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110f4f3 | 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 92 93 94 95 | import Mathlib
import OAI.Analysis.BiholderTransport.LocalFlow.SmoothFlowEndpoint
import OAI.Analysis.BiholderTransport.Calculus.SmoothExtensionNear
namespace OAI
noncomputable section
open Set MeasureTheory Manifold Bundle
open scoped ContDiff Manifold ENNReal NNReal Topology
open Set Filter
open scoped Topology NNReal
open Set Filter
open scoped Topology
open Set Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set
open scoped Topology
open Set Filter Manifold Bundle ContinuousLinearMap
open scoped Topology ContDiff Manifold Bundle
open Set Filter ContinuousLinearMap InnerProductSpace
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open scoped NNReal
namespace WeakMTWTransport
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[CompleteSpace E] [HasContDiffBump E]
lemma exists_smooth_local_flow_on {f : E → E} {S : Set E}
(hS : IsOpen S) (hf : ContDiffOn ℝ ∞ f S) {a : E} (ha : a ∈ S) :
∃ r : ℝ, 0 < r ∧ Metric.ball a r ⊆ S ∧ ∃ Φ : (ℝ × E) → E,
ContDiffOn ℝ ∞ Φ (Ioo (-r) r ×ˢ Metric.ball a r) ∧
(∀ x ∈ Metric.ball a r, Φ (0,x) = x) ∧
∀ t ∈ Ioo (-r) r, ∀ x ∈ Metric.ball a r,
Φ (t,x) ∈ S ∧ HasDerivAt (fun s => Φ (s,x)) (f (Φ (t,x))) t := by
obtain ⟨g,hg,hgf⟩ := exists_smooth_extension_near hS hf ha
obtain ⟨r,hr,Φ,hΦ,hΦ0,hΦder⟩ := exists_smooth_local_flow g hg a
have hcenter : ((0:ℝ),a) ∈ Ioo (-r) r ×ˢ Metric.ball a r :=
⟨⟨neg_neg_of_pos hr,hr⟩,Metric.mem_ball_self hr⟩
have hcont : ContinuousAt Φ (0,a) :=
(hΦ.contDiffAt ((isOpen_Ioo.prod Metric.isOpen_ball).mem_nhds hcenter)).continuousAt
have hΦa : Φ (0,a) = a := hΦ0 a (Metric.mem_ball_self hr)
have hnear : ∀ᶠ z in 𝓝 ((0:ℝ),a), Φ z ∈ S ∧ g (Φ z) = f (Φ z) := by
have h := hcont.eventually (show ∀ᶠ y in 𝓝 (Φ (0,a)), y ∈ S ∧ g y = f y from by
rw [hΦa]
exact (show ∀ᶠ y in 𝓝 a, y ∈ S from hS.mem_nhds ha).and hgf)
exact h
obtain ⟨ε,hε,hεsub⟩ := Metric.mem_nhds_iff.mp hnear
let δ := min (r/2) (ε/2)
have hδ : 0 < δ := lt_min (by positivity) (by positivity)
have hδr : δ < r := (min_le_left _ _).trans_lt (by linarith)
have hδε : δ < ε := (min_le_right _ _).trans_lt (by linarith)
have hdom : Ioo (-δ) δ ×ˢ Metric.ball a δ ⊆ Ioo (-r) r ×ˢ Metric.ball a r := by
intro z hz
exact ⟨⟨by linarith [hz.1.1],hz.1.2.trans hδr⟩,lt_trans hz.2 hδr⟩
have himg : ∀ t ∈ Ioo (-δ) δ, ∀ x ∈ Metric.ball a δ,
Φ (t,x) ∈ S ∧ g (Φ (t,x)) = f (Φ (t,x)) := by
intro t ht x hx
apply hεsub
change dist (t,x) (0,a) < ε
rw [Prod.dist_eq,max_lt_iff,Real.dist_eq,sub_zero]
exact ⟨(abs_lt.mpr ht).trans hδε,lt_trans hx hδε⟩
refine ⟨δ,hδ,?_,Φ,hΦ.mono hdom,?_,?_⟩
· intro x hx
have h0 : Φ (0,x) = x := hΦ0 x (lt_trans hx hδr)
exact h0 ▸ (himg 0 ⟨neg_neg_of_pos hδ,hδ⟩ x hx).1
· intro x hx
exact hΦ0 x (lt_trans hx hδr)
· intro t ht x hx
refine ⟨(himg t ht x hx).1,?_⟩
rw [←(himg t ht x hx).2]
have hz := hdom (show (t,x) ∈ Ioo (-δ) δ ×ˢ Metric.ball a δ from ⟨ht,hx⟩)
exact hΦder t hz.1 x hz.2
end WeakMTWTransport
end
end OAI
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