| import OAI.Geometry.IsometricImmersion.Caps.CapFlowJetBounds |
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| namespace OAI |
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| noncomputable section |
| open Set Filter Function |
| open scoped ContDiff Topology |
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| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted |
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| theorem negative_over_cube_partial {f v : Coord → ℝ} {p : Coord} |
| (hf : DifferentiableAt ℝ f p) (hv : DifferentiableAt ℝ v p) (hne : v p ≠ 0) (i : Fin 2) : |
| coordPartial i (fun x => -f x / (v x)^3) p = |
| -coordPartial i f p / (v p)^3 + 3 * f p * coordPartial i v p / (v p)^4 := by |
| have hn : DifferentiableAt ℝ (fun x => -f x) p := hf.neg |
| have hp : DifferentiableAt ℝ (fun x => (v x)^3) p := hv.pow 3 |
| rw [coordPartial_div hn hp (pow_ne_zero 3 hne) i] |
| have hneg : coordPartial i (fun x => -f x) p = -coordPartial i f p := by simp [coordPartial] |
| have hpow : coordPartial i (fun x => (v x)^3) p = 3 * (v p)^2 * coordPartial i v p := by |
| unfold coordPartial |
| rw [(hv.hasFDerivAt.pow 3).fderiv] |
| norm_num [smul_apply, smul_eq_mul] |
| rw [hneg, hpow] |
| field_simp [hne] |
| ring |
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|
| theorem capChartRho_partial {Y : ℝ → ℝ → ℝ} |
| (hF : ContDiffOn ℝ ∞ (capFlowHeight Y) capChartDomain) |
| {p : Coord} (hp : p ∈ capChartDomain) |
| (hpos : 0 < coordPartial 1 (capFlowHeight Y) p) (i : Fin 2) : |
| coordPartial i (capChartRho Y) p = |
| -coordPartial i (coordPartial 1 (capFlowHeight Y)) p / |
| (coordPartial 1 (capFlowHeight Y) p)^2 := by |
| have hV := ((partial_contDiffOn hF capChartDomain_isOpen 1).contDiffAt |
| (capChartDomain_isOpen.mem_nhds hp)).differentiableAt (by simp) |
| change coordPartial i (fun x => (1 : ℝ) / coordPartial 1 (capFlowHeight Y) x) p = _ |
| rw [coordPartial_div (differentiableAt_const (1 : ℝ)) hV hpos.ne' i] |
| simp [coordPartial] |
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| theorem capChartSigma_partial {Y : ℝ → ℝ → ℝ} |
| (hF : ContDiffOn ℝ ∞ (capFlowHeight Y) capChartDomain) |
| {p : Coord} (hp : p ∈ capChartDomain) |
| (hpos : 0 < coordPartial 1 (capFlowHeight Y) p) (i : Fin 2) : |
| coordPartial i (capChartSigma Y) p = |
| -coordPartial i (coordPartial 1 (coordPartial 1 (capFlowHeight Y))) p / |
| (coordPartial 1 (capFlowHeight Y) p)^3 + |
| 3 * coordPartial 1 (coordPartial 1 (capFlowHeight Y)) p * |
| coordPartial i (coordPartial 1 (capFlowHeight Y)) p / |
| (coordPartial 1 (capFlowHeight Y) p)^4 := by |
| have hV := partial_contDiffOn hF capChartDomain_isOpen 1 |
| have hW := partial_contDiffOn hV capChartDomain_isOpen 1 |
| exact negative_over_cube_partial |
| ((hW.contDiffAt (capChartDomain_isOpen.mem_nhds hp)).differentiableAt (by simp)) |
| ((hV.contDiffAt (capChartDomain_isOpen.mem_nhds hp)).differentiableAt (by simp)) hpos.ne' i |
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| theorem capReciprocal_bounds_of_jets {Y : ℝ → ℝ → ℝ} {C c : ℝ} |
| (hF : ContDiffOn ℝ ∞ (capFlowHeight Y) capChartDomain) |
| (hC : 0 ≤ C) (hc : 0 < c) |
| (hFB : CoordinateBound (capFlowHeight Y) capChartDomain 3 C) |
| (hVlower : ∀ p ∈ capChartDomain, c ≤ coordPartial 1 (capFlowHeight Y) p) |
| {p : Coord} (hp : p ∈ capChartDomain) : |
| 0 < capChartRho Y p ∧ |capChartRho Y p| ≤ 1 / c ∧ |
| (∀ i : Fin 2, |coordPartial i (capChartRho Y) p| ≤ C / c^2) ∧ |
| |capChartSigma Y p| ≤ C / c^3 ∧ |
| (∀ i : Fin 2, |coordPartial i (capChartSigma Y) p| ≤ C / c^3 + 3 * C^2 / c^4) := by |
| have hVp : 0 < coordPartial 1 (capFlowHeight Y) p := hc.trans_le (hVlower p hp) |
| have hpow (n : ℕ) : c^n ≤ |(coordPartial 1 (capFlowHeight Y) p)^n| := by |
| rw [abs_pow, abs_of_pos hVp] |
| exact pow_le_pow_left₀ hc.le (hVlower p hp) n |
| have hW : |coordPartial 1 (coordPartial 1 (capFlowHeight Y)) p| ≤ C := |
| hFB [1, 1] (by norm_num) p hp |
| have hVi (i : Fin 2) : |coordPartial i (coordPartial 1 (capFlowHeight Y)) p| ≤ C := |
| hFB [i, 1] (by norm_num) p hp |
| have hWi (i : Fin 2) : |coordPartial i (coordPartial 1 (coordPartial 1 (capFlowHeight Y))) p| ≤ C := |
| hFB [i, 1, 1] (by norm_num) p hp |
| refine ⟨one_div_pos.mpr hVp, ?_, ?_, ?_, ?_⟩ |
| · exact LowQuotient.abs_div_bound (by norm_num) (by norm_num : |(1 : ℝ)| ≤ 1) hc |
| (by simpa only [abs_of_pos hVp] using hVlower p hp) |
| · intro i |
| rw [capChartRho_partial hF hp hVp i] |
| exact LowQuotient.abs_div_bound hC (by simpa only [abs_neg] using hVi i) |
| (pow_pos hc 2) (hpow 2) |
| · exact LowQuotient.abs_div_bound hC (by simpa only [abs_neg] using hW) |
| (pow_pos hc 3) (hpow 3) |
| · intro i |
| rw [capChartSigma_partial hF hp hVp i] |
| have hfirst : |
| |-coordPartial i (coordPartial 1 (coordPartial 1 (capFlowHeight Y))) p / |
| (coordPartial 1 (capFlowHeight Y) p)^3| ≤ C / c^3 := |
| LowQuotient.abs_div_bound hC (by simpa only [abs_neg] using hWi i) |
| (pow_pos hc 3) (hpow 3) |
| have hnum : |3 * coordPartial 1 (coordPartial 1 (capFlowHeight Y)) p * |
| coordPartial i (coordPartial 1 (capFlowHeight Y)) p| ≤ 3 * C^2 := by |
| rw [abs_mul, abs_mul, abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 3)] |
| calc |
| _ = 3 * (|coordPartial 1 (coordPartial 1 (capFlowHeight Y)) p| * |
| |coordPartial i (coordPartial 1 (capFlowHeight Y)) p|) := by ring |
| _ ≤ 3 * (C * C) := mul_le_mul_of_nonneg_left |
| (mul_le_mul hW (hVi i) (abs_nonneg _) hC) (by norm_num) |
| _ = 3 * C^2 := by ring |
| have hsecond := LowQuotient.abs_div_bound (by positivity) hnum (pow_pos hc 4) (hpow 4) |
| exact (abs_add_le _ _).trans (add_le_add hfirst hsecond) |
|
|
| section ActualFlow |
| variable {q : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ} {M : 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) |
| variable (hM : 0 ≤ M) |
| variable (hq0 : ∀ p ∈ modelSquare, |q p| ≤ (1 : ℝ) / 100) |
| variable (hq1 : ∀ i : Fin 2, ∀ p ∈ modelSquare, |coordPartial i q p| ≤ M) |
| variable (hq2 : ∀ i j : Fin 2, ∀ p ∈ modelSquare, |coordPartial i (coordPartial j q) p| ≤ M) |
| variable (hq3 : ∀ i j k : Fin 2, ∀ p ∈ modelSquare, |coordPartial i (coordPartial j (coordPartial k q)) p| ≤ M) |
| include hq hU hSU hY hrange hstart hode hM hq0 hq1 hq2 hq3 |
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| theorem capReciprocal_bounds_from_q {p : Coord} (hp : p ∈ capChartDomain) : |
| 0 < capChartRho Y p ∧ |capChartRho Y p| ≤ 1 / Real.exp (-2 * M) ∧ |
| (∀ i : Fin 2, |coordPartial i (capChartRho Y) p| ≤ flowCoordinateBound M / (Real.exp (-2 * M))^2) ∧ |
| |capChartSigma Y p| ≤ flowCoordinateBound M / (Real.exp (-2 * M))^3 ∧ |
| (∀ i : Fin 2, |coordPartial i (capChartSigma Y) p| ≤ |
| flowCoordinateBound M / (Real.exp (-2 * M))^3 + |
| 3 * (flowCoordinateBound M)^2 / (Real.exp (-2 * M))^4) := by |
| have hs := cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode |
| have hVlower (x : Coord) (hx : x ∈ capChartDomain) : |
| Real.exp (-2 * M) ≤ coordPartial 1 (capFlowHeight Y) x := by |
| rw [capFlowHeight_partial_one_eq hs hx] |
| exact (cap_flow_initial_deriv_bounds hq hU hSU hY hrange hstart hode hM (hq1 1) hx.2 hx.1).1 |
| exact capReciprocal_bounds_of_jets (capFlowHeight_contDiffOn hs) (flowCoordinateBound_nonneg hM) |
| (Real.exp_pos _) (capFlowHeight_coordinate_bound hq hU hSU hY hrange hstart hode hM hq0 hq1 hq2 hq3) |
| hVlower hp |
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| end ActualFlow |
| end SmoothLocal.Flow |
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| end |
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| end OAI |
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