo hi :=
⟨ha.1.trans_le hv.1, hv.2.trans_lt hb.2⟩
have hgeo (v : ℝ) (hv : v ∈ Icc a b) := inward_interval_geometry hab hspeed hxl hxr hlength hv
have hsource (v : ℝ) (hv : v ∈ Icc a b) :
movingSourceNorm xl xr a speed (actualQHighRemainder g z m) v ≤ (B : ℝ) := by
have hwlow : lengthFloor ≤ inwardRight xr a speed v - inwardLeft xl a speed v := by
unfold inwardRight inwardLeft
have hvT : v - a ≤ T := (sub_le_sub_right hv.2 a).trans habT
have hmul := mul_le_mul_of_nonneg_left hvT (by positivity : 0 ≤ 2 * speed)
linarith
have hwup : inwardRight xr a speed v - inwardLeft xl a speed v ≤ xr - xl := by
unfold inwardRight inwardLeft
have hh : 0 ≤ speed * (v - a) := mul_nonneg hspeed (sub_nonneg.mpr hv.1)
linarith
exact hSource g z (coordinateRectangle radius lo hi) S hg hU hz hSU hxx hcoords
(fun p hp => coordinateMetricBound_to_horizontalTwoJet hgB hp)
(hzB.mono (by norm_num) le_rfl) hdet hden v
(inwardLeft xl a speed v) (inwardRight xr a speed v)
(fun x hx => point_mem_shrinkingSlab hv hx) hwlow hwup (hprev v hv)
have hS := heightQCoefficient_contDiffOn hg hU hz hxx 5
have hInitial : Real.sqrt (∫ x in xl..xr,
energyDensity (heightQCoefficient g z 5) (horizontalJet z (m + 3)) (coordinatePoint x a)) ≤ (I : ℝ) := by
have hcutPoint (x : ℝ) (hx : x ∈ Icc xl xr) : coordinatePoint x a ∈ S := by
apply point_mem_shrinkingSlab (show a ∈ Icc a b from ⟨le_rfl, hab⟩)
simpa only [inwardLeft, inwardRight, sub_self, mul_zero, add_zero, sub_zero] using hx
apply initial_energy_bound_of_cut_jets hU hz hS
(fun x hx => hSU (hcutPoint x hx))
(by nlinarith [mul_nonneg hspeed (sub_nonneg.mpr hab)]) hM
(fun x hx => (le_abs_self _).trans (hLow _ (hcutPoint x hx)).2.2.2.1) (m + 3) A
simpa only [Nat.add_assoc] using hcut
have hEt : Real.sqrt (movingEnergy xl xr a speed (heightQCoefficient g z 5)
(horizontalJet z (m + 3)) t) ≤ (Ebound : ℝ) := by
have httop : shrinkingSlab xl xr a t speed ⊆ S := shrinkingSlab_mono_top ht.2
have hlenT : 2 * speed * (t - a) < xr - xl := by
have hh := mul_le_mul_of_nonneg_left ht.2 (by positivity : 0 ≤ 2 * speed)
nlinarith
have hprop := actual_Q_moving_slab_energy hg hz hD hxx m hradius ht.1 hspeed
hxl hxr ha (htime t ht) hlenT hs0 (by positivity : 0 ≤ M * (m + 4))
(fun p hp => (hLow p (httop hp)).1)
(fun p hp => by simpa only [Nat.cast_add, Nat.cast_ofNat, add_assoc,
show (3 : ℝ) + 1 = 4 by norm_num] using
hLow.high_coefficients hM (m + 3) (httop hp))
(fun p hp => (hLow p (httop hp)).2.2.2.2.2.2.2)
have hR := actualQHighRemainder_contDiffOn hg hU hz hD hxx m
have hforceCont : ContinuousOn (movingSourceNorm xl xr a speed (actualQHighRemainder g z m)) (Icc a t) :=
movingSourceNorm_continuousOn hR hradius
(fun v hv => (hgeo v ⟨hv.1, hv.2.trans ht.2⟩).1)
(fun v hv => (hgeo v ⟨hv.1, hv.2.trans ht.2⟩).2.1)
(fun v hv => htime v ⟨hv.1, hv.2.trans ht.2⟩)
have hInt : (∫ v in a..t, movingSourceNorm xl xr a speed (actualQHighRemainder g z m) v) ≤
(t - a) * (B : ℝ) := by
have hh : (∫ v in a..t, movingSourceNorm xl xr a speed (actualQHighRemainder g z m) v) ≤
∫ v in a..t, (B : ℝ) := intervalIntegral.integral_mono_on ht.1
(hforceCont.intervalIntegrable_of_Icc ht.1) intervalIntegrable_const
(fun v hv => hsource v ⟨hv.1, hv.2.trans ht.2⟩)
simpa only [intervalIntegral.integral_const, smul_eq_mul] using hh
have hduration : 0 ≤ t - a := sub_nonneg.mpr ht.1
have hdurationT : t - a ≤ T := (sub_le_sub_right ht.2 a).trans habT
have hfirst : Real.sqrt (movingEnergy xl xr a speed (heightQCoefficient g z 5)
(horizontalJet z (m + 3)) t) ≤ Real.exp (C * (t - a)) * ((I : ℝ) + (t - a) * (B : ℝ)) :=
hprop.trans (mul_le_mul_of_nonneg_left (add_le_add hInitial hInt) (Real.exp_pos _).le)
exact hfirst.trans (mul_le_mul
(Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hdurationT hC))
(add_le_add le_rfl (mul_le_mul_of_nonneg_right hdurationT B.2))
(by positivity) (Real.exp_pos _).le)
intro i n hn
by_cases hsmall : n ≤ m + 2
· exact (hprev t ht i n hsmall).trans (by exact_mod_cast (le_max_left H (spatialEnergyNormBudget s0 Ebound)))
· have he : n = m + 3 := by omega
subst n
have hn := spatialFirstJet_slice_eLpNorm_two_of_energy hU hz hS
(fun x hx => hSU (point_mem_shrinkingSlab ht hx))
(hgeo t ht).2.2.1.le hs0
(fun x hx => (hLow _ (point_mem_shrinkingSlab ht hx)).1) (m + 3) Ebound hEt i
exact hn.trans (by exact_mod_cast (le_max_right H (spatialEnergyNormBudget s0 Ebound)))
end SmoothLocal.HighEquation
end
end OAI
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