File size: 3,272 Bytes
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 | import OAI.Geometry.IsometricImmersion.Darboux.QInitialEnergy
import OAI.Geometry.IsometricImmersion.Energy.LocalQResidualL2
import OAI.Geometry.IsometricImmersion.Metrics.MetricHorizontalRequests
namespace OAI
noncomputable section
open Set
open scoped ContDiff NNReal
namespace SmoothLocal.HighEquation
open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE SmoothLocal.Hyperbolic
def QLowBounds (g : MetricField) (z : Coord → ℝ) (S : Set Coord)
(s0 M speed : ℝ) : Prop :=
∀ p ∈ S, s0 ≤ heightQCoefficient g z 5 p ∧
|heightQCoefficient g z 2 p| ≤ M ∧ |heightQCoefficient g z 3 p| ≤ M ∧
|heightQCoefficient g z 5 p| ≤ M ∧
|coordPartial 0 (heightQCoefficient g z 4) p| ≤ M ∧
|coordPartial 0 (heightQCoefficient g z 5) p| ≤ M ∧
|coordPartial 1 (heightQCoefficient g z 5) p| ≤ M ∧
|heightQCoefficient g z 4 p| + Real.sqrt (heightQCoefficient g z 5 p) ≤ speed
theorem QLowBounds.mono_set {g : MetricField} {z : Coord → ℝ} {S T : Set Coord}
{s0 M speed : ℝ} (h : QLowBounds g z S s0 M speed) (hTS : T ⊆ S) :
QLowBounds g z T s0 M speed := fun p hp => h p (hTS hp)
theorem QLowBounds.high_coefficients
{g : MetricField} {z : Coord → ℝ} {S : Set Coord} {s0 M speed : ℝ}
(h : QLowBounds g z S s0 M speed) (hM : 0 ≤ M) (ell : ℕ) {p : Coord} (hp : p ∈ S) :
|qHighBTheta g z ell p| ≤ M * (ell + 1) ∧
|qHighBXi g z ell p| ≤ M * (ell + 1) ∧
|coordPartial 0 (heightQCoefficient g z 4) p| ≤ M * (ell + 1) ∧
|coordPartial 0 (heightQCoefficient g z 5) p| ≤ M * (ell + 1) ∧
|coordPartial 1 (heightQCoefficient g z 5) p| ≤ M * (ell + 1) := by
obtain ⟨_, h2, h3, _, h4x, h5x, h5t, _⟩ := h p hp
have he : 0 ≤ (ell : ℝ) := Nat.cast_nonneg ell
have hB : M ≤ M * (ell + 1) := by nlinarith
have hbt : |qHighBTheta g z ell p| ≤ M * (ell + 1) := by
unfold qHighBTheta
calc
_ ≤ |heightQCoefficient g z 3 p| + |(ell : ℝ) * coordPartial 0 (heightQCoefficient g z 4) p| := abs_add_le _ _
_ ≤ M + (ell : ℝ) * M := by
rw [abs_mul, abs_of_nonneg he]
exact add_le_add h3 (mul_le_mul_of_nonneg_left h4x he)
_ = _ := by ring
have hbx : |qHighBXi g z ell p| ≤ M * (ell + 1) := by
unfold qHighBXi
calc
_ ≤ |heightQCoefficient g z 2 p| + |(ell : ℝ) * coordPartial 0 (heightQCoefficient g z 5) p| := abs_add_le _ _
_ ≤ M + (ell : ℝ) * M := by
rw [abs_mul, abs_of_nonneg he]
exact add_le_add h2 (mul_le_mul_of_nonneg_left h5x he)
_ = _ := by ring
exact ⟨hbt, hbx, h4x.trans hB, h5x.trans hB, h5t.trans hB⟩
theorem shrinkingSlab_mono_top {xl xr a b t speed : ℝ} (ht : t ≤ b) :
shrinkingSlab xl xr a t speed ⊆ shrinkingSlab xl xr a b speed := by
intro p hp
exact ⟨⟨hp.1.1, hp.1.2.trans ht⟩, hp.2⟩
theorem SpatialSliceL2Bound.mono_order_budget
{ι : Type*} {F : ι → Coord → ℝ} {theta left right : ℝ}
{N n : ℕ} {H H' : ℝ≥0}
(h : SpatialSliceL2Bound F theta left right N H) (hn : n ≤ N) (hH : H ≤ H') :
SpatialSliceL2Bound F theta left right n H' := by
intro i k hk
exact (h i k (hk.trans hn)).trans (by exact_mod_cast hH)
end SmoothLocal.HighEquation
end
end OAI
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