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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 | import OAI.Geometry.IsometricImmersion.Flows.FlowBounds
namespace OAI
noncomputable section
open Set Filter MeasureTheory Function
open scoped ContDiff Topology Interval
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted
def initialFlowDerivative (Y : ℝ → ℝ → ℝ) (p : ℝ × ℝ) : ℝ :=
deriv (fun s => Y s p.1) p.2
section ActualFlow
variable {q : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ}
variable (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
variable (hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2))
variable (hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
Y s t ∈ Icc (-3 : ℝ) 3)
variable (hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s)
variable (hode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t)
include hq hU hSU hY hrange hstart hode
theorem flowCoordinateMap_mem_modelSquare {p : ℝ × ℝ}
(hp : p ∈ pairRectangle 2 (-2) 2) : flowCoordinateMap Y p ∈ modelSquare := by
have hF := cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode
have hbound : Y p.2 p.1 ∈ Icc (-3 : ℝ) 3 := by
apply isClosed_Icc.mem_of_tendsto
(hF.contDiffAt ((pairRectangle_isOpen 2 (-2) 2).mem_nhds hp)).continuousAt
filter_upwards [(pairRectangle_isOpen 2 (-2) 2).mem_nhds hp] with point hpoint
exact hrange point.2 ⟨hpoint.2.1.le, hpoint.2.2.le⟩
point.1 ⟨hpoint.1.1.le, hpoint.1.2.le⟩
exact coordinatePoint_mem_modelSquare ⟨by linarith [hp.1.1], by linarith [hp.1.2]⟩
hbound
theorem flowPullback_contDiffOn {C : Coord → ℝ} (hC : ContDiffOn ℝ ∞ C U) :
ContDiffOn ℝ ∞ (fun p => C (flowCoordinateMap Y p)) (pairRectangle 2 (-2) 2) := by
have hF := cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode
have hT : ContDiffOn ℝ ∞ (flowCoordinateMap Y) (pairRectangle 2 (-2) 2) :=
(contDiffOn_fst.smul contDiffOn_const).add (hF.smul contDiffOn_const)
exact hC.comp hT (fun p hp => hSU
(flowCoordinateMap_mem_modelSquare hq hU hSU hY hrange hstart hode hp))
theorem initialFlowDerivative_eq_pairPartialY {p : ℝ × ℝ}
(hp : p ∈ pairRectangle 2 (-2) 2) :
initialFlowDerivative Y p = pairPartialY (fun v : ℝ × ℝ => Y v.2 v.1) p := by
have hF := cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode
have hd := ((hF.contDiffAt ((pairRectangle_isOpen 2 (-2) 2).mem_nhds hp)).differentiableAt
(by simp)).hasFDerivAt
exact (hd.comp_hasDerivAt p.2 ((hasDerivAt_const p.2 p.1).prodMk (hasDerivAt_id p.2))).deriv
theorem initialFlowDerivative_contDiffOn :
ContDiffOn ℝ ∞ (initialFlowDerivative Y) (pairRectangle 2 (-2) 2) := by
have hF := cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode
have hd : ContDiffOn ℝ ∞ (pairPartialY (fun v : ℝ × ℝ => Y v.2 v.1))
(pairRectangle 2 (-2) 2) :=
(hF.fderiv_of_isOpen (pairRectangle_isOpen 2 (-2) 2) (by simp)).clm_apply contDiffOn_const
exact hd.congr (fun p hp => initialFlowDerivative_eq_pairPartialY
hq hU hSU hY hrange hstart hode hp)
theorem flowPullback_partial_y {C : Coord → ℝ} (hC : ContDiffOn ℝ ∞ C U)
{p : ℝ × ℝ} (hp : p ∈ pairRectangle 2 (-2) 2) :
pairPartialY (fun v => C (flowCoordinateMap Y v)) p =
coordPartial 1 C (flowCoordinateMap Y p) * initialFlowDerivative Y p := by
have hsY : HasDerivAt (fun s => Y s p.1) (initialFlowDerivative Y p) p.2 :=
(cap_flow_hasDerivAt_initial hq hU hSU hY hrange hstart hode hp.2 hp.1).differentiableAt.hasDerivAt
have hsT : HasDerivAt (fun s => flowCoordinateMap Y (p.1, s))
(initialFlowDerivative Y p • Pi.single 1 (1 : ℝ)) p.2 := by
convert
((hasDerivAt_const p.2 p.1).smul_const (Pi.single 0 (1 : ℝ) : Coord)).add
(hsY.smul_const (Pi.single 1 (1 : ℝ) : Coord)) using 1 <;>
first | rfl | simp
have hmem := hSU (flowCoordinateMap_mem_modelSquare hq hU hSU hY hrange hstart hode hp)
have hdC := ((hC.contDiffAt (hU.mem_nhds hmem)).differentiableAt (by simp)).hasFDerivAt
have hchain : HasDerivAt (fun s => C (flowCoordinateMap Y (p.1, s)))
(coordPartial 1 C (flowCoordinateMap Y p) * initialFlowDerivative Y p) p.2 := by
convert hdC.comp_hasDerivAt p.2 hsT using 1 <;>
first | rfl | simp only [map_smul, smul_eq_mul, coordPartial, mul_comm]
have hB := flowPullback_contDiffOn hq hU hSU hY hrange hstart hode hC
have hdB : HasFDerivAt (fun v : ℝ × ℝ => C (flowCoordinateMap Y v))
(fderiv ℝ (fun v : ℝ × ℝ => C (flowCoordinateMap Y v)) p) p :=
((hB.contDiffAt ((pairRectangle_isOpen 2 (-2) 2).mem_nhds hp)).differentiableAt
(by simp)).hasFDerivAt
have hpair : HasDerivAt (fun s => C (flowCoordinateMap Y (p.1, s)))
(pairPartialY (fun v => C (flowCoordinateMap Y v)) p) p.2 := by
convert hdB.comp_hasDerivAt p.2
((hasDerivAt_const p.2 p.1).prodMk (hasDerivAt_id p.2)) using 1 <;> rfl
exact hpair.unique hchain
theorem cap_flow_initial_second_hasDerivAt {s t : ℝ}
(hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
HasDerivAt (fun u => deriv (fun v => Y v t) u)
(-deriv (fun v => Y v t) s *
(∫ r in 0..t, coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r)) *
deriv (fun v => Y v r) s)) s := by
let B (p : ℝ × ℝ) := coordPartial 1 q (flowCoordinateMap Y p)
have hB : ContDiffOn ℝ ∞ B (pairRectangle 2 (-2) 2) :=
flowPullback_contDiffOn hq hU hSU hY hrange hstart hode (partial_contDiffOn hq hU 1)
have hI := pairPrimitive_hasDerivAt_y (hB.of_le (by simp)) ht hs
have hdiag : pairPrimitive (pairPartialY B) (t, s) =
∫ r in 0..t, coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r)) *
deriv (fun v => Y v r) s := by
apply intervalIntegral.integral_congr
intro r hr
have hr' : r ∈ Ioo (-2 : ℝ) 2 := by
have hz : (0 : ℝ) ∈ Ioo (-2 : ℝ) 2 := by norm_num
exact ⟨(lt_min hz.1 ht.1).trans_le hr.1, hr.2.trans_lt (max_lt hz.2 ht.2)⟩
exact flowPullback_partial_y hq hU hSU hY hrange hstart hode
(partial_contDiffOn hq hU 1) ⟨hr', hs⟩
have hV : HasDerivAt (fun u => Real.exp (-pairPrimitive B (t, u)))
(Real.exp (-pairPrimitive B (t, s)) * -pairPrimitive (pairPartialY B) (t, s)) s := hI.neg.exp
have he : (fun u => deriv (fun v => Y v t) u) =ᶠ[𝓝 s]
(fun u => Real.exp (-pairPrimitive B (t, u))) := by
filter_upwards [isOpen_Ioo.mem_nhds hs] with u hu
exact (cap_flow_hasDerivAt_initial hq hU hSU hY hrange hstart hode hu ht).deriv
have hVs : Real.exp (-pairPrimitive B (t, s)) = deriv (fun v => Y v t) s :=
(cap_flow_hasDerivAt_initial hq hU hSU hY hrange hstart hode hs ht).deriv.symm
rw [hdiag, hVs] at hV
convert hV.congr_of_eventuallyEq he using 1
first | rfl | ring
theorem cap_flow_initial_second_formula {s t : ℝ}
(hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
deriv (fun u => deriv (fun v => Y v t) u) s =
-deriv (fun v => Y v t) s *
(∫ r in 0..t, coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r)) *
deriv (fun v => Y v r) s) :=
(cap_flow_initial_second_hasDerivAt hq hU hSU hY hrange hstart hode hs ht).deriv
theorem flow_initial_second_integral_bound {M : ℝ} (hM : 0 ≤ M)
(hMq : ∀ p ∈ modelSquare, |coordPartial 1 q p| ≤ M)
(hMqq : ∀ p ∈ modelSquare, |coordPartial 1 (coordPartial 1 q) p| ≤ M)
{s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
IntervalIntegrable (fun r => coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r)) *
deriv (fun v => Y v r) s) volume 0 t ∧
|∫ r in 0..t, coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r)) *
deriv (fun v => Y v r) s| ≤ 2 * M * Real.exp (2 * M) := by
let C (p : ℝ × ℝ) := coordPartial 1 (coordPartial 1 q) (flowCoordinateMap Y p) * initialFlowDerivative Y p
have hC : ContDiffOn ℝ ∞ C (pairRectangle 2 (-2) 2) :=
(flowPullback_contDiffOn hq hU hSU hY hrange hstart hode
(partial_contDiffOn (partial_contDiffOn hq hU 1) hU 1)).mul
(initialFlowDerivative_contDiffOn hq hU hSU hY hrange hstart hode)
have hc : ContinuousOn (fun r => C (r, s)) (Ioo (-2 : ℝ) 2) :=
hC.continuousOn.comp (f := fun r => (r, s)) (by fun_prop) (fun r hr => ⟨hr, hs⟩)
have hseg : uIcc (0 : ℝ) t ⊆ Ioo (-2 : ℝ) 2 := by
intro r hr
have hz : (0 : ℝ) ∈ Ioo (-2 : ℝ) 2 := by norm_num
exact ⟨(lt_min hz.1 ht.1).trans_le hr.1, hr.2.trans_lt (max_lt hz.2 ht.2)⟩
refine ⟨(hc.mono hseg).intervalIntegrable, ?_⟩
have hbound (r : ℝ) (hr : r ∈ uIcc (0 : ℝ) t) :
‖coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r)) * deriv (fun v => Y v r) s‖ ≤
M * Real.exp (2 * M) := by
have hr' := hseg hr
have hys := cap_flow_initial_deriv_bounds hq hU hSU hY hrange hstart hode hM hMq hs hr'
have hpos := cap_flow_initial_deriv_pos hq hU hSU hY hrange hstart hode hs hr'
rw [Real.norm_eq_abs, abs_mul, abs_of_pos hpos]
exact mul_le_mul (hMqq _ (flowCoordinateMap_mem_modelSquare (p := (r, s)) hq hU hSU hY hrange hstart hode
⟨hr', hs⟩)) hys.2 hpos.le hM
have hi := intervalIntegral.norm_integral_le_of_norm_le_const
(a := 0) (b := t) (C := M * Real.exp (2 * M))
(fun r hr => hbound r (uIoc_subset_uIcc hr))
have ht2 : |t| ≤ 2 := (abs_lt.mpr ht).le
calc
_ ≤ (M * Real.exp (2 * M)) * |t| := by simpa only [Real.norm_eq_abs, sub_zero] using hi
_ ≤ (M * Real.exp (2 * M)) * 2 := mul_le_mul_of_nonneg_left ht2
(mul_nonneg hM (Real.exp_pos _).le)
_ = 2 * M * Real.exp (2 * M) := by ring
theorem cap_flow_initial_second_bound {M : ℝ} (hM : 0 ≤ M)
(hMq : ∀ p ∈ modelSquare, |coordPartial 1 q p| ≤ M)
(hMqq : ∀ p ∈ modelSquare, |coordPartial 1 (coordPartial 1 q) p| ≤ M)
{s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
|deriv (fun u => deriv (fun v => Y v t) u) s| ≤ 2 * M * Real.exp (4 * M) := by
rw [cap_flow_initial_second_formula hq hU hSU hY hrange hstart hode hs ht, abs_mul,
abs_neg, abs_of_pos (cap_flow_initial_deriv_pos hq hU hSU hY hrange hstart hode hs ht)]
calc
_ ≤ Real.exp (2 * M) * (2 * M * Real.exp (2 * M)) :=
mul_le_mul (cap_flow_initial_deriv_bounds hq hU hSU hY hrange hstart hode hM hMq hs ht).2
(flow_initial_second_integral_bound hq hU hSU hY hrange hstart hode hM hMq hMqq hs ht).2
(abs_nonneg _) (Real.exp_pos _).le
_ = (2 * M) * (Real.exp (2 * M) * Real.exp (2 * M)) := by ring
_ = 2 * M * Real.exp (4 * M) := by
rw [← Real.exp_add]
congr 2
ring
end ActualFlow
end SmoothLocal.Flow
end
end OAI
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