| import OAI.Geometry.IsometricImmersion.Energy.WeightPrimitive |
| import Mathlib.Analysis.ODE.ExistUnique |
| import Mathlib.Analysis.Calculus.MeanValue |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Metric Filter Function |
| open scoped ContDiff Topology NNReal |
|
|
| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted |
|
|
| def modelSquare : Set Coord := Icc (fun _ => (-3 : ℝ)) (fun _ => (3 : ℝ)) |
|
|
| theorem modelSquare_isCompact : IsCompact modelSquare := isCompact_Icc |
|
|
| theorem coordinatePoint_mem_modelSquare {t y : ℝ} |
| (ht : t ∈ Icc (-3 : ℝ) 3) (hy : y ∈ Icc (-3 : ℝ) 3) : |
| coordinatePoint t y ∈ modelSquare := by |
| simp only [mem_Icc] at ht hy |
| constructor <;> intro i <;> fin_cases i <;> simp [coordinatePoint] <;> tauto |
|
|
| def flowField (q : Coord → ℝ) (t y : ℝ) : ℝ := -q (coordinatePoint t y) |
|
|
| variable {q : Coord → ℝ} {U : Set Coord} |
|
|
| theorem flowField_hasDerivAt_y (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) |
| {t y : ℝ} (hp : coordinatePoint t y ∈ U) : |
| HasDerivAt (flowField q t) (-coordPartial 1 q (coordinatePoint t y)) y := by |
| have hd := (hq.contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp) |
| exact (hd.hasFDerivAt.comp_hasDerivAt y (coordinatePoint_hasDerivAt_y t y)).neg |
|
|
| theorem exists_flowField_lipschitz (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) |
| (hSU : modelSquare ⊆ U) : |
| ∃ L : ℝ≥0, ∀ t ∈ Icc (-2 : ℝ) 2, |
| LipschitzOnWith L (flowField q t) (closedBall (0 : ℝ) 3) := by |
| have hc : ContinuousOn (coordPartial 1 q) modelSquare := |
| (partial_contDiffOn hq hU 1).continuousOn.mono hSU |
| obtain ⟨M, hM⟩ := modelSquare_isCompact.exists_bound_of_continuousOn hc |
| let L : ℝ≥0 := ⟨max M 0, le_max_right _ _⟩ |
| refine ⟨L, ?_⟩ |
| intro t ht |
| have hmem (y : ℝ) (hy : y ∈ closedBall (0 : ℝ) 3) : |
| coordinatePoint t y ∈ modelSquare := by |
| have hy' : |y| ≤ 3 := by simpa only [mem_closedBall, Real.dist_eq, sub_zero] using hy |
| exact coordinatePoint_mem_modelSquare ⟨by linarith [ht.1], by linarith [ht.2]⟩ (abs_le.mp hy') |
| apply (convex_closedBall (0 : ℝ) 3).lipschitzOnWith_of_nnnorm_hasDerivWithin_le |
| (fun y hy => (flowField_hasDerivAt_y hq hU (hSU (hmem y hy))).hasDerivWithinAt) |
| intro y hy |
| change ‖-coordPartial 1 q (coordinatePoint t y)‖ ≤ max M 0 |
| rw [norm_neg] |
| exact (hM _ (hmem y hy)).trans (le_max_left _ _) |
|
|
| theorem exists_cap_picard (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) |
| (hSU : modelSquare ⊆ U) (hsmall : ∀ p ∈ modelSquare, |q p| ≤ (1 : ℝ) / 100) : |
| ∃ L : ℝ≥0, IsPicardLindelof (flowField q) |
| (⟨0, by norm_num⟩ : Icc (-2 : ℝ) 2) 0 3 2 (1 / 100) L := by |
| obtain ⟨L, hL⟩ := exists_flowField_lipschitz hq hU hSU |
| refine ⟨L, hL, ?_, ?_, ?_⟩ |
| · intro y hy |
| have hy' : |y| ≤ 3 := by simpa only [mem_closedBall, Real.dist_eq, sub_zero, |
| NNReal.coe_ofNat] using hy |
| exact hq.continuousOn.neg.comp (by unfold coordinatePoint; fun_prop) |
| (fun t ht => hSU (coordinatePoint_mem_modelSquare |
| ⟨by linarith [ht.1], by linarith [ht.2]⟩ (abs_le.mp hy'))) |
| · intro t ht y hy |
| have hy' : |y| ≤ 3 := by simpa only [mem_closedBall, Real.dist_eq, sub_zero, |
| NNReal.coe_ofNat] using hy |
| have hm := coordinatePoint_mem_modelSquare (t := t) (y := y) |
| ⟨by linarith [ht.1], by linarith [ht.2]⟩ (abs_le.mp hy') |
| simpa only [flowField, norm_neg, Real.norm_eq_abs, NNReal.coe_div, |
| NNReal.coe_one, NNReal.coe_ofNat] using hsmall _ hm |
| · norm_num |
|
|
| theorem exists_cap_trajectory (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) |
| (hSU : modelSquare ⊆ U) (hsmall : ∀ p ∈ modelSquare, |q p| ≤ (1 : ℝ) / 100) |
| (s : ℝ) (hs : s ∈ Icc (-2 : ℝ) 2) : |
| ∃ y : ℝ → ℝ, y 0 = s ∧ ContDiffOn ℝ ∞ y (Icc (-2 : ℝ) 2) ∧ |
| (∀ t ∈ Icc (-2 : ℝ) 2, |
| HasDerivWithinAt y (-q (coordinatePoint t (y t))) (Icc (-2 : ℝ) 2) t) ∧ |
| (∀ t ∈ Icc (-2 : ℝ) 2, |y t - s| ≤ (1 : ℝ) / 100 * |t|) ∧ |
| (∀ t ∈ Icc (-2 : ℝ) 2, |y t - s| ≤ (1 : ℝ) / 50) ∧ |
| (∀ t ∈ Icc (-2 : ℝ) 2, y t ∈ closedBall (0 : ℝ) 3) := by |
| obtain ⟨L, hPL⟩ := exists_cap_picard hq hU hSU hsmall |
| have hsball : 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 |
| obtain ⟨alpha, halpha⟩ := ODE.FunSpace.exists_isFixedPt_next hPL hsball |
| have hinit : alpha.compProj 0 = s := by |
| rw [ODE.FunSpace.compProj_of_mem hz, ← halpha] |
| exact ODE.FunSpace.next_apply₀ hPL hsball alpha |
| have hderiv (t : ℝ) (ht : t ∈ Icc (-2 : ℝ) 2) : |
| HasDerivWithinAt alpha.compProj (flowField q t (alpha.compProj t)) |
| (Icc (-2 : ℝ) 2) t := by |
| apply ODE.hasDerivWithinAt_picard_Icc (show (0 : ℝ) ∈ Icc (-2 : ℝ) 2 by norm_num) |
| hPL.continuousOn_uncurry alpha.continuous_compProj.continuousOn |
| (fun _ _ => alpha.compProj_mem_closedBall hPL.mul_max_le) s ht |>.congr_of_mem _ ht |
| intro t' ht' |
| nth_rw 1 [← halpha] |
| rw [ODE.FunSpace.compProj_of_mem ht'] |
| exact ODE.FunSpace.next_apply hPL hsball alpha (t := ⟨t', ht'⟩) |
| have hrange (t : ℝ) : alpha.compProj t ∈ closedBall (0 : ℝ) 3 := |
| alpha.compProj_mem_closedBall hPL.mul_max_le |
| have hdispl (t : ℝ) (ht : t ∈ Icc (-2 : ℝ) 2) : |
| |alpha.compProj t - s| ≤ (1 : ℝ) / 100 * |t| := by |
| have hh := alpha.lipschitzWith.dist_le_mul ⟨t, ht⟩ |
| (⟨0, by norm_num⟩ : Icc (-2 : ℝ) 2) |
| have ha0 : alpha ⟨0, hz⟩ = s := by |
| simpa only [ODE.FunSpace.compProj_of_mem hz] using hinit |
| rw [← ODE.FunSpace.compProj_of_mem ht, ha0] at hh |
| simpa only [Subtype.dist_eq, Real.dist_eq, sub_zero, NNReal.coe_div, |
| NNReal.coe_one, NNReal.coe_ofNat] using hh |
| 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 ⟨alpha.compProj, hinit, |
| ODE.contDiffOn_enat_Icc_of_hasDerivWithinAt hfield hderiv (fun t _ => hrange t), |
| hderiv, hdispl, ?_, fun t _ => hrange t⟩ |
| intro t ht |
| have hdt := hdispl t ht |
| have ht2 := abs_le.mpr ht |
| nlinarith |
|
|
| end SmoothLocal.Flow |
|
|
| end |
|
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| end OAI |
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