| import OAI.Geometry.IsometricImmersion.Energy.MovingSlabEnergy |
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| namespace OAI |
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| noncomputable section |
| open Set MeasureTheory |
| open scoped ContDiff Interval |
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| namespace SmoothLocal.Hyperbolic |
| open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE |
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| theorem scalar_gradient_le_energy {s0 S a b : ℝ} (hs0 : 0 < s0) (hS : s0 ≤ S) : |
| a ^ 2 + b ^ 2 ≤ (2 * (1 + 1 / s0)) * ((a ^ 2 + S * b ^ 2) / 2) := by |
| have hspos : 0 ≤ S := hs0.le.trans hS |
| have hsb : b ^ 2 ≤ (S * b ^ 2) / s0 := (le_div_iff₀ hs0).2 |
| (by nlinarith [mul_le_mul_of_nonneg_right hS (sq_nonneg b)]) |
| have hsb' : b ^ 2 ≤ (1 / s0) * (S * b ^ 2) := by |
| simpa only [div_eq_mul_inv, one_mul, mul_one, mul_comm, mul_left_comm, mul_assoc] using hsb |
| have hi : 0 ≤ 1 / s0 := by positivity |
| nlinarith [mul_nonneg hi (sq_nonneg a), mul_nonneg hspos (sq_nonneg b)] |
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| theorem moving_gradient_integral_le_energy |
| {S v : Coord → ℝ} {radius lo hi xl xr theta0 speed theta s0 : ℝ} |
| (hS : ContDiffOn ℝ ∞ S (coordinateRectangle radius lo hi)) |
| (hv : ContDiffOn ℝ ∞ v (coordinateRectangle radius lo hi)) |
| (hl : inwardLeft xl theta0 speed theta ∈ Ioo (-radius) radius) |
| (hr : inwardRight xr theta0 speed theta ∈ Ioo (-radius) radius) |
| (htheta : theta ∈ Ioo lo hi) |
| (hlr : inwardLeft xl theta0 speed theta ≤ inwardRight xr theta0 speed theta) |
| (hs0 : 0 < s0) |
| (hfloor : ∀ x ∈ Icc (inwardLeft xl theta0 speed theta) (inwardRight xr theta0 speed theta), |
| s0 ≤ S (coordinatePoint x theta)) : |
| movingIntervalIntegral (inwardLeft xl theta0 speed) (inwardRight xr theta0 speed) |
| (fun p => (coordPartial 1 v p) ^ 2 + (coordPartial 0 v p) ^ 2) theta ≤ |
| (2 * (1 + 1 / s0)) * movingEnergy xl xr theta0 speed S v theta := by |
| have hU := coordinateRectangle_isOpen radius lo hi |
| have ht := partial_contDiffOn hv hU 1 |
| have hx := partial_contDiffOn hv hU 0 |
| have he : ContDiffOn ℝ ∞ (energyDensity S v) (coordinateRectangle radius lo hi) := |
| ((ht.pow 2).add (hS.mul (hx.pow 2))).div_const 2 |
| have hgrad : ContDiffOn ℝ ∞ |
| (fun p => (coordPartial 1 v p) ^ 2 + (coordPartial 0 v p) ^ 2) |
| (coordinateRectangle radius lo hi) := (ht.pow 2).add (hx.pow 2) |
| have hconst : ContDiffOn ℝ ∞ |
| (fun p => (2 * (1 + 1 / s0)) * energyDensity S v p) |
| (coordinateRectangle radius lo hi) := contDiffOn_const.mul he |
| have hi := intervalIntegral.integral_mono_on hlr |
| (slice_intervalIntegrable hgrad.continuousOn hl hr htheta) |
| (slice_intervalIntegrable hconst.continuousOn hl hr htheta) |
| (fun x hx => scalar_gradient_le_energy hs0 (hfloor x hx)) |
| simpa only [movingIntervalIntegral, movingEnergy, intervalIntegral.integral_const_mul] using hi |
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| theorem sectionFour_inward_interval |
| {L r theta : ℝ} (hL : 0 < L) (hr : 0 < r) (htheta : theta ∈ Icc (-r) r) : |
| Icc (-(L * r) / 2) (L * r / 2) ⊆ |
| Icc (inwardLeft (-(L * r)) (-r) (L / 4) theta) |
| (inwardRight (L * r) (-r) (L / 4) theta) ∧ |
| 0 < L * r ∧ L * r ≤ |
| inwardRight (L * r) (-r) (L / 4) theta - |
| inwardLeft (-(L * r)) (-r) (L / 4) theta := by |
| have hLt := mul_le_mul_of_nonneg_left htheta.2 hL.le |
| have hLr : 0 < L * r := mul_pos hL hr |
| constructor |
| · intro x hx |
| unfold inwardLeft inwardRight |
| constructor <;> nlinarith [hx.1, hx.2] |
| constructor |
| · exact hLr |
| · unfold inwardLeft inwardRight |
| nlinarith |
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| theorem sectionFour_pulse_in_shrinkingSlab |
| {L r x theta delta tau : ℝ} (hL : 0 < L) (hr : 0 < r) |
| (hwidth : delta / (2 * tau) ≤ r) (hx : |x| ≤ L * r / 16) |
| (htheta : |theta| ≤ delta / (2 * tau)) : |
| coordinatePoint x theta ∈ shrinkingSlab (-(L * r)) (L * r) (-r) r (L / 4) := by |
| have ht : theta ∈ Icc (-r) r := by |
| obtain ⟨htl, htr⟩ := abs_le.mp (htheta.trans hwidth) |
| exact ⟨htl, htr⟩ |
| apply point_mem_shrinkingSlab ht |
| apply (sectionFour_inward_interval hL hr ht).1 |
| obtain ⟨hxl, hxr⟩ := abs_le.mp hx |
| have hLr := mul_pos hL hr |
| constructor <;> nlinarith |
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| end SmoothLocal.Hyperbolic |
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| end |
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| end OAI |
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