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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 | import OAI.Geometry.IsometricImmersion.Flows.PicardFamily
namespace OAI
noncomputable section
open Set Metric Filter Function
open scoped ContDiff Topology NNReal
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted
theorem exists_cap_flow {q : Coord → ℝ} {U : Set Coord}
(hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
(hsmall : ∀ p ∈ modelSquare, |q p| ≤ (1 : ℝ) / 100) :
∃ Y : ℝ → ℝ → ℝ,
ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2) ∧
(∀ s ∈ Icc (-2 : ℝ) 2, ContDiffOn ℝ ∞ (Y s) (Icc (-2 : ℝ) 2)) ∧
(∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s) ∧
(∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t) ∧
(∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
|Y s t - s| ≤ (1 : ℝ) / 100 * |t|) ∧
(∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |Y s t - s| ≤ (1 : ℝ) / 50) ∧
(∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
|Y s t| ≤ (101 : ℝ) / 50 ∧ coordinatePoint t (Y s t) ∈ modelSquare) ∧
(∃ J : ℝ≥0, ∀ t, LipschitzOnWith J (fun s => Y s t) (Icc (-2 : ℝ) 2)) := by
obtain ⟨L, hPL⟩ := exists_cap_picard hq hU hSU hsmall
obtain ⟨Y, hY, hconf, ⟨J, hJ⟩, hcont, htime⟩ :=
exists_confined_continuous_picard_family hPL
have hball (s : ℝ) (hs : s ∈ Icc (-2 : ℝ) 2) :
s ∈ closedBall (0 : ℝ) (↑(2 : ℝ≥0)) := by
simpa only [mem_closedBall, Real.dist_eq, sub_zero, NNReal.coe_ofNat] using abs_le.mpr hs
have hz : (0 : ℝ) ∈ Icc (-2 : ℝ) 2 := by norm_num
have hstart (s : ℝ) (hs : s ∈ Icc (-2 : ℝ) 2) : Y s 0 = s :=
(hY s (hball s hs)).1
have hderiv (s : ℝ) (hs : s ∈ Icc (-2 : ℝ) 2) (t : ℝ)
(ht : t ∈ Icc (-2 : ℝ) 2) :
HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t :=
(hY s (hball s hs)).2 t ht
have hdispl (s : ℝ) (hs : s ∈ Icc (-2 : ℝ) 2) (t : ℝ)
(ht : t ∈ Icc (-2 : ℝ) 2) : |Y s t - s| ≤ (1 : ℝ) / 100 * |t| := by
have hh := (htime s (hball s hs)).dist_le_mul t ht 0 hz
rw [hstart s hs] at hh
simpa only [Real.dist_eq, sub_zero, NNReal.coe_div, NNReal.coe_one, NNReal.coe_ofNat] using hh
have hdispl' (s : ℝ) (hs : s ∈ Icc (-2 : ℝ) 2) (t : ℝ)
(ht : t ∈ Icc (-2 : ℝ) 2) : |Y s t - s| ≤ (1 : ℝ) / 50 := by
have hd := hdispl s hs t ht
have habs := abs_le.mpr ht
nlinarith
have hfield : ContDiffOn ℝ ∞ (uncurry (flowField q))
(Icc (-2 : ℝ) 2 ×ˢ closedBall (0 : ℝ) 3) := by
apply hq.neg.comp (by unfold coordinatePoint; fun_prop)
intro p hp
have hy : |p.2| ≤ 3 := by
simpa only [mem_closedBall, Real.dist_eq, sub_zero] using hp.2
exact hSU (coordinatePoint_mem_modelSquare
⟨by linarith [hp.1.1], by linarith [hp.1.2]⟩ (abs_le.mp hy))
refine ⟨Y, hcont.mono (fun p hp => ⟨hball p.1 hp.1, hp.2⟩), ?_, hstart,
hderiv, hdispl, hdispl', ?_, ⟨J, fun t => (hJ t).mono (fun s hs => hball s hs)⟩⟩
· intro s hs
exact ODE.contDiffOn_enat_Icc_of_hasDerivWithinAt hfield (hderiv s hs)
(fun t _ => hconf s (hball s hs) t)
· intro s hs t ht
have hys : |Y s t| ≤ (101 : ℝ) / 50 := by
have hsum : |Y s t| ≤ |Y s t - s| + |s| := by
simpa only [sub_add_cancel] using abs_add_le (Y s t - s) s
have hd := hdispl' s hs t ht
have hs2 := abs_le.mpr hs
linarith
refine ⟨hys, coordinatePoint_mem_modelSquare
⟨by linarith [ht.1], by linarith [ht.2]⟩ ?_⟩
exact abs_le.mp (hys.trans (by norm_num))
end SmoothLocal.Flow
end
end OAI
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