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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 80 81 82 83 84 85 86 87 88 89 90 91 | import OAI.Geometry.IsometricImmersion.Energy.MovingSlabEnergy
namespace OAI
noncomputable section
open Set MeasureTheory
open scoped ContDiff Interval
namespace SmoothLocal.Hyperbolic
open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE
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)]
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
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
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
end SmoothLocal.Hyperbolic
end
end OAI
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