| import Mathlib |
| import OAI.Analysis.BiholderTransport.LocalFlow.SmoothFlowEndpoint |
| import OAI.Analysis.BiholderTransport.Calculus.SmoothExtensionNear |
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| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory Manifold Bundle |
| open scoped ContDiff Manifold ENNReal NNReal Topology |
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| open Set Filter |
| open scoped Topology NNReal |
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| open Set Filter |
| open scoped Topology |
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| open Set Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set |
| open scoped Topology |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter ContinuousLinearMap InnerProductSpace |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
| open scoped NNReal |
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| namespace WeakMTWTransport |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [CompleteSpace E] [HasContDiffBump E] |
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| 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 |
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| end WeakMTWTransport |
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| end |
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| end OAI |
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