| import OAI.Geometry.IsometricImmersion.Flows.FlowVariation |
| import OAI.Geometry.IsometricImmersion.Calculus.FinitePrimitive |
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|
| namespace OAI |
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| noncomputable section |
| open Set Filter MeasureTheory Function |
| open scoped ContDiff Topology Interval |
|
|
| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted |
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|
| def flowCoordinateMap (Y : ℝ → ℝ → ℝ) (p : ℝ × ℝ) : Coord := |
| coordinatePoint p.1 (Y p.2 p.1) |
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| variable {q : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ} |
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| theorem cap_flow_joint_contDiffOn |
| (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| (hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2)) |
| (hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |
| Y s t ∈ Icc (-3 : ℝ) 3) |
| (hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s) |
| (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) : |
| ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2) := by |
| let D := pairRectangle 2 (-2) 2 |
| let F (p : ℝ × ℝ) := Y p.2 p.1 |
| let B (p : ℝ × ℝ) := coordPartial 1 q (flowCoordinateMap Y p) |
| have hmem : MapsTo (flowCoordinateMap Y) D U := by |
| intro p hp |
| exact hSU (coordinatePoint_mem_modelSquare |
| ⟨by linarith [hp.1.1], by linarith [hp.1.2]⟩ |
| (hrange p.2 ⟨hp.2.1.le, hp.2.2.le⟩ p.1 ⟨hp.1.1.le, hp.1.2.le⟩)) |
| have hdx (p : ℝ × ℝ) (hp : p ∈ D) : |
| HasDerivAt (fun x => F (x, p.2)) (-q (flowCoordinateMap Y p)) p.1 := by |
| exact (hode p.2 ⟨hp.2.1.le, hp.2.2.le⟩ p.1 ⟨hp.1.1.le, hp.1.2.le⟩).hasDerivAt |
| (Icc_mem_nhds hp.1.1 hp.1.2) |
| have hdy (p : ℝ × ℝ) (hp : p ∈ D) : |
| HasDerivAt (fun y => F (p.1, y)) (Real.exp (-pairPrimitive B p)) p.2 := by |
| exact cap_flow_hasDerivAt_initial hq hU hSU hY hrange hstart hode hp.2 hp.1 |
| have hnat : ∀ n : ℕ, ContDiffOn ℝ n F D := by |
| intro n |
| induction n with |
| | zero => |
| have hc : ContinuousOn F D := hY.comp continuous_swap.continuousOn |
| (fun p hp => ⟨⟨hp.2.1.le, hp.2.2.le⟩, ⟨hp.1.1.le, hp.1.2.le⟩⟩) |
| simpa only [Nat.cast_zero, contDiffOn_zero] using hc |
| | succ n hn => |
| have horder : (n : ℕ∞ω) ≤ ∞ := WithTop.coe_le_coe.mpr le_top |
| have hmap : ContDiffOn ℝ n (flowCoordinateMap Y) D := by |
| exact (contDiffOn_fst.smul contDiffOn_const).add (hn.smul contDiffOn_const) |
| have hfx : ContDiffOn ℝ n (fun p => -q (flowCoordinateMap Y p)) D := |
| ((hq.of_le horder).comp hmap hmem).neg |
| have hB : ContDiffOn ℝ n B D := |
| ((partial_contDiffOn hq hU 1).of_le horder).comp hmap hmem |
| have hfy : ContDiffOn ℝ n (fun p => Real.exp (-pairPrimitive B p)) D := |
| ((pairPrimitive_contDiffOn_nat n hB).neg).exp |
| exact contDiffOn_succ_of_continuous_partials (pairRectangle_isOpen 2 (-2) 2) n |
| hfx hfy hdx hdy |
| exact contDiffOn_infty.mpr hnat |
|
|
| theorem exists_smooth_cap_flow |
| (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| (hsmall : ∀ p ∈ modelSquare, |q p| ≤ (1 : ℝ) / 100) : |
| ∃ Y : ℝ → ℝ → ℝ, |
| ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 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) / 50) ∧ |
| (∀ s ∈ Ioo (-2 : ℝ) 2, ∀ t ∈ Ioo (-2 : ℝ) 2, |
| HasDerivAt (fun u => Y u t) |
| (Real.exp (-(∫ r in 0..t, coordPartial 1 q (coordinatePoint r (Y s r))))) s ∧ |
| 0 < deriv (fun u => Y u t) s) := by |
| obtain ⟨Y, hY, htime, hstart, hode, hdisp, hdisp', hrange, hLip⟩ := |
| exists_cap_flow hq hU hSU hsmall |
| have hrange' (s : ℝ) (hs : s ∈ Icc (-2 : ℝ) 2) (t : ℝ) |
| (ht : t ∈ Icc (-2 : ℝ) 2) : Y s t ∈ Icc (-3 : ℝ) 3 := |
| abs_le.mp ((hrange s hs t ht).1.trans (by norm_num)) |
| refine ⟨Y, cap_flow_joint_contDiffOn hq hU hSU hY hrange' hstart hode, |
| hstart, hode, hdisp', ?_⟩ |
| intro s hs t ht |
| exact ⟨cap_flow_hasDerivAt_initial hq hU hSU hY hrange' hstart hode hs ht, |
| cap_flow_initial_deriv_pos hq hU hSU hY hrange' hstart hode hs ht⟩ |
|
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| end SmoothLocal.Flow |
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| end |
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| end OAI |
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