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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 initialSecondFlowDerivative (Y : ℝ → ℝ → ℝ) (p : ℝ × ℝ) : ℝ :=
deriv (fun u => deriv (fun v => Y v p.1) u) p.2
def initialSecondForcing (q : Coord → ℝ) (Y : ℝ → ℝ → ℝ) (p : ℝ × ℝ) : ℝ :=
∫ r in 0..p.1, coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y p.2 r)) *
deriv (fun v => Y v r) p.2
def initialThirdForcing (q : Coord → ℝ) (Y : ℝ → ℝ → ℝ) (p : ℝ × ℝ) : ℝ :=
∫ r in 0..p.1,
coordPartial 1 (coordPartial 1 (coordPartial 1 q)) (coordinatePoint r (Y p.2 r)) *
(deriv (fun v => Y v r) p.2) ^ 2 +
coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y p.2 r)) *
deriv (fun u => deriv (fun v => Y v r) u) 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 flowPullback_hasDerivAt_y {C : Coord → ℝ} (hC : ContDiffOn ℝ ∞ C U)
{s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
HasDerivAt (fun u => C (coordinatePoint t (Y u t)))
(coordPartial 1 C (coordinatePoint t (Y s t)) * deriv (fun u => Y u t) s) s := by
have hB := flowPullback_contDiffOn hq hU hSU hY hrange hstart hode hC
have hd : HasFDerivAt (fun p : ℝ × ℝ => C (flowCoordinateMap Y p))
(fderiv ℝ (fun p : ℝ × ℝ => C (flowCoordinateMap Y p)) (t, s)) (t, s) :=
((hB.contDiffAt ((pairRectangle_isOpen 2 (-2) 2).mem_nhds ⟨ht, hs⟩)).differentiableAt
(by simp)).hasFDerivAt
have hh := hd.comp_hasDerivAt s ((hasDerivAt_const s t).prodMk (hasDerivAt_id s))
change HasDerivAt (fun u => C (coordinatePoint t (Y u t)))
(pairPartialY (fun p => C (flowCoordinateMap Y p)) (t, s)) s at hh
rw [flowPullback_partial_y hq hU hSU hY hrange hstart hode hC ⟨ht, hs⟩] at hh
exact hh
theorem initialSecondFlowDerivative_contDiffOn :
ContDiffOn ℝ ∞ (initialSecondFlowDerivative Y) (pairRectangle 2 (-2) 2) := by
have hV := initialFlowDerivative_contDiffOn hq hU hSU hY hrange hstart hode
have hd : ContDiffOn ℝ ∞ (pairPartialY (initialFlowDerivative Y))
(pairRectangle 2 (-2) 2) :=
(hV.fderiv_of_isOpen (pairRectangle_isOpen 2 (-2) 2) (by simp)).clm_apply contDiffOn_const
apply hd.congr
intro p hp
have hder := ((hV.contDiffAt ((pairRectangle_isOpen 2 (-2) 2).mem_nhds hp)).differentiableAt
(by simp)).hasFDerivAt
exact (hder.comp_hasDerivAt p.2
((hasDerivAt_const p.2 p.1).prodMk (hasDerivAt_id p.2))).deriv
theorem initialSecondForcing_hasDerivAt {s t : ℝ}
(hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
HasDerivAt (fun u => initialSecondForcing q Y (t, u))
(initialThirdForcing q Y (t, s)) s := by
let C (p : ℝ × ℝ) := coordPartial 1 (coordPartial 1 q) (flowCoordinateMap Y p) *
initialFlowDerivative Y p
have hqq := partial_contDiffOn (partial_contDiffOn hq hU 1) hU 1
have hC : ContDiffOn ℝ ∞ C (pairRectangle 2 (-2) 2) :=
(flowPullback_contDiffOn hq hU hSU hY hrange hstart hode hqq).mul
(initialFlowDerivative_contDiffOn hq hU hSU hY hrange hstart hode)
have hi := pairPrimitive_hasDerivAt_y (hC.of_le (by simp)) ht hs
have heq : pairPrimitive (pairPartialY C) (t, s) = initialThirdForcing q Y (t, s) := by
apply intervalIntegral.integral_congr
intro r hr
have hz : (0 : ℝ) ∈ Ioo (-2) 2 := by norm_num
have hr' : r ∈ Ioo (-2 : ℝ) 2 :=
⟨(lt_min hz.1 ht.1).trans_le hr.1, hr.2.trans_lt (max_lt hz.2 ht.2)⟩
have hqder := flowPullback_hasDerivAt_y hq hU hSU hY hrange hstart hode hqq hs hr'
have hVder := (cap_flow_initial_second_hasDerivAt hq hU hSU hY hrange hstart hode hs hr').differentiableAt.hasDerivAt
have hprod := hqder.mul hVder
have hfd : HasFDerivAt C (fderiv ℝ C (r, s)) (r, s) :=
((hC.contDiffAt ((pairRectangle_isOpen 2 (-2) 2).mem_nhds ⟨hr', hs⟩)).differentiableAt
(by simp)).hasFDerivAt
have he := (hfd.comp_hasDerivAt s
((hasDerivAt_const s r).prodMk (hasDerivAt_id s))).unique hprod
change pairPartialY C (r, s) = _ at he
change pairPartialY C (r, s) = _
rw [he]
ring
rw [heq] at hi
exact hi
theorem cap_flow_initial_third_hasDerivAt {s t : ℝ}
(hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
HasDerivAt (fun u => initialSecondFlowDerivative Y (t, u))
(-initialSecondFlowDerivative Y (t, s) * initialSecondForcing q Y (t, s) -
initialFlowDerivative Y (t, s) * initialThirdForcing q Y (t, s)) s := by
have hV : HasDerivAt (fun u => initialFlowDerivative Y (t, u))
(initialSecondFlowDerivative Y (t, s)) s :=
(cap_flow_initial_second_hasDerivAt hq hU hSU hY hrange hstart hode hs ht).differentiableAt.hasDerivAt
have hI := initialSecondForcing_hasDerivAt hq hU hSU hY hrange hstart hode hs ht
have hproduct : HasDerivAt
(fun u => -initialFlowDerivative Y (t, u) * initialSecondForcing q Y (t, u))
(-initialSecondFlowDerivative Y (t, s) * initialSecondForcing q Y (t, s) +
-initialFlowDerivative Y (t, s) * initialThirdForcing q Y (t, s)) s := hV.neg.mul hI
have he : (fun u => initialSecondFlowDerivative Y (t, u)) =ᶠ[𝓝 s]
(fun u => -initialFlowDerivative Y (t, u) * initialSecondForcing q Y (t, u)) := by
filter_upwards [isOpen_Ioo.mem_nhds hs] with u hu
exact cap_flow_initial_second_formula hq hU hSU hY hrange hstart hode hu ht
convert hproduct.congr_of_eventuallyEq he using 1
first | rfl | ring
theorem cap_flow_initial_third_formula {s t : ℝ}
(hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
deriv (fun u => initialSecondFlowDerivative Y (t, u)) s =
-initialSecondFlowDerivative Y (t, s) * initialSecondForcing q Y (t, s) -
initialFlowDerivative Y (t, s) * initialThirdForcing q Y (t, s) :=
(cap_flow_initial_third_hasDerivAt hq hU hSU hY hrange hstart hode hs ht).deriv
theorem initialThirdForcing_integrable_bound {M : ℝ} (hM : 0 ≤ M)
(hMq : ∀ p ∈ modelSquare, |coordPartial 1 q p| ≤ M)
(hMqq : ∀ p ∈ modelSquare, |coordPartial 1 (coordPartial 1 q) p| ≤ M)
(hMqqq : ∀ p ∈ modelSquare, |coordPartial 1 (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 (coordPartial 1 q)) (coordinatePoint r (Y s r)) *
(deriv (fun v => Y v r) s)^2 +
coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r)) *
deriv (fun u => deriv (fun v => Y v r) u) s) volume 0 t ∧
|initialThirdForcing q Y (t, s)| ≤ (2 * M + 4 * M^2) * Real.exp (4 * M) := by
let C (p : ℝ × ℝ) :=
coordPartial 1 (coordPartial 1 (coordPartial 1 q)) (flowCoordinateMap Y p) * (initialFlowDerivative Y p)^2 +
coordPartial 1 (coordPartial 1 q) (flowCoordinateMap Y p) * initialSecondFlowDerivative Y p
have hqq := partial_contDiffOn (partial_contDiffOn hq hU 1) hU 1
have hC : ContDiffOn ℝ ∞ C (pairRectangle 2 (-2) 2) :=
((flowPullback_contDiffOn hq hU hSU hY hrange hstart hode (partial_contDiffOn hqq hU 1)).mul
((initialFlowDerivative_contDiffOn hq hU hSU hY hrange hstart hode).pow 2)).add
((flowPullback_contDiffOn hq hU hSU hY hrange hstart hode hqq).mul
(initialSecondFlowDerivative_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) :
‖C (r, s)‖ ≤ (M + 2 * M^2) * Real.exp (4 * M) := by
have hr' := hseg hr
have hp := flowCoordinateMap_mem_modelSquare (p := (r, s)) hq hU hSU hY hrange hstart hode ⟨hr', hs⟩
have hV := (cap_flow_initial_deriv_bounds hq hU hSU hY hrange hstart hode hM hMq hs hr').2
have hVpos := cap_flow_initial_deriv_pos hq hU hSU hY hrange hstart hode hs hr'
have hVsq : |(deriv (fun v => Y v r) s)^2| ≤ Real.exp (4 * M) := by
rw [abs_of_nonneg (sq_nonneg _)]
calc
_ ≤ (Real.exp (2 * M))^2 := pow_le_pow_left₀ hVpos.le hV 2
_ = Real.exp (4 * M) := by rw [pow_two, ← Real.exp_add]; congr 1; ring
have hW := cap_flow_initial_second_bound hq hU hSU hY hrange hstart hode hM hMq hMqq hs hr'
rw [Real.norm_eq_abs]
calc
|C (r, s)| ≤
|coordPartial 1 (coordPartial 1 (coordPartial 1 q)) (coordinatePoint r (Y s r))| *
|(deriv (fun v => Y v r) s)^2| +
|coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r))| *
|deriv (fun u => deriv (fun v => Y v r) u) s| := by
calc
_ ≤ |coordPartial 1 (coordPartial 1 (coordPartial 1 q)) (coordinatePoint r (Y s r)) *
(deriv (fun v => Y v r) s)^2| +
|coordPartial 1 (coordPartial 1 q) (coordinatePoint r (Y s r)) *
deriv (fun u => deriv (fun v => Y v r) u) s| := abs_add_le _ _
_ = _ := by rw [abs_mul, abs_mul]
_ ≤ M * Real.exp (4 * M) + M * (2 * M * Real.exp (4 * M)) :=
add_le_add (mul_le_mul (hMqqq _ hp) hVsq (abs_nonneg _) hM)
(mul_le_mul (hMqq _ hp) hW (abs_nonneg _) hM)
_ = (M + 2 * M^2) * Real.exp (4 * M) := by ring
have hi := intervalIntegral.norm_integral_le_of_norm_le_const
(a := 0) (b := t) (C := (M + 2 * M^2) * Real.exp (4 * M)) (f := fun r => C (r, s))
(fun r hr => hbound r (uIoc_subset_uIcc hr))
have ht2 : |t| ≤ 2 := (abs_lt.mpr ht).le
calc
_ ≤ ((M + 2 * M^2) * Real.exp (4 * M)) * |t| := by
simpa only [Real.norm_eq_abs, sub_zero, initialThirdForcing, C,
flowCoordinateMap, initialFlowDerivative, initialSecondFlowDerivative] using hi
_ ≤ ((M + 2 * M^2) * Real.exp (4 * M)) * 2 := mul_le_mul_of_nonneg_left ht2
(mul_nonneg (by positivity) (Real.exp_pos _).le)
_ = (2 * M + 4 * M^2) * Real.exp (4 * M) := by ring
theorem cap_flow_initial_third_bound {M : ℝ} (hM : 0 ≤ M)
(hMq : ∀ p ∈ modelSquare, |coordPartial 1 q p| ≤ M)
(hMqq : ∀ p ∈ modelSquare, |coordPartial 1 (coordPartial 1 q) p| ≤ M)
(hMqqq : ∀ p ∈ modelSquare, |coordPartial 1 (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 => initialSecondFlowDerivative Y (t, u)) s| ≤
(2 * M + 8 * M^2) * Real.exp (6 * M) := by
rw [cap_flow_initial_third_formula hq hU hSU hY hrange hstart hode hs ht]
have hW := cap_flow_initial_second_bound hq hU hSU hY hrange hstart hode hM hMq hMqq hs ht
have hI := (flow_initial_second_integral_bound hq hU hSU hY hrange hstart hode hM hMq hMqq hs ht).2
have hJ := (initialThirdForcing_integrable_bound hq hU hSU hY hrange hstart hode hM hMq hMqq hMqqq hs ht).2
have hV : |initialFlowDerivative Y (t, s)| ≤ Real.exp (2 * M) := by
unfold initialFlowDerivative
rw [abs_of_pos (cap_flow_initial_deriv_pos hq hU hSU hY hrange hstart hode hs ht)]
exact (cap_flow_initial_deriv_bounds hq hU hSU hY hrange hstart hode hM hMq hs ht).2
calc
_ ≤ |initialSecondFlowDerivative Y (t, s)| * |initialSecondForcing q Y (t, s)| +
|initialFlowDerivative Y (t, s)| * |initialThirdForcing q Y (t, s)| := by
simpa only [abs_mul, abs_neg] using
abs_sub (-(initialSecondFlowDerivative Y (t, s)) * initialSecondForcing q Y (t, s))
(initialFlowDerivative Y (t, s) * initialThirdForcing q Y (t, s))
_ ≤ (2 * M * Real.exp (4 * M)) * (2 * M * Real.exp (2 * M)) +
Real.exp (2 * M) * ((2 * M + 4 * M^2) * Real.exp (4 * M)) :=
add_le_add (mul_le_mul hW hI (abs_nonneg _) (by positivity))
(mul_le_mul hV hJ (abs_nonneg _) (Real.exp_pos _).le)
_ = (2 * M + 8 * M^2) * (Real.exp (2 * M) * Real.exp (4 * M)) := by ring
_ = (2 * M + 8 * M^2) * Real.exp (6 * M) := by
rw [← Real.exp_add]
congr 2
ring
end ActualFlow
end SmoothLocal.Flow
end
end OAI
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